Sukkah, Daf Het, Part 3
Introduction
More math! We continue to try to figure out how R. Yohanan came up with the number 24 for the required circumference of the round sukkah. Yesterday’s section ended with the conclusion that the sukkah really only needed to have a circumference of 16.8 cubits but that R. Yohanan was simply approximating when he said it needed to be 24 cubits.
אימור דאמרינן לא דק פורתא, טובא מי אמרינן לא דק?
Say that we say that [where the discrepancy is] small he might approximate, but could such an assumption be made [where the discrepancy is] big?
If R. Yohanan gave a number that was close to 16.8, we could say that he was issuing an approximation. But 24 cubits is just too far off of 16.8. It’s hard to imagine that he was simply approximating. Thus we are back to square one-why does the circular sukkah have to be so large.
אמר ליה מר קשישא בריה דרב חסדא לרב אשי: מי סברת גברא באמתא יתיב? תלתא גברי בתרתי אמתא יתבי. – כמה הוו להו – שיתסר, אנן שיבסר נכי חומשא בעינן! לא דק.
Mar Kashisha the son of R. Hisda said to R. Ashi: Do you think that a man occupies one cubit? [The fact is that] three men occupy two cubits. How much then does this [amount to for twenty-four men]? Sixteen cubits; and we [really] demand here sixteen and four fifths. [Because, as has been said, R. Yohanan] gave only an approximate figure.
Mar Kashisha now adjusts one of the key figures in the original calculation. We thought that each person takes up 1 cubit, but in reality three men can fit into 2 cubits. So for 24 men to be able to sit around this sukkah, they only need 16 cubits. This is still not exactly what R. Yohanan said, but it’s close. R. Yohanan’s figure is 16 and the size of the sukkah that fits around a 4 cubits square sukkah is 16.8.
אימור דאמרינן לא דק לחומרא, לקולא מי אמרינן לא דק?
Say that we give approximate figures in order to create a stringency, but could such an assumption be made where this makes the law lenient?
The problem is that R. Yohanan’s approximation leads to a leniency. R. Yohanan says it only needs to be 16 cubits around so that 24 people could sit around it. But in reality it should be slightly bigger.
אמר ליה רב אסי לרב אשי: לעולם גברא באמתא יתיב, ורבי יוחנן מקום גברי לא קחשיב. – כמה הוו להו – תמני סרי, בשיבסר נכי חומשא סגיא! – היינו דלא דק, ולחומרא לא דק.
R. Assi said to R. Ashi: In truth, a man occupies a cubit-space, but R. Yohanan does not include the space occupied by the men. How many [cubits] does this[amount to]? Eighteen; while sixteen and four-fifths would be enough. That is [then] what was meant [when it was stated] that he only gave an approximate figure; and in this case it is in the direction of stringency.
R. Assi now changes the assumption we have been making all along that the people were sitting outside of the sukkah. If each person takes up a cubit’s space, this leads to the diameter being 8 cubits (1/3 of the circumference). But if they sit inside the sukkah, then we can reduce a cubit in each direction, bringing the diameter down to 6 cubits. This leads to a circumference of 18. So if he really needed to require 16.8, the circumference that would allow an inner circle of 16, R. Yohanan was slightly stringent. And it is okay to be offer an approximation if it leads to a stringency.
רבנן דקיסרי, ואמרי לה דייני דקיסרי אמרי: עיגולא דנפיק מגו ריבועא – רבעא, ריבועא, דנפיק מגו עיגולא – פלגא.
The rabbis of Caesarea (and some say, the judges of Caesarea) say: The circumference of a circle inscribed in a square is a quarter; but the square inscribed within that circle is a half.
The rabbis of Caesarea provide a formula for measuring a circle within a square and a square that is inside a circle.
The circle inside a square has a circumference that is 1/4 less than the square. This matches what we said before. So if the perimeter of the square is 16, than the diameter of the inscribed circle is 4. Multiply that by 3 and you get 12, which is 1/4 less than 16.
A square that is inside a circle has a circumference that is half that of the circle.
ולא היא, דהא קחזינן דלא הוי כולי האי.
But this is not correct, for we see that these are not so much bigger.
The Talmud rejects the rabbis of Caesarea’s last formulation. If the square is 4 square cubits, then its 5.6 cubits, which is also the diameter of the inscribing circle. The circumference of the circle is 3 times this amount, meaning 16.8. Indeed it is difficult to imagine that the rabbis of Caesarea (or R. Yohanan) made such a gross error in calculation
The Gaon of Vilna explained that in actuality this was not an error at all. What these rabbis meant to say was that if you put another square around this circle, that square would have a circumference that is 50 percent larger than the inner square. This outside square would have sides that are 5.6. Its circumference would be 24 (5.6 x 4). This square is now 50 per cent larger than the inner square, whose circumference is 16. This, according to the GRA is what these rabbis really intended to say.
This matches R. Yohanan’s number above. R. Yohanan meant to say that this circular sukkah must be large enough so that when you put a kosher sukkah inside (16 square cubits), the square formed outside of the sukkah would be 24 cubits.
Interestingly, we see that R. Yohanan’s math was actually correct. It was later rabbis that seemed to have misunderstood his words. Perhaps, we might surmise that in Eretz Yisrael, where R. Yohanan and the rabbis of Caesarea lived they understood math better than they did at a later period in Babylonia. This makes sense considering the fact that the Greeks were well known to have been excellent mathematicians.