Sukkah, Daf Het, Part 2
Introduction
Today’s section continues to deal with the size of the round sukkah. Yesterday we concluded that the perimeter of the square needed to be greater than a circle. But today we see that it still won’t get us from a circumference of 12 to a circumference of 24, the number required by R. Yohanan for the round sukkah to be kosher.
מכדי, כמה מרובע יותר על העיגול – רביע, בשיתסר סגי!
But consider: By how much is a square greater than its [inscribed] circle?
By a quarter. Should it then not suffice if only sixteen [men can be seated around it]?
A square is only 25 percent larger than an inscribed circle. Thus if the circle has a diameter of 4, its circumference is 12 (assuming that pi is 3). A square placed around this circle will have sides that are 4 cubits (= to the diameter), leading to a circumference of 16 (4 x 4). So why then does the circle need to be 24 cubits?
הני מילי בעיגול דנפיק מיגו ריבועא, אבל ריבועא דנפיק מגו עיגולא – בעיא טפי, משום מורשא דקרנתא.
That is so in the case of a circle inscribed within a square, but if a square is to be inscribed within a circle a greater circumference is required on account of the projection of the corners.
The above was assuming that the square was around the circle. But if R. Yohanan was talking about a round sukkah that could fit inside a sukkah of four squared cubits in it, the sukkah must be bigger because of the projection in the corners (the places where the circle doesn’t fill the square.
מכדי, כל אמתא בריבועא אמתא ותרי חומשא באלכסונא, בשבסר נכי חומשי סגיא! –
But consider: If the side of a square is a cubit, its diagonal is approximately one and two fifths cubits.
Should not then [a circumference equivalent to] sixteen and four fifths [cubits] suffice?
The problem is that this still doesn’t add up to 24. According to rabbinic calculation, a hypotenuse is 1.4 times the side of a square (in reality it is the square root of 2, but 1.4 is close). So the diameter of the circle (which is equal to the hypotenuse of the inscribed square) is 4 x 1.4 5.6. That would mean that the circumference of the circle is 16.8 (3 x 5.6). We still have not gotten too close to 24.
לא דק.
[R. Johanan] gave only an approximate figure.
The Talmud answers that R. Yohanan wasn’t precise in his figure. Admittedly, not a particularly satisfying answer.