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Math Problem for Talmudic Rabbis: Building the Right Size Sukkah

Pi, irrational numbers, and squaring the circle are all brought to bear to find justifications for tradition

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(Photoillustration Tablet Magazine; original photo Sergio Fabris/Wikimedia Commons)
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For a thousand years, traditional Jewish education was focused on learning the Talmud. But as I’ve often found in reading Daf Yomi, learning the Talmud also requires mastering secular arts and sciences such as mathematics, astronomy, and medicine. In fact, a complex geometry problem came up early in this week’s reading in Tractate Sukka, raising the question: How well did the rabbis know the value of pi?

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.

All this geometry was hard enough for me to follow in the Koren Talmud’s expanded and leisurely English translation. In the original Aramaic, I imagine it would be quite impossible to understand. Take, for instance, the sentences in which the Gemara considers Rav Asi’s proposal. Here is the word-for-word translation of the Aramaic: “How many are there? Eighteen. Seventeen minus one-fifth sufficient. This is where he was not precise, and he is not precise stringency.” To expand this into comprehensible English, the Koren editors add at least 150 words, explaining each step of the mathematical reasoning. It is a reminder that reading the Talmud the way I have been doing, alone, is not the traditional method because it is simply not feasible. A teacher, or at least a translator, is absolutely necessary to make sense of the rabbis’ compressed reasoning.

With this mathematical knot untangled, Tractate Sukka goes on to discuss several other rules for building a valid sukkah. Many of these come down to the question of intentionality: Was the structure built with the intention of using it to celebrate the holiday, or was it repurposed or made with recycled materials for convenience’s sake? Can one, for instance, use an “old sukkah,” defined as one that was built 30 days prior to the holiday? Beit Shammai says no, while Beit Hillel, always more lenient, allows it. However, both rabbis agree that “if he established it for the sake of the festival”—that is, if you declared while building your booth that it was for Sukkot—then there is no time limit; you could even use a building that was a year old.

A similar logic is involved in the question of whether you can build your sukkah under a tree. Could you use the branches of the tree as a kind of ceiling, to provide the shade required in a sukkah? The answer is no: The branches are already growing, so they were not prepared specifically for the sukkah. If, however, you “lower” the branches of the tree—say, by pulling them down and weaving them into the roofing—then they can be used, so long as the majority of the shade is still coming from the roof and not the tree.

But this takes for granted that we know what can be used in ordinary sukkah roofing. Traditionally, Jews use cut branches and vines to cover the sukkah; but where, exactly, is the legal basis for this practice? In Sukka 11a, the mishna explains: “Anything that is susceptible to ritual impurity, or its growth is not from the ground, one may not roof his sukka with it. And anything that is not susceptible to ritual impurity and its growth is from the ground, one may roof with it.” This principle rules out all animal products, such as hides, and all manufactured vessels, which are capable of becoming tamei, ritually impure. It also rules out all actual food products, since food too can become tamei. What is left is plant matter that cannot be eaten—stalks, vines, branches, leaves.

This rule is nowhere stated directly in the Torah, but Rabbi Yochanan finds a hint in Deuteronomy 16:13, which reads: “You shall prepare for you the festival of Sukkot for seven days as you gather from your threshing floor and your winepress.” This suggests that the waste products of threshing and wine-making are what should be used on Sukkot. And in Sukka 12a, Rav Chisda cites a verse from much later in the Bible, from the Book of Nehemiah, which confirms that this was indeed the practice of the ancient Israelites.

Here, as so often, the Talmud raises the actual practices of Judaism to the level of theory, finding a legal principle behind what may well have evolved as a folk custom. In this sense, the Talmud, which stands at the heart of traditional Judaism, is anything but a traditional text. It is never content to justify a Jewish practice simply because it has always been done that way. Rather, it brings reason to bear to find a justification for tradition—a method of cultivating the intelligence even more powerful than learning geometry.

***

To read Tablet’s complete archive of Daf Yomi Talmud study, click here.

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Math Problem for Talmudic Rabbis: Building the Right Size Sukkah

Pi, irrational numbers, and squaring the circle are all brought to bear to find justifications for tradition

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