Eruvin 112

תנו רבנן המרבע את העיר עושה אותה כמין טבלא מרובעת וחוזר ומרבע את התחומין ועושה אותן כמין טבלא מרובעת

Our Rabbis taught: If [a circular] town is to be [circumscribed by a] square<span class="x" onmousemove="('comment',' In connection with the calculations of the permitted Sabbath limits around it.');">¹</span> [the sides must be] drawn in the shape of a square tablet. The Sabbath limits also are then drawn in the shape of a square tablet.<span class="x" onmousemove="('comment',' This is explained infra.');">²</span> When the measurements<span class="x" onmousemove="('comment',' In connection with the calculations of the permitted Sabbath limits around it.');">¹</span>

וכשהוא מודד לא ימדוד מאמצע הקרן אלפים אמה מפני שהוא מפסיד את הזויות אלא מביא טבלא מרובעת: שהיא אלפים אמה על אלפים אמה ומניחה בקרן באלכסונה

are taken one should not measure the two thousand cubits<span class="x" onmousemove="('comment',' The permitted distance in all directions from the imaginary square round the town.');">³</span> from the middle point of the town corner,<span class="x" onmousemove="('comment',' I.e., extending the diagonals of the imaginary square to the length of two thousand cubits and joining them so as to form a larger square.');">⁴</span> because, thereby, one loses the corners.<span class="x" onmousemove="('comment',' As will be shown presently.');">⁵</span> One should rather imagine<span class="x" onmousemove="('comment',' Lit., 'bring'.');">⁶</span>

נמצאת העיר משתכרת ארבע מאות אמות לכאן וארבע מאות אמות לכאן נמצאו תחומין משתכרין ח' מאות אמות לכאן ושמונה מאות לכאן נמצאו העיר ותחומין משתכרין אלף ומאתים לכאן ואלף ומאתים לכאן

that a square tablet of th size of two thousand cubits by two thousand cubits is applied to each corner diagonally,<span class="x" onmousemove="('comment',' One extremity of the diagonal of the imaginary tablet touching in turn each of the four corners of the imaginary square, the diagonal of the latter forming a straight line with that of the former.');">⁷</span> so that the town gains thereby four hundred cubits in each corner,<span class="x" onmousemove="('comment',' Lit., 'towards here and . . towards here'. The town spoken of here (as stated by Abaye infra) is one that is circular in shape and the diameter of which is two thousand cubits. By enclosing it in an imaginary square the diagonal of which (on the rule that the diagonal of a square exceeds its side by two fifths approx.) the town is extended in each of its four corners by ( (2000 X 2/5) /2) = 4000/10 = 400 cubits (cf. foll. n.) .');">⁸</span> the Sabbath limits gain eight hundred cubits in each corner,<span class="x" onmousemove="('comment',' A line of two thousand cubits is by two fifths (cf. prev. n.) , less than the diagonal of a two thousand cubits square. 'A square tablet of the size of two thousand cubits by two thousand cubits applied to each corner diagonally' would consequently add to each corner two thousand cubits plus (2000 x 2) /5 = 800 cubits.');">⁹</span> while the town and the Sabbath limits together gain twelve hundred cubits<span class="x" onmousemove="('comment',' I.e., the total of 400 and 800 cubits in each of the inner and outer corners respectively.');">¹⁰</span>

אמר אביי ומשכחת לה במתא דהויא תרי אלפי אתרי אלפי

in each corner.<span class="x" onmousemove="('comment',' Tosef. 'Er. IV.');">¹¹</span> This<span class="x" onmousemove="('comment',' The various measurements and gains just described.');">¹²</span> is possible, Abaye explained. in a town of the size of two thousand by two thousand cubits.<span class="x" onmousemove="('comment',' Cf. preceding notes.');">¹³</span>

תניא אמר רבי אליעזר ברבי יוסי תחום ערי לוים אלפים אמה צא מהן אלף אמה מגרש נמצא מגרש רביע והשאר שדות וכרמים

It was taught: R'Eliezer son of R'Jose stated: The limit of the allotted land beyond the confines of the levitical cities<span class="x" onmousemove="('comment',' In addition to the cities themselves the Levites were allowed stretches of land around them for use as open spaces, fields and vineyards as will be specified below.');">¹⁴</span> was two thousand cubits.<span class="x" onmousemove="('comment',' In an outward direction round each city.');">¹⁵</span> Deducting from these<span class="x" onmousemove="('comment',' The 2000 cubits mentioned.');">¹⁶</span> an open space of one thousand cubits,<span class="x" onmousemove="('comment',' Immediately behind and around each city.');">¹⁷</span>

מנא הני מילי אמר רבא דאמר קרא (במדבר לה, ד) מקיר העיר וחוצה אלף אמה סביב אמרה תורה סבב את העיר באלף נמצא מגרש רביע

such open space would represent a quarter of the entire area<span class="x" onmousemove="('comment',' This will be explained presently.');">¹⁸</span> the remainder of which consisted of fields and vineyards.<span class="x" onmousemove="('comment',' Tosaf. 'Ar. ad fin., Sotah 27b.');">¹⁹</span> Whence is this<span class="x" onmousemove="('comment',' That a strip of one thousand cubits around each levitical city must be reserved as an open space.');">²⁰</span> deduced? - Raba replied: From Scripture which says.

רביע פלגא הוי אמר רבא בר אדא משוחאה אסברה לי משכחת לה במתא דהויא תרי אלפי אתרי אלפי תחום כמה הויא שיתסר קרנות כמה הויין שיתסר דל תמניא דתחומין וארבעה דקרנות כמה הוי תריסר

[And the open land,.] from the wall of the city and outward a thousand cubits round about,<span class="x" onmousemove="('comment',' Num. XXXV, 4, dealing with the cities of the Levites.');">²¹</span> the Torah has thus enjoined, 'Surround the city by an open space of one thousand cubits'.' Such an open space [it was said] would represent a quarter of the entire area' - 'A quarter'! Is it not in fact one [in the neighbourhood] of a half?<span class="x" onmousemove="('comment',' One thousand cubits of open space in every two thousand cubits allowed: 1000/2000 = 1/2. The actual area of the open space on the present assumption would, of course, be less than a half of the total area, since an inner belt of the width of a thousand cubits is smaller in area than one of equal width around it.');">²²</span> - Raba replied: The surveyor Bar Adda<span class="x" onmousemove="('comment',' Aliter: The son of the surveyor Adda'. Aliter: 'Raba b. Adda replied: A surveyor.'');">²³</span>

נמצא מגרש רביע טפי מתלתא נינהו

explained this to me. Such a proportion is possible in the case of a town whose area is two thousand by two thousand cubits. For what is the area of its limits?<span class="x" onmousemove="('comment',' Sc. the stretch of land two thousand cubits in width around it.');">²⁴</span> Sixteen [million square cubits].<span class="x" onmousemove="('comment',' 2000 by 2000 cubits on each of its four sides: 2000 X 2000 X 4 = 16,000,000 square cubits.');">²⁵</span>

אייתי ארבעה דמתא שדי עלייהו אכתי תילתא הוי

What is the area of the corners?<span class="x" onmousemove="('comment',' The corner spaces between the limits just described.');">²⁶</span> Also sixteen [million square cubits].<span class="x" onmousemove="('comment',' The area of each corner being 2000 X 2000 square cubits the total area of the four corners is 2000 X 2000 X 4 = 16,000,000 square cubits.');">²⁷</span> Deducting [for the open spaces] eight [million square cubits]<span class="x" onmousemove="('comment',' Since the Torah enjoined to surround the whole city with a strip of one thousand cubits wide, one 1000 by 2000 cubits on each of the four sides 2,000,000 X 4 = 8,000,000 sq. cubits.');">²⁸</span> front the limits, and four [million square cubits]<span class="x" onmousemove="('comment',' One 1000 by 1000 cubits in each of the four corners = 1,000,000 X 4 = 4,000,000 sq. cubits.');">²⁹</span>

מי סברת בריבועא קאמר בעיגולא קאמר

from the corners, to what area would this space amount? To one of twelve [million square cubits].<span class="x" onmousemove="('comment',' 8,000,000 sq. cubits and 4,000,000 sq. cubits (cf. prev. two nn.) amount to 12,000,000 sq. cubits.');">³⁰</span> Would then 'such an open space represent a quarter'? Is it not in fact more than a third of the entire area?<span class="x" onmousemove="('comment',' 12,000,000/32,000,000 being equal to 3/8.');">³¹</span>

כמה מרובע יתר על העגול רביע דל רביע מינייהו פשו להו תשעה ותשעה מתלתין ושיתא ריבעא הוי

- Take the four [million square cubits] of the town area itself and add to them.<span class="x" onmousemove="('comment',' To the 32,000,000. This brings the total up to 36,000,000.');">³²</span> Does not this, however, still amount to a third?<span class="x" onmousemove="('comment',' 12,000,000/36,000,000 = 1/3. Why then was it described as 'a quarter'?');">³³</span> - Do you imagine that a quadrilateral town was spoken off? No, a circular town was meant.

אביי אמר משכחת לה נמי במתא דהויא אלפא באלפא תחומין כמה הוו תמניא קרנות כמה הוי שיתסר

For by how much does the area of a square exceed that of a circle? By one quarter [approximately] - Deduct a quarter from the measurements given and there would remain nine [million square cubits];<span class="x" onmousemove="('comment',' The city that was originally assumed to have an area of 2,000 X 2,000 = 4,000,000 sq. cubits, being circular in shape has only an area of 4,000,000 X 3/4 = 3,000,000 sq. cubits approx. The belt of open spaces around it, which was originally assumed to have an area of 12,000,000 sq. cubits would similarly amount to 4,000 (city, 2,000, and open spaces on two of its sides 2,000) by 4,000 X 3/4 (difference between area of sq. and circle) 3,000,000 approx. (area of circular city) . 4,000 X 4,000 X 3/4 - 3,000,000 = 12,000,000 - 3,000,000 = 9,000.000 sq. cubits.');">³⁴</span> and nine [million] represents one quarter of thirty six [million].<span class="x" onmousemove="('comment',' The latter figure represents the total area in sq. cubits of the city and the entire stretch of open spaces, fields and vineyards allowed to each levitical city. The shape of the city does not affect this outer area which always extends to a perpendicular distance of 2,000 cubits from it in all directions of the city.');">³⁵</span> Abaye said: This<span class="x" onmousemove="('comment',' That the open space shall represent a quarter of the area of the land allowed around each city of the Levites.');">³⁶</span> is also possible in the case of a town that has an area of a thousand by a thousand cubits For what are its limits?<span class="x" onmousemove="('comment',' The stretch of land allowed around it.');">³⁷</span> Eight [million square cubits].<span class="x" onmousemove="('comment',' Area of 1,000 by 2,000 cubits on each of its four sides equal to 2,000,000 X 4 = 8,000,000 sq. cubits.');">³⁸</span> What is the area of the corners? Sixteen [million square cubits].<span class="x" onmousemove="('comment',' Each corner having an area of 2,000 by 2,000 sq. cubits the area of the four corners amounts to 2,000 X 2,000 X 4 = 16,000,000 sq. cubits.');">³⁹</span>