Recent years have brought some extremely “vibrant” and often downright hostile discussions about the correct approach to apparent contradictions between the Torah, Chazal and modern scientific knowledge.

This applies across the board from astronomy, medicine, geography, physics, mathematics, and through archeology, even to history.

There are those like my good friend, the famous “Zoo Rabbi,” Rabbi Nathan Slifkin, who after his books touching on the subject were banned by various Chareidi authorities, has made it a major part of his life’s work to restore the popularity of “Rationalist Judaism.”

This, loosely speaking, encompasses the approach of many of the Geonim, the Rambam, and many other great early authorities that statements made by Chazal which appear to conflict with nature and science are not to be taken literally, and that when Chazal express their views on scientific matters, they are basing themselves on the accepted science of the time, and not on neo-prophetic revelation.

In contrast, Rabbi Moshe Meiselman, a renowned Chareidi Rosh Yeshiva who also has an academic background, has written his own work “Torah, Chazal, and Science” with the primary intention of both condemning and refuting this view, as well as attempting to prove that even the oft-quote protagonists amongst the Rishonim have been misunderstood.

In all humility, in a series of shiurim of my own on Agada, and a Hebrew analysis that I am still working on, I have brought numerous sources, including the introduction to the Talmud attributed to Rabbeinu Shmuel haNagid (printed at the back of most traditional versions of Masechta brachos,) which explicitly say that Agada is not comparable to halacha in its divine source and authority, but rather consists of Chazal’s own interpretations of the pesukim.

I also brought the words of the Ran (drasha 5) in his Derashos that seem to say the exact opposite and define anyone who does not believe that every word of Chazal’s Agadot are sourced at Sinai as a heretic.

As complex and sometimes aggressive this debate tends to be, it is exponentially more problematic when Chazal’s apparently out of date scientific knowledge forms the basis for practical halachik rulings.

In such cases, agreeing to disagree is not even an option, as huge areas of halacha are affected and a practical decision must made. What in one view invalidates a Sukkah or an Eruv, for example, can be essential to making it valid in the other. What renders a fish permitted according to one view, might render it forbidden according to another.

The above applies both ways, but although some authorities do indeed take into account discrepancies between modern scientific knowledge and that which Chazal were presumed to have, it is virtually always when it results in a stringency and not in a leniency.

It is not my intention to take sides in this longstanding and critical debate, but rather, as is my way in general, to examine the relevant sugyas on their own merit, together with the way the Rishonim interpret them, and see what we can learn from them

In our mishna, we are told that the minimum width of the pole used to “close” the open side of a מבוי is 1 handbreadth.

If the pole is round, we are to view it as if it is square and go by the width of its diameter.

As directly measuring the diameter of a solid cylinder is tricky, the Mishna advises us to measure its circumference, something far more practical and rely on a universal ratio between the circumference of a circle and its diameter to calculate the diameter.

The ratio given by the Mishna is the number three, according to the formula:

“כל שיש בהקיפו שלשה טפחים יש בו רוחב טפח”

“any (circle) whose circumference is 3 handbreadths has a diameter of one handbreadth.

As such, it follows that so long as the circumference of the round pole is at least 3 tefachim, we can assume that the diameter meets the minimum width of 1 tefach (handbreadth).

The same principle is employed (Sukkah 7b) to measure the diameter of a circular Sukkah to ensure it meets the minimum width of 4 amos. In the same sugya, the square-root of 2 is also assumed to be 1.4.

Every student of basic mathematics is immediately faced with the fact that the Mishna’s ration of 3 to 1 appears extremely inaccurate.

The universal ration between the circumference and diameter of any circle is of course the constant pi, a little more than 3.14, which has been known for some time already to be an irrational number.

Tosfos on our daf is so bothered by this apparent contradiction that after pointing out that it seems that our Gemara understood our Mishna’s ratio of 3 to be precise, based on the continuation of the sugya and another sugya in Bava Basra, he notes that this requires further investigation, since the mathematical experts hold that 3 is certainly not the precise ratio.

One should note that Tosfos leaves this question unanswered- he does not suggest explicitly that either Chazal or the contemporary mathematicians were wrong!

In contrast, both the Rambam and Tosfos haRosh on this Mishna are adamant that the Mishna is simply giving an estimation, and each have their own approaches as to why and how this is acceptable.

Whereas this approach certainly seems more logical, we obviously need to learn the sugya and its parallel sugyas properly to see if this fits into the flow of the Gemara.

Please join me on this exciting journey:

The Gemara opens its analysis on this part of the Mishna towards the bottom of the first side of today’s daf.

It asks the simple question: מנא הני מילי – from where are these words?

The very fact that the Gemara is looking for a verse to prove a mathematical reality that should be known to all is itself indicative of something deeper at work.

The Gemara answers that we derive this from the description of the circular ים (lit “sea” but probably referring to a water feature)) that Shlomo haMelech made, which had a diameter of 10 amos and a circumference of 30.

By describing the precise measurements of this circular feature, the passuk seems to be telling us that the ration of a circle’s circumference to its diameter is 3.

Once again, the fact that a verse is brought to teach us a simple mathematical fact seems very strange.

This question is strengthened by the fact that the ancient Greeks were very familiar with the concept of PI, and although they could not measure it precisely (though they might have suspected it was an irrational number,) it seems from my research that they certainly knew that it was more than 3, and could approximate it to at least 2 decimal points as around 3.14 .

It is hardly likely that Chazal, who took their mathematics very seriously, were unaware of this common knowledge of their time.

The Gemara then asks how we account for the width of the rim itself, which needs to be added to the actual diameter before working out the ratio with the circumference.

The Gemara responds that the possuk also tells us that the rim was extremely narrow (and thus negligible in the calculations.)

Seemingly unsatisfied by the assumption that the passuk was even nominally inaccurate in its workings, the Gemara points out that however narrow the rim was, it still would widen the exterior diameter slightly and effectively change the ratio accordingly.

The Gemara concludes that the circumference of 3 tefachim mentioned in the passuk was also measured from the inside, excluding the rim.

By now, it seems blatantly obvious that Chazal seem to take this measurement extremely precisely, and Tosfos’ observations to this affect are more than understandable.

It is harder to understand the Rambam’s approach, where he claims that any fraction that cannot be accurately measured is rounded off by Chazal.

If this is true, why were Chazal so bothered by the fact that the passuk could be doing exactly the same thing?

The Tosfos haRosh goes further and interprets the flow of the Gemara entirely differently in a way that he feels backs up his claim that we are dealing with approximations.

He understands that the Gemara’s original question, “from where are these words” is not referring to the precise value of PI but rather to the very rule that it is permissible to rely on approximations.

He understands that this leniency is sourced from the very passuk that described the properties of the circle in a way that is clearly an approximation, and quite a large one at that.

He does not say how large an approximation is needed, not under which circumstances it becomes acceptable- it could be that he agrees with Rambam who limits this to an irrational number, but is also possible that he would hold the same for other improper fractions that are hard to work with.

What remains is to understand how both the Rosh and the Rambam would explain the continuation of the sugya which certainly seems to be require precision rather than approximation.

Furthermore, even if we are able to reinterpret the rest of the flow of the sugya in a way that fits with this, or to distinguish between the approximations that are permitted and the one’s the Gemara adamantly seems to reject, we are faced with a very strong difficulty from another related sugya (Bava Basra 14b)

There, the Gemara describes how in addition to the tablets, a sefer Torah was also placed in the ark that rested in the holy of holies.

Based on the view that the circumference of a Torah needs to be 6 handbreadths, the Gemara uses our ratio to show how the 2 tefach wide Torah could fit into the 2 empty tefachim that remained in the Ark after the tablets where placed therein.

The Gemara then notes that an item with a precise width of 2 cannot fit into a precise space of 2 (presumably due to friction.)

It therefore concludes that the Torah was rolled in a way that was not precisely round (the last part was folded on top of the “cylinder”), and the width therefore was less than a third of its circumference.

It seems clear once again that the Gemara is assuming the value of PI to be precisely 3- after all, if it were more than three, a circumference of 6 would produce a width of less than 2 which would easily fit in the remaining space.

Hopefully to be continued

These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha.