INSIGHTS INTO THE DAILY DAF
prepared by Kollel Iyun Hadaf of Yerushalayim
Rosh Kollel: Rabbi Mordecai Kornfeld
OPINIONS: The Gemara cites Rava's ruling about one who made two Eruvin for two different people at different times, and the food of both Eruvin became lost. One Eruv was made while it was still day (on Erev Shabbos), and one was made during Bein ha'Shemashos. The first Eruv became lost during Bein ha'Shemashos, and the second Eruv become lost after Bein ha'Shemashos (when it was definitely nighttime). Rava teaches that both Eruvin are valid. What type of Eruv is Rava discussing?
(a) RASHI explains that Rava is discussing a case of Eruv Techumin. According to Rashi, Rava apparently rules like Rebbi Yosi (35a) who says that an Eruv Techumin whose effectiveness is in doubt is nevertheless a valid Eruv.
(b) TOSFOS (DH Sheneihem) says that Rava is not referring to Eruv Techumin, because in this case even Rebbi Yosi would agree that such an Eruv would not be valid. There is no doubt about the Eruvin in this case; we know for certain when the Eruvin were made and when they were lost. The only doubt is whether Bein ha'Shemashos is day or night. In such a case of uncertainty, Rebbi Yosi would not be lenient to validate the Eruv. Rather, Rava is discussing Eruvei Chatzeros, the laws of which are generally more lenient than those of Eruvei Techumin. An Eruv Chatzeros made even during Bein ha'Shemashos is valid. (See also Insights to Eruvin 35:2.)
QUESTION: The Mishnah (76a) teaches that in order for a window in the wall between two Chatzeros to be considered a Pesach (opening) between the Chatzeros and provide the Chatzeros with the choice to join together with one Eruv, the window must measure at least four by four Tefachim and be within ten Tefachim of the bottom of the wall.
What must be the dimensions of the window if it is not square but round? Rebbi Yochanan (76a) states that a circular window "must have 24 Tefachim in its circumference, and two Tefachim and a bit of the window must be within ten Tefachim of the bottom of the wall, so that if a square would be inscribed in the circle a part of it would be within ten Tefachim of the ground." Rebbi Yochanan maintains that a circle drawn around a square with sides of four Tefachim (and a perimeter of 16 Tefachim) has a circumference of 24 Tefachim.
The Gemara (76b) concludes that Rebbi Yochanan's geometrical calculations are based on the formula of the Judges of Kesari. The Judges of Kesari taught that the circumference of a circle inscribed inside of a square is 25% smaller than the square's perimeter, and the circumference of a circle circumscribed around the outside of a square is 50% larger than the square's perimeter. Accordingly, the circumference of the circle drawn around the 16-Tefach perimeter of a square is 50% larger than the square's perimeter, or 24 (50% of 16 added to 16 is 24).
As mathematics demonstrates, and as the Gemara itself in Sukah (8a) points out, this formula is clearly incorrect. According to the formula used by the Chachamim (see ), the actual relationship of the perimeter of an inscribed square to the circle around it is 3 X (1.4 x S), where 3 = the value of Pi, and S = the length of a side of the square. The ratio that the Chachamim use for the relationship between the side of a square and its diagonal (which is also the diameter of the circumscribed circle) is 1:1.4. Therefore, the circumference of a circle circumscribed around a square with sides of 4 Tefachim is 3 X (1.4 X 4), or 16.8 -- and not 24!
How did the Judges of Kesari make such a mistake, and why did Rebbi Yochanan follow them?
(a) TOSFOS (DH v'Rebbi Yochanan) suggests that the Judges of Kesari were not giving the relationship of the perimeter of the inner square to a circle around it. Rather, they were giving the relationship of the area of the inner square to an outer square drawn around the circle that encloses the inner square. This is what they mean when they say that "when a circle is drawn around the outside of a square, the outer one's (i.e., the outer square's) perimeter is 50% larger than the inner one's." (The picture printed in Tosfos in the Vilna Shas is slightly misleading. In the picture that appears in the TOSFOS HA'ROSH (see Graphic), the inner square is rotated 45 degrees from the orientation of the outer square. This is a clearer demonstration of Tosfos' point.) The area of the inner square is exactly half of the area of the outer square.
Tosfos concludes that Rebbi Yochanan misunderstood the intention of the Judges of Kesari, and he made his statement regarding the relationship of the circumference of a circle to the perimeter of a square based on his misunderstanding.
However, the VILNA GA'ON (Hagahos ha'Gra here and in OC 372) takes issue with the conclusion of Tosfos and asserts that even Rebbi Yochanan is referring to the perimeter of a square with a circle inscribed (which, in turn, circumscribes a square of four by four Tefachim). That perimeter is indeed nearly 24 Tefachim. (According to the formula used by the Chachamim's to calculate the diagonal of a square, it is 22.4 Tefachim. The geometric calculation yields 22.6 Tefachim.) When Rebbi Yochanan says that two Tefachim of the window must be within ten Tefachim of the ground, he is referring to the length of the arc that begins from the lowest point of the circle that was drawn around the square window (four by four), to the point where it intersects with the bottom of the square window. (Actually, as the Vilna Ga'on points out, a bit more of the arc must be within ten Tefachim of the ground -- 0.1 Tefach according to the formula of the Chachamim, or 0.121 Tefach according to the geometric calculation.)
(b) The RITVA explains that the calculations of the Judges of Kesari and Rebbi Yochanan are accurate. When Rebbi Yochanan mentions a "round" window, he does not mean a circular window with an imaginary square inscribed within it. Rather, he is referring to a window made in the shape of a square with four semi-circles protruding from the four sides (like a four-leaf clover; see Graphic). In such a case, the perimeter of the window (i.e., the arcs of the four semi-circles) indeed is 50% larger than the perimeter of the square around which the arcs are drawn. In order to ensure that the square inside the clover-shaped window reaches within ten Tefachim from the ground, at least two Tefachim and a bit of the radius of the bottom semi-circle must be within ten Tefachim (since the radius of each semi-circle is two, or half of one side of the square, which is four). Alternatively, two and a bit Tefachim plus four Tefachim of the perimeter of the semi-circle must be within ten Tefachim from the ground (as Rashi explains, end of 76a), since the perimeter of each semi-circle is six Tefachim.
(c) RASHI does not explain how to justify the formula of the Judges of Kesari or how to understand Rebbi Yochanan's calculation. He seems to have no difficulty with them. Perhaps Rashi understands that the Judges of Kesari were proposing a Halachic stringency: When we determine a value (such as the circumference of a circle) by using the diagonal of a square for the purpose of a practical application in Halachah, we consider the diagonal to be equal to the sum of two consecutive sides of the square or rectangle (since the two sides connect one end of the diagonal to the other). Thus, if the sides of the inscribed square are each 4 Tefachim, then the diagonal is considered to be 8 Tefachim. Accordingly, the circle around that square would have a diameter of 8 Tefachim, and thus its circumference would be 24 Tefachim and not 16.8 (which is the length of the circumference based on the actual diameter of the square).
The reason for this is to prevent one from mistakenly using the length of the diagonal in a case in which he is supposed to use the sum of the lengths of two sides. In addition, reality does not allow for the application of pure mathematics (as the ratio of the diagonal to the sides of a square is an irrational number; moreover, it is not possible to measure an angle exactly). Therefore, the formula for determining the diagonal of a square for purposes of Halachic applications is the sum of the lengths of two sides.
If this is the reason why Rashi is not bothered by the apparent inaccuracy of the formula of the Judges of Kesari, then we may suggest that Rashi is consistent with his own opinion as expressed elsewhere (Shabbos 85a, Eruvin 5a, 78a, 94b), where Rashi seems to determine the Halachic length of the diagonal of a rectangle by adding two consecutive sides. TOSFOS in all of those places argues with Rashi. Rashi may hold that this computation of the length of the diagonal may be relied upon for rulings that involve Halachos d'Rabanan. (M. KORNFELD)
(d) Perhaps it is possible to propose an entirely new explanation, according to which the Judges of Kesari and Rebbi Yochanan are entirely correct.
When Rebbi Yochanan says that a circular window "must have 24 Tefachim in its circumference," he does not mean that the circumference must be 24 Tefachim, but that there must be 24 square Tefachim inside the circle. In other words, he means that the area of the circle must be 24 square Tefachim!
The area of a circle is calculated by multiplying Pi by the radius squared. The radius of the circle drawn around a square with sides that are each 4 Tefachim long is half of the diagonal (5.6), which is 2.8. Using the Halachic estimate of the value of Pi as 3, we arrive at the following calculation: 3 X (2.8)(2.8) = 23.52, or approximately 24.
This is what Rebbi Yochanan means when he says that the circle must have within its circumference an area of 24. (He rounds up to 24 as a stringency.)
What does Rebbi Yochanan mean when he says that there must be two and a bit within a height of ten from the ground? 24 square Tefachim is the area of the circle. Within that area is an inscribed square of 4 by 4, which has an area of 16 square Tefachim. What is the area of the four sectors that are outside of the square? They represent the difference between the area of the circle and the area of the square, which is 24 - 16 = 8, and thus each sector has an area of 2 Tefachim. This is what Rebbi Yochanan means when he says that in order to get the inscribed square of 4 by 4 Tefachim below a height of ten Tefachim, at least 2 Tefachim and a bit of the area of the circular window must be below ten Tefachim!
(David Garber and Boaz Tzaban of Bar Ilan University, who have been printing articles on geometrical themes in Chazal for a number of years, pointed out to us that the ME'IRI here suggests this solution, citing it in the name of the BA'AL HA'ME'OR. It can be traced further back to a responsum of the RIF in Temim De'im #223. An Acharon, TESHUVOS GALYA MASECHES #3, offers this solution as well. Note, also, that according to the mathematics of Chazal, the ratio of the area of a circle to the area of the square in which the circle is inscribed is a ratio of 3:4. The outer square is double the area of the inner square (which is 4 by 4 Tefachim). Thus, the outer square has an area of 32 square Tefachim. Accordingly, the calculation for the area of the circle is exactly 24 Tefachim (or 3/4 of the outer square), and not just approximately 24, as we concluded using the equation of Pi X r X r (this is because the area of the outer square (32) is exactly double the area of the square drawn inside of the circle (16), and 3/4 of 32 is 24). The Me'iri uses the word "Shibur" or "Tishbores" to refer to the calculation of area.)