In Shabbos 85a, when discussing Rav's opinion that the plot is not surrounded by other fields, the Gemara asks "v'haika makom kranos," which Rashi interprets to mean that you plant seeds in the corners, not all the way across the plot, such that for two sides next to each other there's a two-tefach distance in one direction and a one-tefach distance in another direction; according to Rashi, the hypotenuse of such a triangle would be the necessary three. This is not correct, however, as Tosafos indicates; the hypotenuse is the square root of the sum of the legs' squares, which means it's only sqrt(5) - about 2.24!
Rashi makes a similar comment in Eruvin 5a, where he states that courtyards placed at a corner of a 4x4 tefach square still have a straight line of four tefachim, the hypotenuse of the triangle whose legs are 3 and 1. He likewise notes in Eruvin 94a that a roof that collapsed with legs 5 and more than 5 would yield a breach of more than 10 amos; once again, the hypotenuse is only slightly more than 7.
Rashi clearly knew of the statement that "if a side of a square is one tefach, its diagonal is 1.4" (Sukkah 8a, et al), which is derived from the Pythagorean Theorem, so how could he have made such a mistake? Obviously there has to be some way to make his math make sense.
Putting aside mathematical theorems, logically speaking it doesn't make sense to say that a side of a triangle can equal the sum of its legs.
It doesn't make sense to say that all three of the aforementioned sugyos go according to the Dayanei D'Keisari (Sukkah 8b, Eruvin 76b), who Rashi explain similarly in both of those locations, since the Gemara in Sukkah says that we don't pasken like them, and that their math doesn't make sense (at least in reference to perimeter, as per Tosafos ad loc).
I had seen on your site (insights to Eruvin 76) that Rashi was going with a halachic stringency, but it depends on the context whether he's being stringent or not. In reference to the round Sukkah and window, he is being machmir, but in reference to klayim, he is allowing a setup that is assur! Furthermore, in reference to Eruvin, it doesn't make sense to say that he's being stringent, because we hold that safeik d'rabanan l'kula; assuming that Rashi held this way because he wasn't sure how to evaluate the hypotenuse of the triangle, he would have had a doubt to settle leniently!
I have been working on this question for the past few months. None of the rebbeim I've asked so far - from my shul or yeshiva ketanah - have been able to answer my question yet; given the high-quality information on your site, I was hoping you'd be able to help.
Thank you so much,
Daniel
Daniel Filreis, Atlanta, GA
I am going to attempt to answer your important questions, first on a "localized" basis, and then I may be able to make some more general comments on the way of Rashi in his commentary on Shas, which may help us to understand why in several places Tosfos writes that Rashi "Lo Dak" -- he was not precite in the mathematical data he gave us.
1) The Yad David answers the question of Tosfos (Shabbos 85b) on Rashi (85a), where Rashi's calculation was inaccurate.
(As interesting historical background, it is worthwhile pointing out that the Yad David was written by Rav David Zitsheimer, whom Napoleon sought to appoint as the head of the new "Sanhedrin" which he wanted to establish. Although that Sanhedrin never materialized, the author of the Yad David was still recognized as one of the leading Talmidei Chachamim of his time.) The Yad David (first edition to Maseches Shabbos, on Rashi 85a) writes that Rashi is consistent with his opinion in the Mishnah (84b, DH v'Achas) that the crucial factor concerning Kil'ayim is not "Yenikah" (whether the different species acquire nourishment from each other) but rather that there should be a "Heker" -- it should be noticeable that the different seeds are not being grown together in a mixed fashion, but rather it is clear that each one is growing separately. Therefore, if there is a two-Tefach distance in one direction and a one-Tefach distance in the other direction, this is a sufficient "Heker" that a distinction is being made between the different species, and even though there is less than three full Tefachim between them, and it is possible that there is a physical "Yenikah" between the seeds and they grow together, nevertheless the person who looks at the plant bed can see that the different seeds are not actually mixed up together.
2) The Sefas Emes answers the question of Tosfos in Eruvin (5a, DH d'Pasach) on Rashi DH b'Keren. He writes that in reality there is no dispute between Rashi and Tosfos. When Rashi writes that one Tefach was broken from one wall and three Tefachim from the other wall, this does not mean that this was the only space that was broken down. In fact, some of the area of the "Patzimim" (the door-posts) was also broken, and as a result a total length of four Tefachim was left open on the diagonal. This is what Tosfos means when he writes that Rashi "Lo Dak." He does not mean that he is disagreeing with Rashi but rather that Rashi did not tell us about the entire area that was broken, only about the area broken from the walls themselves.
3) See also what Rabbi Kornfeld writes in Insights to Sukah 8:1:b: http://www.dafyomi.co.il/sukah/insites/su-dt-008.htm .
There is still more to write on this subject but I will close here for now.
Kol Tuv,
Dovid Bloom
I will now address the question on Rashi to Eruvin 94b, DH v'Kiruyav.
1) Tosfos to Eruvin 94b (DH v'Kiruyav) asks your question and writes that Rashi "was not concerned with being accurate on this." This suggests that Rashi certainly knew that the hypotenuse of a right-angled triangle with sides of 5 and 5 is not 10, but Rashi was not concerned to give us accurate figures, but instead just gave an approximate number.
2) This is stated more clearly by the Tosfos ha'Rosh who writes that Rashi gave his numbers "merely as a Siman (a sign) that the gap in the wall must be more than 10." Rashi certainly knew that the diagonal would be only slightly over 7, but he wrote 5 + 5 because this is an easy way of remembering that the Pirtzah must be 10. I hope in a further reply to suggest why Rashi did not see the need to be more scientific with his measurements, but at least we do see for the time being that Rashi's way of writing in his commentary on Shas is sometimes merely to be approximate.
3) However, the Sefas Emes learns that Rashi was precise with his figures and there is in fact a Halachic dispute between Rashi and Tosfos concerning what constitutes a "Pirtzah" -- a gap in a wall. Tosfos maintains that if the Pirtzah is in the north-east corner, one must have a gap of 10 Amos on the diagonal, and therefore a gap of 5 on the north side and 5 on the east side is not sufficient because this is barely more than 7 on the diagonal. However, Rashi maintains that one does not require 10 Amos on the diagonal to be considered a Pirtzah. According to Rashi, one looks at the sum of the two open sides, and if there is a total of 10 Amos between the two combined gaps, this is considered a Pirtzah even if the diagonal is less than 10 Amos open. Rashi's opinion is that one looks length-wise, not diagonal-wise.
Kol Tuv,
Dovid Bloom
I am now going to try to explain the thinking behind what I cited in the previous reply (in #1) in the name of Tosfos, that Rashi was not concerned about being accurate about the length of the hypotenuse of a right-angled triangle, and in the name of the Tosfos ha'Rosh that Rashi wrote that the breach in the two right-angled sides were both 5 long, as a mere "Siman" to show that the total length was 10; even though Rashi was clearly aware that the hypotenuse was a lot less than that, Rashi wrote these figures in order that we should remember that the breach must be at least 10, since everyone knows that 5 + 5 equals 10.
In my two previous replies I cited major Mefarshim, while in this reply I may suggest one or two insights of my own, but I will attempt to base it on classical sources.
1) The first source is the Beis Yosef on the Tur (Orach Chaim 10:6, end of DH ul'Inyan Halachah), who writes that Rashi is a commentator, not a Halachic decisor. I suggest that this means that the main task of Rashi is to help us understand the Gemara while we are learning it. Rashi's chief goal is not to issue Halachic rulings, a task more fitting for the Rambam.
2) The next source is the Tosfos Yom Tov on Peah 2:2. He writes that the way of Rashi in many places is to explain the Mishnah according to the original logic of the Gemara, not according to the conclusion of the Gemara. The Tosfos Yom Tov writes that this is because when one is learning a Mishnah it is appropriate to understand it according to the way the Gemara understands at the beginning of the Sugya, before one gets to the end of the Sugya.
It seems to me that the same message comes through. Rashi is chiefly a commentator trying to help us learn and grasp the logic and arguments of the Gemara as we progress in our study of the Gemara.
3) Now, we have Tosfos saying that Rashi was not concerned to give accurate figures, and the Tosfos ha'Rosh saying that Rashi writes that the breaches were 5 and 5 as a Siman that the total breach was 10, even though it is clear that it was only slightly more than 7. I suggest that the same logic is working here. Rashi, in his commentary on the Talmud, is not always most concerned with giving us precise mathematics, even though he certainly knows what the precise math is, but he was trying to help us understand the flow of the Gemara as we are learning the page.
4) I suggest that with the above approach we can resolve at least one difficulty on Rashi on Shas. In Berachos 2b, Rashi (DH Bein) writes that, according to Rebbi Yehudah, the Bein ha'Shemashos twilight period lasted half a "Mil" (9 minutes before the stars emerge, according to most opinions). Rebbi Akiva Eiger there in Gilyon ha'Shas asks that in Shabbos 34b we learn that it is either two-thirds of a Mil or three quarters of a Mil.
I suggest that, according to Tosfos in Eruvin 94b, we may assert that the role of Rashi is not necessarily to provide us with accurate figures when this is not the primary topic of the Sugya at hand, but rather he suffices to give us an approximate time length for Bein ha'Shemashos according to Rebbi Yehudah, since this is adequate for the understanding of the Sugya in Berachos 2b.
Kol Tuv,
Dovid Bloom
Thank you so much for getting back to me. I really appreciate your taking time out to answer my questions.
However, I now have a couple of follow-up questions.
1. You had mentioned the Yad Dovid, who said that the issue in the sugya in Shabbos was not that we required the three tefachim of yenikah but rather a hefresh. According to that logic, why is this any worse than Rosh Tor? If one field that juts out into another field isn't an issue of klayim, why is there an issue when the two fields are separate from one another?
2. In respect to the Sfas Emes on Eiruvin 5a, when I learned the sugya it had seemed that it was an issue of courtyards opening onto the mavoi, not an issue of pirtzos. Why would the patzimim be broken at all, if nothing was broken in the first place?
3. According to Rabbi Kornfeld's comment in Sukkah, why would the Dayanei D'Keisari feel that a halachic stringency is needed by Halachos D'Rabanan? Why wouldn't we apply the rule of safeik d'Rabbanan likula and say that we would use a rough estimate of the diagonal when there's a doubt how to measure it or whether one should use the sum of the sides or the length of the diagonal? Furthermore, the Dayanei D'Keisari are brought up in respect to Sukkah as well, and Rashi clearly states that they would hold in respect to R' Yochanan's Sukkah that there would be a circumference of 24 and not 16.8. That indicates that they would hold of their stringency not only in sefeikos d'Rabanan, but also sefeikos d'Oraisa, whereas Rabbi Kornfeld said that their ruling was only shayich in respect to d'Rabanan's. I had made reference to this in my original question ("I had seen on your site (insights to Eruvin 76) that Rashi was going with a halachic stringency..."); I guess I didn't explain myself clearly enough.
4. I thought I had mentioned this in my original question, but I seem to have neglected to mention it. Rashi makes a similar comment on Eiruvin 78a, in respect to being me'areiv two chatzeiros by leaning a ladder on either side of the 10-tefach-tall dividing wall. Rav Yehuda says in the name of Shmuel that it needs to be 14 tefachim, Rav Yosef requires 13 tefachim, Abaye holds that it's 11 tefachim, and we pasken like Rav Huna berei d'Rav Yehoshua's opinion of 7 tefachim. In respect to the first three opinions, Rashi says that the ladders' bases are four tefachim from the wall - 10+4=14, 9+4=13, and 7+4=11. Tosafos argues, based on "kol amsa b'ribua, amsa u'trei chumshi b'alachsona," that the base of the ladder is however far from the wall a distance equal to however high up the wall it is. Rashi, therefore, seems to be going l'shitaso - according to his mistaken math.
Thank you so much. Have a good Shabbos,
-Daniel
Daniel, thank you for response.
1) On the contrary, it seems to me that what the Yad David says is quite similar to the idea behind Rosh Tor. (By the way, I found that the Sefer Chefetz Hashem on Maseches Shabbos, by the Or ha'Chayim ha'Kadosh, also writes something similar to the Yad David to explain Rashi's Shitah.) The Tosfos Yom Tov in Kil'ayim 2:7 cites the Kesef Mishnah who says that the Torah's prohibiton of Kil'ayim applies only when one actually throws the seeds of different species together on the field. All other forms of Kil'ayim are prohibited mid'Rabanan because it looks as if the seeds have been sown simultaneously. Therefore, Rosh Tor is permitted because one can see that the edge of the wheat field juts into the barley field but it is clear that they are really two separate fields.
Similar thinking lies behind Rashi's opinion on 85a. The regular Shi'ur of 3 full Tefachim is not necessary because 1 Tefach in one direction and 2 Tefachim in the other direction is a sufficient distance so that people should not think that the seeds were thrown together. We cannot be quite as lenient as Rosh Tor (where the wheat and barley are actually together) because in the latter Din one sees two distinct fields, one of wheat and the other of barley, so they clearly have not been sown together. However, less than 3 full Tefachim is enough in Shabbos 85a because this is sufficient space between the two species.
2) I would like to suggest that the rule of Safek d'Rabanan l'Kula is not relevant here. I am basing this on the Poskim in Shulchan Aruch YD 98:3. There, the Shulchan Aruch writes that if a forbidden food became mixed with a permitted food and the mixture is in front of us but we cannot discern whether there is 60 times more permitted than forbidden food in the mixture, the law is that it is forbidden even if the forbidden food is only prohibited mid'Rabanan. The Shach (98:9) writes that the reason is that even though there is a doubt, this is a doubt that arises from "Chisaron Yediyah" -- our knowledge is lacking, so this is not considered a genuine Safek. The Shach adds that this is "Da'as Shotim" -- this is the understanding of foolish people, because if we would be wiser we would have no doubt about the question.
In the same way it is possible to know what the hypotenuse of a right-angle triangle really is. If we happen not to know, this is not considered a Safek and, therefore, even for an Isur d'Rabanan we would have to be stringent and say Safek l'Chumra. B'Ezer Hashem, I will attempt later to answer the other questions.
3) Now to Eruvin 5a: My apologies for my mistake about Pirtzos, as you are correct in saying that nothing was broken. I will attempt again to explain how the Sefas Emes defends Rashi from the question of Tosfos.
The Sefas Emes explains that in fact there is no dispute between Rashi and Tosfos. Rashi does not mean to say that the opening to the "Mavuy" from the courtyard is composed only of 1 Tefach on the middle wall and 3 Tefachim on the side wall. Rather, one also has to take into account the Patzimim (which might make what is missing in the side wall into 3 1/2 Tefachim and what is missing in the middle wall into almost 2 Tefachim, as the Rashba writes). All Rashi is saying is that it is only 1 Tefach in the middle wall and 3 Tefachim from the side wall which is totally open but the Patzimim also occupy space in these walls. Therefore, with a combination of the open space and the Patzimim one will reach a diagonal of 4 Tefachim which is more than what one would reach if it was only 1 and 3 Tefachim.
Again, the understanding behind this is that the aim of Rashi in his commentary is not chiefly to stress the mathematical accuracy. What is more important is to give us a general idea of how the opening to the Mavuy is made up of an open part in the side wall and in the middle wall, but Rashi did not go into exactly how much the Patzimim contribute to the opening.
4) Now to Eruvin 78a: Again, the Sefas Emes comes to the defense of Rashi, and suggests that Rashi does not mean that one pulls the legs of the ladder 4 Tefachim away from the wall. Rather, when Rashi refers to 4, he means the extra 4 Tefachim that Rav Yehudah in the name of Shmuel says that the ladder needs. In other words, in theory a ladder 10 Tefachim high should be sufficient to climb up a wall that is 10 Tefachim high. However, Shmuel says that this is a very uncomfortable way of climbing up the wall, and therefore
maintains that one should add 4 Tefachim to the theoretical minimum size of the ladder. Now Rashi explains what one does with these extra 4 Tefachim. He tells us that the aim of these extra 4 Tefachim is so that one can pull the ladder away from the wall so that it will not be a steep climb. Rashi does not tell us how far away one needs to pull the legs of the ladder from the wall, but in fact he will agree with Tosfos that one withdraws the ladder 10 Tefachim away from the wall (or, to be more accurate, the square root of 96).
G'mar Chasimah Tovah,
Dovid Bloom
Thank you so much.
One last question: how do you read the Sfas Emes into Rashi in Eiruvin 78a? He says that Shmuel requires a 14-tefach ladder "because we need to drag the legs of the ladder 4 from the wall, because a vertical ladder is not easy to climb." Likewise, for R' Yosef he explains that "when you drag the legs from the wall 4 [tefachim], its head will not be separated from the top of the wall [more than a] tefach..." and by Abaye he says that "when you move the feet four [tefachim from the wall], the top of the ladder will be less than 3 tefachim from the top, and it will be considered closed." He seems to imply that you need to move it four from the wall.
Gemar Chasimah Tovah,
-Daniel
Yes, Daniel, I see what you mean that the Sefas Emes does not seem to fit into what Rashi writes in order to explain the opinion of Rav Yosef and Abaye.
1) Today I thought of a different way of explaining Rashi. The beginning of my idea came from the Keren Orah here who cites the Talmud Yerushalmi, in this chapter, near the end of Halachah Alef. The Keren Orah is very brief and does not explain how the Yerushalmi can help us answer the question of Tosfos on Rashi but I will try to explain a little more.
2) The Yerushalmi states that Rebbi Yasa said in the name of Rebbi Bun bar Kahana that it must be drawn 4 Tefachim away from the wall "Kedei Makom." The Korban ha'Edah writes that this means that there must be space by which to go up the ladder, because a vertical ladder is not comfortable to ascend.
3) The Yerushalmi certainly seems to fit better with the simple reading of Rashi's explanation of Shmuel's opinion (without the Sefas Emes) that the ladder must be drawn 4 Tefachim (not 10 Tefachim as per Tosfos) away from the wall. In addition, we notice that Rashi seems to understand that Rav Yosef and Abaye also conform with the Yerushalmi because he writes that they both also require that the ladder be drawn 4 Tefachim away from the wall. However, we now have to answer the question of Tosfos on Rashi.
4) I want to suggest that the numbers 14, 13.01, and 11.01 mentioned in the Gemara do not actually refer to the length of the ladder in the conventional understanding. Instead, what they mean is the sum of the horizontal and vertical sides of the triangle of which the ladder itself represents the hypotenuse. Since an essential feature of a ladder used to combine two courtyards is that it must be comfortable to go up, this means that the 4 Tefachim which the ladder is distanced from the wall is considered part of the measurements of the ladder itself.
5) Accordingly, when Shmuel says 14, this means 4 + 10, and when Rav Yosef says 13.01, this means 9.01 + 4, and Abaye's 11.01 is 7.01 + 4, with 4 always being a constant, i.e. the distance on the floor of the ladder from the wall.
6) It emerges that Rashi is very different from Tosfos since the numbers mentioned in the Gemara do not refer to a hypotenuse but rather to the sum of the other two sides.
Kol Tuv,
Dovid Bloom
Here is a very simple comment from one of the Rishonim to explain Rashi. Tosfos Rabeinu Peretz (78a) goes in the footsteps of Tosfos and writes that Rashi was inaccurate, because if one distances the ladder 10 Tefachim from the wall, then the 14-Tefach ladder will also reach the top of the wall. Therefore, he writes that it seems "d'l'Simana b'Alma Naktei" -- Rashi wrote that the ladder should be kept 4 Tefachim away from the wall, not because this is an accurate figure, because in fact it should be kept 10 Tefachim away from the wall, but rather it is a convenient "sign" because if one says 4 Tefachim this is a slightly simpler way of fitting in with the fact that the other measurements are 14 and 10.
This is similar to what we saw above in a previous reply concerning what the Tosfos Rosh (94b) writes about what Rashi seems to say that if the two right-angled sides are both 5 then the hypotenuse is 10. The Tosfos Rosh writes that Rashi clearly did not mean this literally but "l'Simana b'Alma Naktei" -- Rashi was just giving us a "sign" of easily digestable numbers even though it is clear to everyone that the geometry is not meant to be accurate.
Kol Tuv,
Dovid Bloom