Yerushalmi Eruvin 5:3 ירושלמי עירובין ה׳:ג׳

MISHNAH: One only measures with a rope of 50 cubits, neither less nor more. One should only measure holding up to his heart⁷The measure should be made strictly horizontal. The two people holding the measuring tape hold it on their breasts, far from the ground, to hold it as exactly horizontal as possible.. If he was measuring and came to a valley or a wall he circumvents it and returns to his measurement, on condition that he not exit the Sabbath domain⁸If one measures, e. g., N to S, and comes to an obstacle, one should go exactly E or W until the obstacle disappears, resume measuring due S, and then return on an exactly EW path to the original line and resume measuring.. If he cannot circumvent it, about this said Rebbi Dositheos ben Yannai in the name of Rebbi Meïr: I heard that one strip-measures⁹One measures with a short rope of 4 cubits held horizontally. in the mountains¹⁰As explained in the Halakhah, one measures with a rope of 4 cubits which can be held horizontal also on steep inclines. The 2’000 cubits which one determines are not actual distances but the length of the projection of a path onto the plane locally representing the surface of a flat Earth..

HALAKHAH: Paragraph. “One only measures with a rope of 50 cubits,” etc. How much is the measure of the Sabbath domain? 40 ropes. “Neither less” since it can be tightened and increases, “nor more” since it gets loose and loses¹¹⁶Babli 58 a.. Some Tannaim state, one measures with a linen rope, but some Tannaim state, with a chain rope¹¹⁷A metal chain., and some Tannaim state, with a rope of four cubits. ¹¹⁶Babli 58 a. Rebbi Joshua said, there is no truer measure than with chains, but what can I do since in Prophets it is written ropes, a linen twine was in his hand and a measuring rod¹¹⁸Ez. 40:3..

If from him to a river were 75 cubits. Two Amoraim, one said, he measures with a rope of 50 cubits and returns 25 cubits; but the other one said, he measures with a rope of 50 cubits and the remainder he measures with a rope of four cubits. If the river was narrow on top and wide below, up to 50 cubits one considers it as if full of dust and pebbles; otherwise one sees it inclining upwards and inclining downwards¹¹⁹If a canyon is up to 50 cubits wide on top one simply measures on top and disregards what is below; otherwise one has to measure the slopes.. If the river was bent, Rebbi Ḥasdai says, with telescopes he estimates in the plain and does the same with a mountain¹²⁰If one has a secure base of given length it is more accurate anyhow to determine lengths by measuring angles with a theodolite.. Rebbi Aḥa, Rebbi Ḥinena, Rebbi Jeremiah in the name of Rebbi Samuel bar Rav Isaac: only if the measure results in the mountain four hand-breadths as a place¹²¹In mountainous terrain one measures with ropes of 4 cubits length. It is considered an incline which needs special treatment only if the difference in height for the length of the rope is 4 hand-breadths, ⅔ of a cubit. Let α be the angle of inclination, this means sin α = .167 or α = 4.5°. Since cos α = .958, the error committed by measuring the slope instead of its horizontal projection is 1.42%.. Rebbi Yose, Rebbi Abuna in the name of Rav Jehudah, Rebbi Yudan has it in the name of Rav: If one holds it high above his head and the other by his feet, everybody agrees that one strip-measures¹²²While the Sages state in the Mishnah that one does not strip-measure at mountains, i. e., that one measures the distance on the ground, not the horizontal projection, they agree that if the slope is very steep the error would be too large and one must work with projections. To translate the rules given here into numbers would require exact data about the length of a cubit and the average height of a man; these are not available. For a rough estimate one may assume that the average height of a man be 3.5 cubits. Then if in a team of geometers one holds his end of the rope of 4 cubits on top of his head while the other, climbing up the mountain, has to hold his end at his feet in order to keep the rope horizontal, the slope is given by tan α = 3.5/4 = 0.875 or α = 41^°11’. In this case the distance measured on the slope would be 5⅓ cubits; measuring on the slope instead of with an horizontal rope would decrease the Sabbath distance by more than a third. In this case the Sages have to agree with R. Meïr.. The high one over his head and the low one on his hips, everybody agrees that one strip-measures¹²³Probably that means tan α = .25, α ~14°, the error in measurement is ~ 5%.. Where do they disagree? On his heart; Rebbi Meïr says one strip-measures but the Sages say one does not strip-measure¹²⁴If the difference is only half a cubit, tan α = .125, α ~ 7°; the error in measurement is ~ 1.2%, which is not worth the bother; the position of the Sages is justified..

¹²⁵This text is from Soṭah 5:5, explained in detail in Notes 120–123 and Makkot 2:7 (Notes 141–142). Rebbi Abba in the name of Rebbi Jehudah, Rebbi Zeˋira in the name of Mar Uqba: One strip-measures only with a rope of 50 cubits. Rebbi Zeˋira in the name of Rav Ḥasdai: One strip-measures (only in mountains but)¹²⁶With the other two sources read: :Not for Levitic cities. not for the place of breaking the calf’s neck. This would be acceptable for him who says, 1000 cubits of open space and 2000 cubits of Sabbath domain. But for him who says, 1000 cubits of open space and 2000 cubits of fields and vineyards, did they not learn the Sabbath domain from the Levitic cities? For the main thing one does not strip-measure; does one strip-measure for the derivative?

¹²⁷Num. 35:3. From where that one does not bury in Levitic cities? Rebbi Abbahu in the name of Rebbi Yose bar Ḥanina: And their open spaces shall be for their animals … and all their lives¹²⁸This paragraph is copied here only as an appendix to the preceding; it does not belong here.. They were given for living; they were not given for burials.