Is It Amazing That Rambam Knew That Pi Is Irrational?

(For those who are sick of Pi, feel free to skip this post! But it does clarify some important points. The reason why I am spending so much time on this topic is in a post that I drafted last week but which I am waiting another few days to publish.)

First, I have to correct a mistake that I made. Back in 2011, on Pi Day (3.14), I wrote that Rambam was the first person in recorded history to explicitly describe pi as being an irrational number (i.e. a number that cannot be expressed by a fraction). But I now discovered that it seems that he was preceded by others.

Boaz Tsaban and David Garber, in an article that is helpfully referred to in Rabbi Meiselman's book, note that "Various ancient Greek writers, including Hero, Eutocius, and Simplicius, understand the difficulty of finding an exact value for the ratio, and seem to realize the possibility of its being irrational," although they did not say so definitively. It is thus certainly no surprise that after centuries of failed efforts to calculate number precisely, people would conclude that it is indeed irrational.

The fifth-century Indian mathematician Aryabhata wrote that “Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached.” The 15th century commentator Nilakantha Somayaji interprets the original words as saying that not that is this an approximation, but that the value is irrational.

Then, at the turn of the ninth century, the Persian mathematician Muhammad Al-Khwarizmi notes that there are several different methods for calculating Pi. A marginal note (I am not sure when it was written) observes that "It is an approximation not a proof, and no one stands on the truth of this, and no one but Allah knows the true circumference of the circle, as the line is not straight and has no beginning and no end, we merely attempt to approximate and discover the root, but even the root has no definition as no one may know its exact value but Allah, and the best of these approximations that is to multiply the diameter by three and seventh as it is faster and simpler and only Allah might know it true."

The Muslim scholar Abu al-Rayhan al-Biruni, who lived from 973-1048, was familiar with Aryabhata's works. In The Encyclopedia of the History of Arabic Science, [Roshdi Rashed ed.], vol. 2, London/New York, 1996, p. 454), Boris Rosenfeld and Youschkevitch note that al-Biruni described Pi as irrational. In their discussion of medieval Arabic science, they further note that "...Arabic mathematicians repeatedly expressed their belief that the ratio of the length of a circumference to its diameter was irrational... Subsequent European mathematicians were also sure that pi is irrational but only J. H. Lambert, a native of Alsace, in 1766 succeeded in proving this."

In light of all this, it can hardly be seen as surprising that, just as the early Greeks seem to have suspected and just as the early Indian and Muslim scholars were certain, Rambam was likewise certain that pi is irrational.

Now, here is where matters become really strange. Rabbi Meiselman, in Torah, Chazal and Science p. 155, says that Rambam "was not merely repeating an accepted piece of information, since this fact was as of yet unknown to the rest of the world." Rabbi Meiselman concludes that Rambam deduced it from the Talmud, seemingly from the Talmud's approximating Pi as three rather than using a fraction. As such, Rabbi Meiselman presents this as evidence that Rambam, and in turn Chazal, possessed wisdom that was ahead of their time, and was somehow derived from the Torah or some other such supernatural source.

But Rabbi Meiselman himself, while he does not appear to be aware about Aryabhata, acknowledges in a footnote that al-Biruni already knew it! Rabbi Meiselman claims that "there is no reason to assume that the Rambam saw this work." This statement is very strange. The article from which Rabbi Meiselman took his information about al-Biruni just says that "it is not known" if Rambam saw it. Seeing that al-Biruni was a famous scientist, it is certainly highly plausible that Rambam saw it! And furthermore, it is obviously something that was known in the medieval period! How can Rabbi Meiselman write that "this fact was as of yet unknown to the rest of the world"?

Rabbi Yaakov Menken, in his review and defense of Rabbi Meiselman's book, likewise makes these strange and counterfactual claims (aside from repeatedly slandering me by falsely claiming that I denied Rambam's knowledge of pi and his attribution of that knowledge to Chazal.) He writes that "The field of mathematics did not leap forward in the centuries between the Gemara and the Rambam. It stretches credulity to imagine that Chazal made a rough estimate for no known reason, the fact that Pi is irrational then became common knowledge, and the Rambam then projected this common knowledge back upon Chazal." Accordingly, he concludes that not only was Rambam ahead of his time, but even Chazal are shown to be way ahead of their time! To quote Rabbi Menken: "The Rambam’s statement itself is evidence that Chazal possessed knowledge of the physical world beyond what was known to other cultures.”

Yet every single statement here is wrong. The field of mathematics most certainly did leap forward in the centuries between the Gemara and the Rambam. While it is certainly possible that Chazal knew pi to be irrational - after all, even the early Greeks suspected it - it does not at all stretch credulity to posit that Chazal did not know this, and gave the rough estimate of three for any number of reasons (such as those posited by Rishonim other than Rambam). It likewise does not at all stretch credulity to imagine that the irrationality of Pi became common knowledge in Rambam's time - even Rabbi Meiselman himself acknowledges that it was already stated by al-Biruni. Nor does it remotely stretch credulity to posit that Rambam projected this common knowledge back upon Chazal. People do this all the time - just look at how many recent figures have claimed that Chazal actually believed that the sun shines on the other side of the world at night. Rambam in particular is acknowledged by everyone (including the Vilna Gaon!) to have projected Greco-Islamic philosophy back into many statements in Chazal and Tanach (does anyone actually think that Maaseh Merkavah is really about Greek metaphysics?!).

And what is Rabbi Menken positing instead? That Rambam derived from Chazal that pi is irrational? Where exactly did he derive this from? Rabbi Menken does not seem to want to say, as Rabbi Meiselman does, that the source is the Gemara approximating Pi as three. But Rabbi Menken refuses to say where else the Rambam got it from. Is it some lost Gemara that nobody has ever heard of?!

It is clear that it is extremely plausible, even overwhelmingly likely, that that Chazal made a rough estimate of what they thought was a difficult (though not necessarily irrational) number for the reasons given by the other Rishonim. It is close to a fact that the irrationality of Pi later became common knowledge. And it is extremely normal and in character for Rambam to project such knowledge back upon Chazal.

Would anyone claim that the fact of the 15th century commentator Nilakantha Somayaji deriving the irrationality of pi from the fifth-century Indian mathematician Aryabhata is evidence that Aryabhata was in possession of a supernatural source of knowledge?! What stretches credulity is that Rabbi Meiselman and Rabbi Menken see Rambam's statement as "evidence that Chazal possessed knowledge of the physical world beyond what was known to other cultures"!