It's Pi Day! And in honor of this year's celebration I decided to do a bit of historical research. While π—the ratio between a circle's circumference and its diameter—has long been known and approximated, even since ancient times, only since the Eighteenth Century has it been proven to be an irrational number. Prior to that, various approximations were given, such as even the nice round number of 3.

I was recently reading the Wikipedia page on approximations of π and noticed that it included a note saying that Maimonides—the Jewish physician and scholar who lived nearly 1,000 years ago—seems to have alluded to its irrationality in his writing. The source took me to the book The Ancient Tradition of Geometric Problems and using Amazon's helpful Look Inside feature, I traced this putative statement to the commentary of Maimonides on the Mishnah, a collection of Jewish Law which forms part of the Talmud (for those playing at home, it's in his comments on Eruvin 1:5).

I was able to track down this commentary, with the relevant section below (it's in Hebrew, and written in what is known is Rashi script). In addition, I have included my own, hopefully reasonably accurate, translation of the important points:

The ratio between a diameter of a circle and its distance around is not known. We cannot speak about it precisely...This ratio’s reality cannot be found, but it is known in approximation...we can place an approximation of this as one to three and a seventh...

Maimonides uses the approximation for π of 22/7 which is about 3.14 and is a good approximation. In addition, he does seem to imply that any value is necessarily an approximation and can't be known precisely. So it could be argued that this is indeed evidence that he thought there might be some sort of irrationality to π, though that might be imputing too much and could simply be that Maimonides thought that π was hard to calculate. Furthermore, this is no doubt an imperfect English translation of a Hebrew text which was itself translated from the original Arabic, so that's an additional complication.

Nevertheless, these kinds of discussions are certainly intriguing and it is exciting that scholars of the Middle Ages were perhaps already intuiting that π could never be calculated as a ratio.

Happy Pi Day!

Top image:Wikimedia Commons/Public Domain

*Thanks to Rabbi Daniel Rockoff for help with the text! I have translated the entire section and am happy to share it with those who contact me.

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