In an earlier post (Eruvin 14,) we saw how the Mishna teaches that the ratio between the circumference and the diameter of a circle is 3.
The Gemara derived this from the ים של שלמה (circular water-feature) which was 10 amos wide and 30 amos in circumference.
We raised the obvious issue that the actual value of this ratio is π, an irrational number equal to slightly more than 3.14, and we saw two basic approaches amongst the Rishonim:
- The Tosfos brought evidence that Chazal were being precise in their measurements, pointed out that the mathematical experts hold that it is not precise, and leave it as a difficulty.
- The Rambam and Tosfos haRosh both understand that this is an approximation.
On our daf, we encounter this ratio once again.
Our Mishna discusses how large a “window” in the boundary wall between two neighbors’ properties needs to be for them to be able to make one eruv between the two of them.
It rules that the window needs to be at least four by four tefachim to qualify as a פתח
(opening) and that at least part of it needs to be within 10 tefachim of the ground.
Rabbi Yochanan brings up the case of a round window and how large it needs to be.
We should recall that in the context of daf 13b, the Mishna ruled that a round pole used for the beam of a מבוי does not have to be large enough to contain a tefach-wide square beam within it- it merely needs to be a tefach wide at the diameter, or 3 tefachim in circumference. (רואים כאלו היא מרובעת)
The Gemara initially seems to have thought that the same should be the case with our window, and that so long as it is 4 tefachim at the diameter, or 12 tefachim in circumference, it counts as if it was a square window of 4 by 4 tefachim- after all, this could just be a symbolic opening in any case.
Yet There is a strong argument to be made that this case should be different seeing as the opening might actually need to function as a פתח, and one could never squeeze through a circular hole that is only 4 tefachim wide at the diameter.
Rabbi Yochanan, citing the famous 3 to 1 ratio, rules that it needs to be 24 tefachim in circumference, and that slightly more than 2 of them need to be below the 10 tefachim line of the wall, so that if the circle is squared, part of the square will be under the line.
This rather cryptic statement of Rabbi Yochanan has pages and pages of commentary trying to explain.
After attempting to make some sense of it myself with the little high-school math I remember and some diagram, I was immediately overwhelmed by the complexity and length of the discussion.
I knew it would not be one day’s work to even scratch the surface, but decided to take my time and try get at least some idea of what is going on, and what we can take from it into the general topic of Torah and Science that we keep coming back to.
First – the “simple” flow of the Gemara, (if there is such a thing:)
- The Gemara notes the usual 3 to 1 rule, and notes that in order to get a diameter of 4 tefachim, the circle should only need to be 12 tefachim in circumference.
- The Gemara answers that this rule replies to a circle, but for a square, more is needed.
We need to understand what the Gemara means to say, as it is clear that we are dealing with a circle and not a square.
One possibility is that although we are dealing with a circle, Rabbi Yochanan requires a circle with a circumference equal to a square of 4 by 4, in order for it to be considered an equivalent valid opening to the square.
- The Gemara answers that a square that circumscribes a circle is only a quarter more than the circle itself , so 16 tefachim should be sufficient. (It does not say what attribute of the square is a quarter more than the circle, but the Gemara seems to assume that this is the circumference, and to be referring to a quarter of the resulting square.)
- The Gemara answers that this is the case with a square that circumscribes a circle (is inscribed by a circle). The internal circle is 3 (PI) times the diameter in circumference, namely 12 tefachim, whereas the square is 4 times its width, or 16 tefachim.
However, what we need here is a square of 4 times for tefachim to be able to fit inside the circle, which means the circumference of the circle needs to be even more to cover the parts of the circle outside the square.
- The Gemara uses another apparent approximation, the length of the hypotenuse of a right-angled triangle formed by cutting a square in two by its diagonal. Although this is the square route of two (an irrational number) times by the width, the Gemara treats it as 1.4 (1 and two fifths.) This would also be the circumference of the required circle.
Using this, it works out that circumference of our circle need only be 16.8 tefachim, in order to be able to have square of 4 by 4 tefachim inscribed in it. (using precise modern measurements, this would be 4Ö2*π, rounded to 17.77)
So why does Rabbi Yochanan require such a large circle!
- The Gemara replies that Rabbi Yochanan was following the judges of Caesarea (some versions say “The Rabbis of Caesarea) who said that “a circle inside a square is a quarter, a square inside a circle is a half.”
This cryptic statement is itself subject to interpretation of course.
There are multiple ways to learn the flow of the Gemara, starting from the requirement for “2 of them and a bit” to be below the 10 tefachim line, and ending with this view of the judges of Caesarea, and it would take pages and pages to go through.
Some essential reading in in the Rishonim include Rashi, Tosfos, Rashba, Ritva, and the Meiri who has a particularly extensive treatment of the subject.
As we have already addressed the issue of PI being approximated by 3 by Chazal, we shall not focus on that right now, although it would be in place to analysis whether the flow of the sugya here indicates that Chazal were aware of this approximation and using it intentionally or not.
What we see here in addition to this is another “approximation” of Chazal (also encountered elsewhere), namely the square root of 2, but even more significantly, that it is used in combination with the approximation of PI, creating quite a large combined “rounding error.”- after all, there is a significant different between a circumference of 16.8 and one of 17.6 (see picture and formula,) and even according to the view of Rambam and Rosh that approximation is sometimes acceptable, relying on a double approximation seems to be a significantly greater novelty.
Another fascinating issue here is the question of the accuracy of the mathematical knowledge of Chazal and the Rishonim.
The simple explanation of the sugya, as understood by Rashi and Tosfos (see also the parallel sugya on Sukkah 8a,) seems to be that Rabbi Yochanan relied on a mistaken mathematical formula used by the judges (or Rabbis) of Caesarea which according to Rashi seems to calculate the diagonal as twice the width, rather than 1.4 times the width of the Gemara or the root of 2 used by mathematicians.
Tosfos is bothered by how the judges of Caesarea could have erred in something that is so obviously easy to ascertain. Interestingly enough, he seems less bothered by the fact that Rabbi Yochanan followed in their error. He even goes so far to suggest that it was not the judges who erred, but Rabbi Yochanan who erred in his interpretation of what they said!
In contrast, the Rashba seems more bothered by the fact that Rabbi Yochanan could have made such an era, and the Gra notes that we should not “chas veshalom” say that they made any era, choosing to interpret their ruling differently to most of the Rishonim.
In addition, Tosfos disagrees with Rashi’s interpretation of the “2 and a bit” rule stated earlier given that his claims are not mathematically correct and proposes another explanation which using the correct formula also does not seem mathematically correct (though in fairness to both Rashi and Tosfos, they are merely commenting on the meaning of Rabbi Yochanan and the judges of Caesarea, not on mathematical reality!)
The temptation to simply say that Rishonim did not understand basic mathematics should not be taken lightly.
Rashi’s ability to make complex calculations is well known, and the Tosfos were brilliant enough to provide their own proof that the length of the hypotenuse of an isosceles right-angled triangle is slightly more than 1.4 times the length of its other sides- It is hard for us to imagine that people with such minds did not know mathematical formula that even the ancient Greeks were aware of so long ago.
Yet despite the above, it might not be necessary to assume that all Rishonim were well-versed in all mathematical formula and knowledge that was known to man at the time.
Medieval Europe was not the most “enlightened” part of the world by any means, communications were not what they are today, and much knowledge that the ancient Greeks had access to was inherited by the Islamic world to the South and East, rather than France and Germany.
The Rishonim might have been brilliant enough to work out mathematical theory on their own had they dedicated their time to it, but they clearly had other priorities and did not go all the way.
It is also not so outlandish to posit that whereas some Chazal were very exposed to and familiar with the scientific knowledge of the Greek and Roman worlds, others were less so, and sometimes needed to be corrected by their colleagues or even later authorities.
This does not even have to contradict the view of the Ramban, discussed in earlier posts, that ascribes a form of wisdom-induced “ruach hakodesh” to great Talmidei-Chachomim.
As we mentioned before, this does not necessarily mean that they ALWAYS experienced this “ruach hakodesh” nor that this “ruach hakodesh” has anything to do with awareness of scientific facts through supernatural means- rather it seems from the examples given in the relevant sugya (Bava Basra ) that it has more to do with intellectual “siyata dishmaya” which allows them to come up with ideas only far greater people would normally have come up with, while still basing these ideas on the information available to them at the time.
However, I am extremely hesitant at over-using such explanations- while they can possibly account for a lack of precision regarding the root of 2 or the value of PI, Chazal and the Rishonim certainly knew how to learn things from simple observation, and larger errors, such as viewing the hypotenuse of an isosceles right-angled triangle as the sum of the two other sides, essentially treating root 2 as the same as 2, simply defy rational explanation.
As such, I tend to believe that at least in such cases, the errors made were more strategic than mathematical, and that for reasons of convenience, the judges of Caesaria chose to be stringent and require the maximum possible length of the third side of ANY “triangle” (ie approaching a straight line with 180 degrees between the two sides) which is the sum of the other two sides. It is important to stress that this is not necessarily Rashi’s view, but his explanation of their view, but it could also explain other issues where Rashi takes this approach (or make it more difficult!)
The Gra, of course, has his own novel approach to the sugya, and whereas it seems somewhat forced in the text and out of line with most of the Rishonim on the subject, the Gra most certainly was a complete expert in all of Torah as well as in mathematics, making a study of his approach particularly appealing.
There are indeed SO many questions raised in this sugya and the way that the Rishonim handle them, and its awfully frustrating to have to leave the discussion for a different forum and start catching up on the daf, but such is life- maybe we shall get to revisit this discussion sooner than we think!
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 halach