Jerusalem Talmud Sotah 5:3

משנה: בּוֹ בַיּוֹם דָּרַשׁ רִבִּי עֲקִיבָה וּמַדּוֹתֶם מִחוּץ לָעִיר אֶת פְּאַת קֵדְמָה אַלְפַּיִם בָּאַמָּה. וּמִקְרָא אַחֵר הוּא אָמַר מִקִּיר הָעִיר וָחוּצָה אֶלֶף אַמָּה סָבִיב. אֵיפְשָׁר לוֹמַר אֶלֶף אַמָּה שֶׁכְּבָר נֶאֱמַר אַלְפַּיִים אַמָּה וְאֵיפְשָׁר לוֹמַר אַלְפַּיִם אַמָּה שֶׁכְּבָר נֶאֱמַר אֶלֶף אַמָּה. וְלָמָּה נֶאֱמַר אֶלֶף וְלָמָּה נֶאֱמַר אַלְפַּיִם. אֶלָּא אֶלֶף אַמָּה מִגְרָשׁ וְאַלְפַּיִם אַמָּה תְּחוּם שַׁבָּת. רִבִּי אֱלִיעֶזֶר בְּנוֹ שֶׁל רִבִּי יוֹסֵי הַגָּלִילִי אוֹמֵר אֶלֶף אַמָּה מִגְרָשׁ וְאַלְפַּיִים שָׂדוֹת וּכְרָמִים. MISHNAH: On the same day³³According to the Babli (Berakhot 28a), any Mishnah which starts “on the same day” was formulated on the day Rabban Gamliel was deposed. There is no proof that this ever was a Galilean tradition., Rebbi Aqiba explained: ¹⁰⁷Num. 35:5, speaking of the cities to be given to the Levites.“You shall measure outside the city 2000 cubits to the East,” but in another verse it says ¹⁰⁸Num. 35:4.“outside the city wall, one thousand cubits all around.” It is impossible to say 1000 cubits because already it was said 2000 cubits and it is impossible to say 2000 cubits because already it was said 1000 cubits. Why was 1000 said, and why 2000? But 1000 cubits open space¹⁰⁹Not for living and not for agriculture. and 2000 cubits Sabbath domain¹¹⁰To say that not only for levitic cities but for all cities a belt 2000 cubits wide belongs to the city. Therefore, when it says “Nobody shall leave his place of the Seventh Day” (Ex. 16:29), it is implied that on the Sabbath a city dweller may walk in a belt 2000 cubits wide around the city.. Rebbi Eliezer, the son of Rebbi Yose the Galilean, said: 1000 cubits open space and two thousand fields and vineyards¹¹¹He holds that the determination of the Sabbath domain as 2000 cubits outside the city wall is rabbinic, but that the belt 2000 cubits wide was given to the Levites and that the outer 1000 cubits were used for agriculture. {Sadducee practice limited walking on the Sabbath to 1000 cubits outside the city, agreeing with R. Eliezer; CD X:21.}.

הלכה: בּוֹ בַיּוֹם דָּרַשׁ רִבִּי עֲקִיבָה וּמַדּוֹתֶם מִחוּץ לָעִיר וגו׳. רִבִּי אֱלִיעֶזֶר בְּנוֹ שֶׁל רִבִּי יוֹסֵי הַגָּלִילִי אוֹמֵר. אַלְפַּיִים אַמָּה תְּחוּם עָרֵי הַלְּוִיִּם. צֵא מֵהֶן אֶלֶף אַמָּה מִגְרָשׁ. נִמְצָא רְבִיעַ מִגְרָשׁ וְהַשְּׁאָר שָׂדוֹת. רִבִּי יִרְמְיָה רִבִּי שְׁמוּאֵל בַּר רַב יִצְחָק בְּשֵׁם רַב. רְבִיעַ מֵאֶלֶף. אָמַר רִבִּי יִצְחָק. וַאֲפִילוּ תֵימַר. רְבִיעַ מִכָּל־צַד. רְבִיעַ הוּא. מָאן דִּמְרַבַּע אַרְבַּע גַּרְבִּין בָּעֵי אֲשִׁיתְתֵּיסָר. רִבִּי מָנָא מְשָׁעֵר כְּהָדָא לְבֵנָה. רִבִּי אָבִין מְשָׁעֵר כְּהָדָא רְצוּעָה. רִבִּי אוֹשַׁעְיָה מְשָׁעֵר כְּהָדָא דִיסְקָרִיָן. אָמַר רִבִּי יוֹסֵי בֵּירַבִּי בּוּן. חַמְשִׁין עַל חַמְשִׁין בֵּית סְאָה. מֵאָת עַל מֵאָת בֵּית אַרְבַּעַת סְאִין. כְּהָדָא רֵישׁ גָּלוּתָא אִיטְלַק עִילּוֹי חַד טְרִיקְלִין אַרְבָּעִין עַל אַרְבָּעִין דִּימַלִּינֵיהּ חִיטִּין. אָתָא לְגַבֵּי רַב הוּנָא. אָמַר לֵיהּ. פַּייְסוֹן דְּיִסְבּוֹן מִינָּךְ עֶשְׂרִין עַל עֶשְׂרִין כְּדוֹן. וְעֶשְׂרִין עַל עֶשְׂרִין בָּתָר זְמָן. וְאַתְּ מִיתֲגָר פַּלְגָּא. HALAKHAH: “On the same day, Rebbi Aqiba explained: “You shall measure outside the city”, etc. Rebbi Eliezer, the son of Rebbi Yose the Galilean, said, 2000 cubits the domain of the levitic cities. Deduct from them 1000 cubits open space, there results one quarter open space and the remainder fields¹¹²This is difficult to understand. A little geometry is needed here (cf. H. Guggenheimer, Applicable Geometry, Huntington NY 1977.) Given an oval (a plane convex figure) of area A and perimeter L, the area of the parallel figure in distance h, obtained by adding to the figure every exterior point of distance ≤ h is
A(h)= A+hL+ πh².
The excess area of the parallel figure over the original oval is
E(h) = hL + πh².
However, if the city has rectangular shape, in tractate Erubin the Sabbath domain is determined as the rectangle whose sides are 2000 cubits from the (straight) walls of the city. In that case, the rectangle defined by lines in distance h has surface area
A(h) = A + hL + 4h²
with excess E(h) = hL + 4h². Let the unit of h be 1000 cubits. In both cases, the condition E(2) = 4E(1) yields 2L = 0. It follows that L = 0: the original oval was a point. The open space is one quarter of the area of the fields only if there is no town; it is a limit case. The parallel text of the entire Halakhah is in Erubin 5, fol. 22d; Makkot 2:7, fol. 32a.
The Babli, Eruvin 56b, explains the problem away by replacing the condition E(2) = 4E(1) by A(2) = 4E(1), computing A(2) by the second formula and E(1) by the first, and assuming π = 3; clearly recognizing the impossibility of the original statement.. Rebbi Jeremiah, Rebbi Samuel bar Rav Isaac in the name of Rav: A quarter of 1000¹¹³This is a radical re-interpretation of the verse: The open space is only 250 cubits wide on each side of the town. Then the problems disappear.. Rebbi Isaac said, even if you say a quarter from every side, it is a quarter. He who wants to make a square from four amphoras needs sixteen¹¹⁴A row of length n defines a square of area n². For the problem at hand, the remark is irrelevant.. Rebbi Mana estimated it from a brick, Rebbi Abin estimated it from a strip, Rebbi Oshaiah estimated it from a dish¹¹⁵Rebbis Mana and Abin derived the second formula of Note 112 by supposing the town walls to form a rectangle; Rebbi Oshaia supposed the town to be of circular shape and derived the first formula. Neither of them accepted the 1:4 ratio.. Rebbi Yose ben Rebbi Abun said, 50 by 50 form a bet se’ah¹¹⁶An equivalent statement in Babli Eruvin 23b.; 100 by 100 are four bet se’ah¹¹⁷Since 22 = 4.. As it happened to the Resh Galuta when it was imposed on him¹¹⁸By the government, Parthian or Persan. to fill a hall of 40 by 40 cubits with wheat. He came to Rav Huna who said to him, negotiate that they should take from you 20 by 20 now and 20 by 20 later; you will earn half¹¹⁹If the government officials are so ignorant of geometry that they do not realize that only half the surface area will be covered..

רִבִּי בָּא בְשֵׁם רַב יְהוּדָה רִבִּי זְעִירָא בְשֵׁם רִבִּי עוּקְבָא. אֵין מְקַדְּדִין אֲלָּא בְחֶבֶל שֶׁלַּחֲמִשִּׁים אַמָּה. רִבִּי זְעִירָא בְשֵׁם רַב חִסְדַיי. אֵין מְקַדְּדִין לֹא בְעָרֵי הַלְּוִיִּם וְלֹא בִמְקוֹם עֲרִיפָה בַנַּחַל. נִיחָא כְּמָאן דָּמַר. אֶלֶף אַמָּה מִגְרָשׁ וְאַלְפַּיִים אַמָּה שָׂדוֹת וּכְרָמִים. בְּרַם כְּמָאן דָּמַר. אֶלֶף אַמָּה מִגְרָשׁ וְאַלְפַּיִים אַמָּה תְחוּם שַׁבָּת. כְּלוּם לָֽמְדוּ תְחוּם שַׁבָּת לֹא מֵעָרֵי הַלְּוִיִּם. לְעִיקָּר אֵין מְקַדְּדִין וְלִטְפֵילָה מְקַדְּדִין. Rebbi Abba in the name of Rav Jehudah, Rebbi Ze‘ira in the name of Rav Uqba: One strip-measures¹²⁰One determines the distance not on the ground but as if from a plane map of the domain. This means that on an incline one keeps the measuring rope horizontal.
Mishnah Erubin 5:4 states that “one measures only with a rope of 50 cubits, no less and no more”. The Babli, Erubin 58b, disagrees with the Yerushalmi and notes that once one has to strip-measure, i. e. that the rope has to be held horizontally, away from the ground, a rope of 4 cubits is required. From a practical point of view, this seems to be the superior way of measuring. For מקדדין (cf. Arabic قدّ “to make strips”), the Babli has מקדרין. only with a rope of 50 cubits. Rebbi Ze‘ira in the name of Rav Ḥisda: One strip-measures neither for the levitic cities nor from the distance for breaking the neck in the wadi^121,Breaking a calf's neck in a wild place to atone for an unsolved murder, Deut. 21:1:9.¹²²In the Babli Erubin 58b this is a statement by Rav Nahman in the name of Rav Abba bar Abuh.. This is 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 cities¹²³Since the Mishnah, Erubin 5:4, requires that Sabbath domains be measured as one would do on a map, it seems impossible that R. Eliezer ben R. Yose the Galilean accept R. Aqiba’s opinion that the extent of the Sabbath domain is fixed in the Torah by the extent of the levitic cities, since the methods of measurement are different in both cases. The Babli, Note 122, gives as reason that for levitic cities and the breaking of the calf’s neck one has to measure on the ground, that these are measurements prescribed by the Torah. The obvious inference is that the distance indicated for the Sabbath domain is rabbinic. (The opponents of R. Aqiba hold that the biblical Sabbath distance is the diameter of the encampment of the Israelites in the plains of Jericho, which in both Talmudim is described as 12 mīl, 24 000 cubits; cf. Šebi‘it 6:1, Note 28). Therefore, it is permissible to extend the Sabbath domain even of a levitic city further than its political boundaries.? For the main thing one does not strip-measure; does one strip-measure for the derivative?

וּמְנַיִין שֶׁלֹּא הָיוּ קוֹבְרִין בְּעָרֵי הַלְּוִיִּם. רִבִּי אַבָּהוּ בְשֵׁם רִבִּי יוֹסֵי בַּר חֲנִינָה. וּמִגְרְשֵׁיהֶם יִהְיוּ לִבְהֶמְתָּם וְלִרְכוּשָׁם וּלְכָל־חַייָתָם. לִבְהֶמְתָּם וְּלְחַייָתָם נִיתְּנוּ וְלֹא נִיתְּנוּ לִקְבוּרָה. From where that one does not bury in a levitic city? Rebbi Abbahu¹²³Since the Mishnah, Erubin 5:4, requires that Sabbath domains be measured as one would do on a map, it seems impossible that R. Eliezer ben R. Yose the Galilean accept R. Aqiba’s opinion that the extent of the Sabbath domain is fixed in the Torah by the extent of the levitic cities, since the methods of measurement are different in both cases. The Babli, Note 122, gives as reason that for levitic cities and the breaking of the calf’s neck one has to measure on the ground, that these are measurements prescribed by the Torah. The obvious inference is that the distance indicated for the Sabbath domain is rabbinic. (The opponents of R. Aqiba hold that the biblical Sabbath distance is the diameter of the encampment of the Israelites in the plains of Jericho, which in both Talmudim is described as 12 mīl, 24 000 cubits; cf. Šebi‘it 6:1, Note 28). Therefore, it is permissible to extend the Sabbath domain even of a levitic city further than its political boundaries. in the name of Rebbi Yose bar Ḥanina: “Their open spaces shall be for their animals, their property, and their lives.”¹²⁴Num. 35:3. They were given for their animals and their lives; the were not given for burial¹²⁵The cemetery of the Levites has to be on the territory of the surrounding tribe..