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Dedicated by Gitle Bekelnitzky of Queens N.Y. and family to the memory of her father in law, Shraga Feibush ben Yoel Yitzchak, and husband, Simcha Gedalya ben Shraga Feibush. "TORAH FROM THE INTERNET," by RABBI MORDECAI KORNFELD, is available now at a HEBREW BOOKSTORE near you, or from the publisher JUDAICAPR@AOL.COM
I would like to thank David Garber and Boaz Tzaban of Bar Ilan University, who have been publishing articles on Talmudical geometry for a number of years, for their invaluable help in preparing this week's essay. Another important resource was Rabbi Chaim Banish's "Midot v'Shi'urei Torah" pp.43-48.
Parashat Masei 5758
"Give the Levi'im the inheritance of their fathers -- cities to dwell in and open areas around the cities.... You shall measure outside the city towards the eastern corner 2000 Amot, to the southern corner 2000 Amot, to the western corner 2000 Amot, to the northern corner 2000 Amot, with the city in the center. These will be the open areas surrounding the city" (Bamidbar 35:2,5).In this week's Parasha, the Torah describes the cities designated for the Levi'im to live in. Each city, we are told, is to be surrounded by a broad, open area 2,000 Amah (approx. one kilometer) in width. These verses are cited as one of the sources for the 2000 Amah radius that a person may walk in any direction from his place of dwelling on Shabbat (the "Techum Shabbat"). Up to 2,000 Amot from one's city, or house, is considered to be part of his dwelling-space; walking further than that is "going traveling," and prohibited (Ra'avad, cited by the early commentators to Eruvin 36a). Our Sages teach (Eruvin 56b) that when determining the 2000 Amot limit of the Techum Shabbat, one does not draw a circle around the person's place of residence or city, but rather one makes a square. This is also learned from the above-quoted verse in this week's Parasha, which mentions "corners" with regard to measuring the 2000 Amot around the cities of the Levi'im. That is, the 2000 Amot are first measured outwards in a straight line from each of the four sides of the city (assuming the city is square). Next, the band around the city is completed by extending the 2000-Amah border from each of the four sides until they meet at the corners, forming a square around the city. As a consequence, the distance one may walk on Shabbat in the direction of the corner of the square Techum around the city is actually more than 2000 Amot. That is, at the corners one may walk the length of the diagonal of a square whose sides are 2000 by 2000. The Gemara in Eruvin calculates this figure to be 2,800 Amot, using the simple formula "diagonal = side x 1.4" (as expressed in Eruvin 57a and other places). How correct is this figure in light of modern geometry? We know (using the Pythagorean theorem) the true formula for the diagonal of a square to be "diagonal = side x (square root of 2)." The square root of 2 is an irrational number which may be approximated as 1.41421... -- a slightly larger figure than the number 1.4 used by our Sages. This would yield a diagonal of about 2,828 Amot for our Techum Shabbat. Thus, the oversimplified formula for calculating the diagonal which is presented in the Talmud is inexact. This geometric imprecision can be seen in other Talmudic calculations as well. For example, in Eruvin 14a, our Sages determine the relationship between the perimeter of a circle and its diameter based on a verse (I Melachim 7:23) which discusses a circular basin that King Solomon built in the Temple. The basin is described as being 10 Amot in diameter and 30 Amot around. From there the Sages determined that the relationship between the diameter of a circle and its circumference is a relationship of 1:3. As an extension of this rule, the Sages concluded that the relationship between the circumference of a circle and the perimeter (or sum of the sides) of a square drawn around it is 3:4 (since the diameter of the circle is equal to the length of a side of the square). Similarly, they determined that the relationship of the *area* of the circle to the area of the square that is drawn around it is also 3:4 (Eruvin 14b). Modern geometry tells us that the correct relationship between the diameter and the circumference of a circle is 1:pi, and not 1:3. Pi is an irrational number whose value is approximately 3.1419... -- a bit less than 3 1/7. This yields a figure some 4.5% larger than the figure given by the Sages. Based on these examples, one might conclude that our Sages unwarily used a flawed geometry in their calculations, and did not realize that their formulae were approximations at best.
However, to the contrary, the early commentators point out that it was quite clear to the Sages that these calculations were inexact. The Rambam, in his Commentary on the Mishnah (Eruvin 1:5 and 2:5) explains that the Sages found it inconvenient to work with irrational numbers since it is mathematically impossible to determine their exact value. (Using smaller and smaller fractions will just yield close approximations, but will not give an exact value for irrational numbers.) Our Sages preferred to use 1.4 instead of the square root of 2 when calculating the diagonal of a square because, doing so makes the calculations much more readily usable, especially for the layperson not trained in mathematics. The Rosh (Tosfos ha'Rosh to Eruvin 14a, see also Tashbetz 1:165) points out that the same reasoning explains why our Sages used the number 3 as a substitute for the true value of pi. Although it is visually evident that pi is more than 3, they preferred to use the number 3 to simplify the calculations for their students. In fact, when the Gemara asks what the relationship between the diameter of a circle and its circumference is, the Talmud does not suggest that we take out a ruler and measure it. Rather, it proves the relationship from a verse! Why is it necessary to prove a geometric measurement from a verse? The Rosh answers that the Talmud means to ask how we know that although the true relationship between the diameter and circumference is an irrational number, *for Halachic calculations* we may use the number 3. The verse is cited as a proof that although 3 is not the exact number for the circumference, nevertheless it is sufficient to use that number in our Halachic calculations. (In a number of places, the early commentaries in fact offer mathematical proofs to show that the numbers 1.4 and 3 used by our Sages in these calculations are inexact (e.g. Tosfot Eruvin 56b DH Kamah; 57a DH Kol). The Rosh [~1400 C.E.], who appears to have been in contact with mathematicians, mentions the Pythagorean theorem in his commentary to Kilayim 5:5.) There is a disagreement among the early and later commentators as to how to convert the reasoning of the Rosh and the Rambam into Halachic terms.
(a) The Tashbetz (ibid.) mentions that some understand this to mean
that Halachic precision and geometric precision are two separate matters.
When determining a Halachah, it is the Torah that delineates to us how to
practice its laws; in our case there may have been a Halachic tradition
telling us that for Halachah matters, it is sufficient to substitute the
number 3 for pi when calculating the circumference of a circle -- or 1.4
for radical 2 in the calculation of a diagonal -- and there is no need to
be more precise than that. This does not mean that it is the true
measurement. Rather, as far as the Halachah is concerned we are to use
these measurements and not the real measurements. The Torah wanted to
simplify our Halachic calculations. This is also the opinion of the
Maharal (Gur Aryeh, Eruvin 14a) and it is mentioned in numerous Halachic
authorities as well (see Shach, Nekudot ha'Kesef to Yoreh Deah 30:2; Aruch
ha'Shulchan, O.C. 32:75 and Y.D. 30:13; Rav Tzadok ha'Kohen in Tiferet
Tzvi to Y.D. 30, end of note 5). In fact, the Magid Mishnah (Hilchot Eruvin 3:2) also appears to rule this way, in contradiction to his words in Hil. Shabbat that we mentioned earlier (b). Perhaps the Magid Mishnah differentiates between the measurement for pi, for which we have a Torah source that it may be estimated (the basin of King Solomon), and the measurement of radical 2, for which we have no Torah source allowing us to approximate. When it comes to radical 2, we must use the exact geometric calculation. If so, his is a fourth opinion.
One Talmudist and mathemetician, Matityahu ha'Kohen Munk (Frankfurt-London, in "Sinai," Tamuz 1962, and "ha'Darom," 1967) pointed out that there is actually a hint in the verse that discusses the basin of King Solomon to the inexactitude of the Talmudic calculation for the diameter of a circle and to a closer approximation for pi. The verse (Melachim I 7:23) that the Gemara cites discussing the basin built by King Solomon states that the basin was 10 Amot wide, and a line ("Kav") of 30 Amot encircled it. According to the Mesorah, the word "Kav," meaning line, is pronounced differently than it is spelled. Instead of being spelled, as usual, "Kuf, Vav," it is spelled "Kuf, Vav, *Heh*" (that is, a silent Heh is added which is not prounounced). The Gematria (a system for assigning numerical values to Hebrew letters) of the word "Kav" is 106, while the Gematria of the word "Kaveh" is 111. Perhaps the point of this Masoretic anomaly is to hint that the line ("Kav") measuring the circumference of the basin was not really exactly 30 Amot. Rather, the measurement given in Melachim is slightly off, by a ratio of 106 to 111 (Kuf-Vav:Kuf-Vav-Heh). This value is an extremely close representation of the real value for pi (106/111 = 3/ 3.1415094, which is accurate to within .00001 of pi)!
Some Rabbinic geometry, though, is even more puzzling, and seems impossible to reconcile. In the Gemara (Eruvin 76b and Sukah 8a) an opinion of the Rabbis of Kesari is mentioned. They assert that when one draws a circle around a square which is 4 by 4, the circle's circumference is 24. Even if we use the rabbinic approximations discussed earlier, a square of 4 by 4 has a diagonal of 5.6, which is also the diameter of the circle drawn around it. When we multiply that diameter by three (the Sages' approximation for pi), then we get 16.8 -- nowhere near 24! How did the Rabbis of Kesari come up with the number 24? In fact, in Sukah 8b, the Talmud itself seems to reject their opinion, saying that it is obviously incorrect as anyone can see by drawing a circle around a square. However, in Eruvin the Gemara seems to remain in support of this statement. How could we possibly reconcile such a method for measuring with the geometric facts, and with the accepted Rabbinic method for geometric calculation? It is obvious that the rabbis of Kesari could not have made such a blatant mistake, offering a measurement which is off by approximately 50% of the true value. A number of solutions have been suggested to explain what the Rabbis of Kesari really meant.
(a) One suggestion, that of the Vilna Ga'on (in his annotations to
Eruvin 76b, and in Shulchan Aruch Orach Chaim 372) based on the words of
Tosfos (Eruvin 76b), is that the Rabbis of Kesari were not discussing the
circumference of the circle itself that is drawn around the square that is
4 by 4. Rather, they were discussing the perimeter of a *square* that is
drawn around the circle (which, in turn, contains within it a square of 4
by 4). Since each side of the outer square is equal to the diagonal of the
inner square which is 5.6, then the perimeter of that outer square is
22.4. The number 24 that was given was an approximation. How is that? If a small square is placed diagonally (like a diamond) inside a larger square, with its tips touching the middle of the sides of the outer square, it becomes evident that the inner square is exactly half of the area of the outer square, since it is cutting each quadrant of the outer square in half. We know that the inner square's area is 16, so the outer square must have an area of 32. If so, the area of a circle drawn inside the outer square (and consequently around the inner square) will have a relationship of 3:4 to the area of the outer square. Since the area of the outer square was 32, the area of the circle is exactly 24! This is what the Rabbis of Kesari meant.
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