On Daf 58, the sugya of measuring the techum:
(a) The Rashba asks on Rashi's definition of Chut ha'Mashklokes Yored Kenegdo that even 4 amos in 50 amos, more so in 2000 amos, would be very difficult to walk. I'd go further, and say it's absolutely impossible, a sheer cliff. How is this going to work?
(b) Is Makdir just another way of measuring horizontal distance? Rashi's description of the meaning of the word sounds that way, as if one makes a tunnel through the mountain and then measures along it. But, Rashi twice says that Makdir saves "k'Chatzi Komas Ish" for every 4 amos. Not sure how much that is, but if a man is three amos tall, that's 9 tefachim. Could be he's only being approximate. But if one uses the Lishna Kama d'Rava, one does Kidur for Mislaket 10 Mitoch 4 Amos and for anything steeper; either all the way to Yored Chut ha'Mashkoles, or (according to Tosefos about Rashi) until an indeterminate stage called Lo Neicha Tashmishtei.
Misklaket 10 Mitoch 4 Amos is already a little too steep; measuring along the slope 4 amos yields a slope that would have a horizontal 24 tefachim for a rise of 11 tefachim. You'd lose 2 tefachim each time. More so if it's steeper.
According to the Lishna Basra d'Rava (and we hold that way), it says that you do Kidur for Misklaket 10 Mitoch 5 Amos and not for Misklaket 10 Mitoch 4 Amos. I'd guess that 4 amos is actually the boundary, and five just means anything more than four. Five is actually not too bad; you get a horizontal 24 tefachim for a rise of about 8.5 tefachim.
Rashi also says that for a deep valley Yeser me'Alpayim, you can't do Havla'ah but only Kidur. I can't understand why that would be if Kidur is just a way to do horizontal measurement when Havla'ah is impossible. It sounds here as if it's preferable.
On the next mishnah, Tos. quotes Rach that Mumcheh means 'straight'; you must go straight out from the city and not go to the side, even if it's smoother. Tosefos gives the reason (Maharsha says that the Rosh's girsa is clearer) because there might be hills or valleys next to the city so one might need to be Makdir or Medidah Yafah. I understand the second one, since it involves either measuring along slopes, or skipping them - it makes a difference. But again, why would one prefer to be Makdir if it's just an approximation of horizontal measurement: go to the side and measure horizontally!
So from all this, I'm wondering if Kidur is more than just a way to do horizontal measurement; (to some extent) it actually takes the slope into account; just not as much as Medidah Yafah.
(c) Tosefos and the Shulchan Aruch talk about using long poles to do Havla'ah on a mountain. It could be pretty high, especially if you're going to allow more than Misklaket 10 Mitoch 4 Amos. Given that one must stretch the rope b'Chol Kocho, it sounds very difficult, but I guess possible. If Misklaket 10 Mitoch 4 Amos is the limit, I figure they won't be more than 11.4 amos high if the slope down is the same. Twice that if one side is a sheer cliff.
On ground Misklaket 10 Mitoch 6 Amos, the Mishnah B'rurah brings from the Rashba that one measures Medidah Yafah straight along the ground without Havla'ah. The Rashba actually says, anything more than 5, that is, any slope more gentle than that. How is that actually done? How long a rope do we use? Obviously, there's no limit to how long the route could be if one measured every tiny bump. Rashi's language on 35b: "v'Lo Moded Kol Madrono Kemos she'Hu..." -- and he does not measure all of its slope as is, with a rope of 50 on smooth ground, because that would be an excessive Chumra" -- sounded like maybe one still uses a 50 amos rope, only go up and down the hills and valleys with it. Not sure how that would work; sometimes you'll end up skipping a valley entirely.
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(d) On the same subject, we're supposed to do exactly that for Eglah Arufah and Ir Miklat and (I guess) for the Techum d'Oraisa of 12 mil. Eglah Arufah I think I understand: You measure from somewhere on the meis (see the gemara in Sotah) to the nearest corner of the city wall or its Ibur or its Techum or such. It's a single measurement. But what about an Ir Miklat? Are we trying to create a rectangular techum? If so, it depends very much where you start. And if not, each starting point will yield a different distance, depending on the bumps in the way. I would think you would have to measure over and over, starting from every spot on the boundary of the city.
(e) Chevel Pishtan - learned from a posuk. How do we do it today, Google Maps? Won't that yield a different result? Do you cut 50 amos pieces off a loose rope, then pull it tight? Or is the 50 amos cutting done on a tight rope as well? If so, why would it matter what type of rope or its length: you've taken the stretch into account already.
(f) Keneged Libo. The Ritva says Lav Dafka, it just means as opposed to one high and one low. Can I assume, though, that they are trying to keep the rope at the same height? For instance, what if one of them is tall and the other short?
(g) Presumably they needed to have a way to make sure they are keeping to a straight line, like marking the path as they go. They also need a way to make perfect right angles, when "Holech l'Makom she'Yachol l'Havli'o u'Mavli'o, v'Tzofeh v'Chozer l'Midaso.
Do we know how they did these things?
Thanks much, Michoel Reach, Baltimore USA
(a) Tosfos (DH v'Chi) also raises this question, but still stays with Rashi's explanation. The Rosh (Keitsad Me'avrin 6) explains that once the horizontal distance is four Amos, which is considered a "Davar Chashuv," the Chachamim required that this be included in the Techum. In other words, we cannot ignore this part of the Techum when it is of a significant size. It is true that it is very difficult to measure the slope, but nonetheless it cannot be ignored.
(b)
1. Kidur is certainly supposed to measure the horizontal distance as accurately as possible over a slope. However, it is less accurate than using a rope of fifty Amos.
2. I believe that Rashi means that if one person holds the rope at his toes and the other at half his height, then they avoid including in the measurement of the Techum an elevation equivalent to half the height of a man. This is because the hypotenuse represents both the horizontal and the vertical distance. Although this is obviously not mathematically correct -- as the length of the hypotenuse is not the sum of the other two sides, nevertheless Rashi uses this formula as a rough estimate. See also Eruvin 5a and Rashi (DH b'Keren) and Tosfos (DH d'Pasach) there. The amount saved will obviously also depend on the gradient of the slope, as you say, but Rashi refers to the case where one holds the rope level to his heart and one holds the rope level to his legs.
3. In the case of a deep valley, the Ritva (DH Amar Rav Yosef) explains that we can only be Mavli'a when the area we are Mavli'a is not more than the total Techum Shabbos. Otherwise we end up with the Tafel more than the Ikar Techum.
4. In the next Mishnah, the reason why Rabeinu Chananel insists that we measure opposite the city is that this is the place that Chazal fixed for measuring, and since discrepancies may occur through different methods of measuring the true measure is the one arrived at by measuring opposite the city. This is despite the fact that a more accurate measure of 2000 Amos may be achieved by measuring elsewhere. The true Techum is according to the measure in the place Chazal fixed and according to the methods suitable in that place.
(c)
1. I am not sure I understand all of your points here. You are correct to say that it could be very high, and I presume that when the mountain is so high as to make Havla'ah impossible, we are unable to do Havla'ah.
2. However, I do not understand why you say that the height depends on the rule of Mislaket Arba Amos Mitoch Asarah. This is just the measure of the gradient and does not relate to the height.
3. I am unclear how you arrived at a figure of 11.4 Amos. What do you mean that if one side is a sheer cliff then the height is double?
4. The Rashba (58a, DH Higi'a l'Kosel) indeed says that an elevation of ten Tefachim in every six Amos should be measured as if it were flat ground; in other words, here we have to measure the slope and not only the horizontal distance. The Rashba also says that with an elevation of ten Tefachim in four Amos (or steeper), we are satisfied with an estimation, and ten Tefachim in five Amos needs Kidur.
5. The measure of an elevation of ten Tefachim in six Amos would therefore be with a rope of fifty Amos, as every measure of flat ground. I did not understand what you mean about skipping a valley. Rashi on 35b says that for steeper inclines we do not need to measure the slope. Please clarify these points and I will try and help.
(d)
1. According to the Poskim who hold twelve Mil is a Techum d'Oraisa (see Mishnah Berurah 404:7), we would not do Kidur or Havla'ah as these are leniencies which are used only in d'Rabanan laws. This is the reason for the fact that we do not do Kidur for Eglah Arufah and Arei Miklat.
2. The Sefas Emes (56b, DH b'Gemara) says that the Rambam holds that the Techum of Arei ha'Leviyim is round. You are correct that we would need numerous measurements of the Techum.
(e)
1. The Me'iri in his Chidushim (not to be confused with his Sefer Beis ha'Bechirah) on the Mishnah (57b) explains that even though the Gemara (42a) says that we can measure the Techum by counting 2000 average paces, this is not the ideal method and only permissible b'Sha'as ha'Dechak. See also the Bi'ur Halachah (399:1, DH Ein). If we could measure accurately by some other method, this may also be acceptable, but maps will not be as accurate as ropes, and furthermore the true measure of the Techum takes into account the discrepancies and vagaries of measuring with ropes as pointed out above (b-4). Some Rabanim do use maps nowadays, but they leave themselves enough leeway to ensure that even if their estimate is inaccurate no Michshol will arise.
2. The Ga'on Yakov says that the rope is measured and cut when not taut.
(f) The Chidushei ha'Me'iri says that even if one is small and the other tall, they both place the rope against their hearts. He gives two reasons for this. First, Chazal took this into account, and even if discrepancy occurs, this is still considered to be an accurate measure. Second, the discrepancy when measuring fifty Amos will be minute.
(g) Certainly this is an important part. I do not know how they did this, but I presume that they had ways of ascertaining that they were traveling in a straight line, presumably by making use of the sun and landmarks.
I hope this is of help. If you have any comments please let me know.
Dov Freedman
Thank you so much for your answer; there was a lot here. I have a few questions on some of the answers, and am including some drawings to illustrate some of them.
(a) I am not asking on the halacha, but I'm trying to understand practically how this works. Since the time of the gemara, many shteiblach must have been measuring their techumin. This is halacha l'ma'aseh according to the Shulkhan Arukh 398: any valley that is too wide for hav'la'ah and which has four amos total horizontal in the descent, even if the descent is very steep indeed, uses kidur to measure the slope. Do we have teshuvos about such towns which discuss how practically to scale those slopes, or whether we sometimes consider it unreasonable or dangerous?
(b) 3. I am not sure I understand the Ritva's point here. If kidur is just a not-quite-as-accurate way of doing hav'la'ah when hav'la'ah is impractical, how would it help to do it here? The Ritva makes it sound as though there's a ma'alah to kidur: It doesn't make the slope "tafel".
4. I am not sure how this question is answered. Of course the measure set by Chazal is the one we want. The question is do we understand why did they set that one? I guess you are answering that there is a requirement to use "the city's" measurement where possible, rather than a measure from "the techum of the city".
(c) For this I am including scanned pictures, as I doubt my powers of description. Here the maximum width of the mountain is 50 amos: we never do hav'la'ah farther than that. If the maximum slope of the mountain is Mislaket 10 Mitoch 4 Amos, and we assume it's symmetrical - slope up, then slope back down - that will give a maximum height of the mountain and thus of the poles. (Picture #1)
And if we don't insist on the hill coming back down, we could just make it a single slope of the same Mislaket 10 Mitoch 4 Amos, ending up higher than it started (Picture #2). In that case, the maximum height would be just doubled. Since that maximum is around 23 amos, say thirty-some feet, I guess it is possible to be mavlia with poles. As I said, it sounds difficult to stretch the rope very tight, and again I'm wondering if anyone has discussed the difficulties.
As for what I said about skipping a valley when doing medidah yafah, I can try to describe it for the Rashba's Mislaket 10 Mitoch 6 Amos, but it is much easier for me to construct interesting impacts if I describe it for the d'oraisa case, say of Ir Miklat. I am assuming: that we still use the rope of 50 amos, as discussed, and that it still must be pulled tight, and that we do that (in the d'oraisa case) even for steep inclines.
First imagine that we are measuring along a sequence of valleys (or hills) that are each 50 amos deep along the slope (Picture #3). If the first valley begins exactly 50 amos from where we start measuring, each rope will reach perfectly to the bottom of its valley, and the next to the top of the next. After two valleys, we will have measured 250 amos (50 before we got to the valleys), of which 200 amos was along the slopes.
Now say they add a house to the Ir Miklat, so we now start measuring about 10 amos closer (also shown in Picture #3). We will have a starting point for the second rope about 10 amos down into the first valley (pt. A). It doesn't reach the bottom, and we are required to stretch it tight (aren't we?) What choice will we have but to find the nearest point on the ground (which as I've drawn it is in the _third_ valley - pt. C) which is 50 amos away from the starting point? And even for that I'm going to need an unusual brand of hav'la'ah, setting up poles in the first and third valleys so that I can connect them - at a slant - with the rope at the appropriate angle.
This time, after two valleys we are not even up to 100 amos. Just adding one house added more than 150 amos to the techum.
[The truth is that the house doesn't need to be 10 amos closer, a little more than one amah is actually enough due to measuring along the diagonal from the house to pt. A...]
Now if you'll tell me that on a techum d'oraisa I don't _need_ to stretch the rope tight, but can follow the curves of the hills and valleys wherever they go, then I won't have this oddity. I would measure 40 amos down that first valley, and 10 amos back up - with one rope, and so forth. Perhaps stretching the rope tight is only done for d'rabanans, to make the techum a little more lenient, but for d'oraisas you leave the length of the rope as it is?
Or, you could tell me that we still stretch the rope tight, but we are allowed to give it bends. You'd stretch it tight down 40 amos, holding it at the 40 amah point on the rope (at the bottom of the valley), then bend it back up the other side of the valley, again stretching it tight for the last ten amos?
But the truth is that both of these suggestions seem to defeat the point of using a 50 amah rope as described by the gemara - that it gives just the right amount of stretch.
[If we are dealing with a techum d'rabanan and the Rashba's Mislaket 10 Mitoch 6 Amos, the effect will be much more subtle. But imagine moderate bends in the slope that are less than 50 amos long; since the rope must be tight it won't be able to see them easily, and the exact result will depend on whether the rope begins and ends within a feature, or jumps right over it. But I don't think it will make major differences.]
(d) I'm wondering if it's really enough to measure straight out from the Ir Miklat. If there is a valley in front of part of the east side of the city, say the northern half, and we have to measure down its slope and back up, we might reach 2000 amos on the northern half of the east techum long before we reach 2000 amos from the southern half (Picture #4. [I don't think the S'fas Emes's round Ir Miklat will make a difference to this point, and it was easier for me to draw it square.]) Do we then say that the techum of the northern half has been reached, even though the rotzeach can get back to town by walking much less than 2000 amos - that is, by going a few steps sideways to the southern half and from there into town? He is, after all, within 2000 amos of the city. [I know that the Rach said on daf 58b that we measure straight, but as you mentioned before, Tosefos says it just means that you start from the side and not the side.]
Here too I wonder if this has ever been discussed practically. There seem to be some nafka minos from techum d'oraisa; at least the poskim mention it every so often, like in cases of safek. I wonder if towns ever tried to measure it; it sounds very difficult.
Thanks very much for your time and attention,
Michoel Reach