Pi, the number that defines the ratio of a circle’s circumference to its diameter, was already known to the ancient Egyptians and Babylonians. Greek and Roman mathematicians computed it to four decimal places—3.1416. The rabbis, in Sukka 7b, have a more rough-and-ready measurement: “For every three handbreadths circumference,” they say, “there is a diameter of one handbreadth,” effectively defining pi as 3. The matter comes up because the rabbis are arguing about the proper dimensions of a circular sukkah. A standard square sukkah must be four cubits by four cubits; it seems to follow that a circular sukkah ought to have a diameter of four cubits. Using the rabbis’ value of pi, this would yield a circle with a circumference of 12. Estimating that each person in the sukkah requires one cubit of space—a tight fit, since a cubit is only about 18 inches, but people were smaller then—this would mean that the smallest valid circular sukkah could fit 12 people along the inside wall.
Why is it, then, that Rabbi Yochanan requires a circular sukkah to be able to fit 24 people? Why should a circular sukkah have to be so much larger than a square sukkah in order to be legally valid? This is the issue the Gemara takes up in Sukka 8a and in doing so wades into the problem of “squaring the circle,” one of the insoluble enigmas of ancient geometry. Because pi is an irrational number, extending to an infinite number of decimal places, it’s impossible to draw a square with exactly the same area as a given circle. In Talmudic terms, this means that one can’t take a circular sukkah and figure out exactly how big a square sukkah would have to be to include the same area.
However, it is possible to approximate the answer, and by calculating the square root of two, the rabbis find that a four-by-four square sukkah is about the same size as a circular sukkah with a circumference of 16.8. Such a sukkah could fit 16 people (plus one extra-thin person if you squeezed him in) with their backs to the walls. But this figure is still substantially lower than Rabbi Yochanan’s requirement of 24 people. The rabbis are again stymied: What was the reason for Yochanan’s figure?
Mar Keshisha takes another shot at the problem. So far, the discussion has proceeded on the assumption that each person takes up one cubit of space, so that Yochanan’s requirement of 24 people was equivalent to 24 cubits. But what if, Mar Keshisha asks, this is too generous an estimate? What if a person actually only needs two-thirds of a cubit, about a foot of space? In that case, a sukkah that held 24 people would actually only have a circumference of 16 cubits. This is much closer to the figure of 16.8. But as the Gemara points out, now we have the opposite problem, since Rabbi Yochanan’s figure is actually smaller than the correct size. In other words, Yochanan is being too lenient, allowing people to get away with building a circular sukkah that is slightly too small.
Finally, Rav Asi comes with a solution to the problem. Instead of calculating a circle big enough to hold 24 people, he suggests, we should be calculating the size of a circle formed by 24 people—that is, the circle is drawn inside the ring of people, not outside them. Assuming once again that a person takes up one cubit, this means that we can subtract two cubits from the diameter of the circle—one for the person sitting on each side of the circle—which means that we now have a diameter of six, instead of eight. Taking pi as three, this yields a circumference of 18—which would be the correct figure for a circular sukkah, according to Yochanan. Once again, Yochanan’s figure is not precise—he requires 18 cubits, where the true figure is 16.8—but this time we find him erring on the side of stringency. He insists that we make a sukkah that is slightly too big, rather than one that is slightly too small. This kind of stringency is all right, according to rabbinic law, and so the Gemara finally accepts Rav Asi’s interpretation.
