Yerushalmi Eruvin 1:3 ירושלמי עירובין א׳:ג׳

MISHNAH: The beam about which they spoke must be wide enough to carry a tile which is half a brick of three hand-breadths⁸Therefore the tile is 1.5׳3 hand-breadths².; therefore it is sufficient for a beam to be one hand-breadth wide to accept the tile lengthwise⁹Since a hand-breadth is counted as 4 thumb-breadths, the tile will extend over the beam by one thumb-breadth on each side..

HALAKHAH: “The beam about which they spoke,” etc. Rebbi Abba said, to accommodate a row¹⁷⁴Greek δόμος, ὁ. of tiles lengthwise. It was stated: Rabban Simeon ben Gamliel says, to accommodate a row of tiles crosswise. What is the difference between them? Rebbi Ashian said, strength is between them. In the opinion of the rabbis, forty tiles. In the opinion of Rabban Simeon ben Gamliel, twenty tiles¹⁷⁵The formulation seems self-contradictory. The length of the beam is assumed to be the standard 10 cubits which is 60 hand-breadths. A tile is 3 hand-breadths long and 1.5 hand-breadths wide. If the tiles are put on the beam lengthwise, the beam has to carry 20 tiles, if crosswise, there are 40 tiles. Of the medieval commentators, Rashba (col. 105, Note 446) quotes the text as given here but adds “it seems that the formulation is slightly corrupt.” His student Ritba (ad 13b, ed. Warsaw 22a) switches the places of 40 and 20. The same is true for Meïri (ed. Herschler, col. 68a) but since he gives a paraphrase rather than a copy of the text it is not clear how he read it. In any case, the Babli disagrees, 14a..

Two beams, on each one half of a hand-breadth; how much can be between them? Rebbi Zeˋira said, half a hand-breadth. What is the difference between half a hand-breadth on either side and half a hand-breadth in the middle¹⁷⁶This follows the rabbis in the preceding paragraph. If two beams protrude from the walls on each side, they can be counted as one only if 20 tiles can be put on them. Therefore the gap between them must be bridged by one tile which according to R. Zeˋira must lie safely on both sides but according to R. Hoshaia only minimally.? Rebbi Hoshaia stated, a hand-breadth. Two beams, on each a little more than a third of a hand-breadth and between them less than a third of a hand-breadth¹⁷⁷This is a more restrictive interpretation of the preceding. As explained in the following sentence, we have a general rule that gaps can be disregarded if they are minor (for walls less that 3 or 4 hand-breadths) and in addition the solid material is wider than the gap. When applied to beams, the author of the present statement holds that on each side the solid part must be larger than the gap. To accommodate one hand-breadth, on each side one needs 1/3+ε of solid material to justify a gap of 1/3 -ε for arbitrary ε>0.. This parallels what Rebbi Joḥanan said, what is standing and empty space combine for four [hand-breadths], on condition that what is standing be more than the empty space¹⁷⁸This refers to walls; cf. Chapter 7(24b l.55), Chapter 8(25a l.32), Babli 16a..

If a beam protrudes from one wall but does not touch the second wall, or there were two opposite ones, if the gap between one and the other is three hand-breadths it needs another beam, otherwise it does not need another beam. There are Tannaim who state, four. He who says three, that a human may pass through. He who says four, that it be a place¹⁷⁹This explains the general disagreement whether one may disregard a gap only up to 3 or up to 4 hand-breadths, cf. Notes 80,81. 3 hand-breadths is the minimum width of an adult; 4 hand-breadth everywhere is the minimal size of a domain in the rules of the Sabbath, cf. Note 74. Babli 14a..

If one of the heads of a beam is higher that twenty cubits and the other lower than twenty cubits, one looks that if one would cut within three hand-breadths it would be seen lower than twenty cubits it is permitted¹⁸⁰If the beam except the last 3 hand-breadths near the second wall is below 20 cubits the beam is acceptable since the higher part could be cut and it still would be qualified under the rule of the preceding paragraph. Since the rules of Eruvin are rabbinic, the possibility alone is sufficient for qualification., otherwise it is forbidden. If one of the heads of a beam is lower than ten hand-breadths and the other higher than ten hand-breadths, one looks that if one would cut within three hand-breadths it would be seen lower¹⁸¹Obviously one has to read: higher. Then the argument is parallel to that of the preceding sentence. than ten hand-breadths it is permitted, otherwise it is forbidden.

If two beams are one on top of the other. Rebbi Yose ben Rebbi Jehudah says, one looks at the lower one as if it were higher or the higher one as if it were lower, on condition that the upper one not be higher than twenty cubits and the lower not be lower than ten hand-breadths¹⁸²Babli 14a, Tosephta 1:5. Mishnah 3 requires that the beam be wide enough for a tile to be placed there. The case considered here is of two beams, each one too narrow, but which would be acceptable if they were parallel on the same level. R. Yose ben R. Jehudah asserts that if the beams could be imagined to be moved vertically to become parallel they are acceptable even if on different height levels.. It was found stated that Rebbi Yose ben Rebbi Jehudah follows his father’s argument and disagrees with him. He follows his fathers argument since he says, one looks at the lower one as if it were higher or the higher one as if it were lower¹⁸³Since R. Jehudah in Mishnah 4 is explicit that the beams do not have to be strong he is the author of Mishnah 5 which introduces the principle of “one looks as if” used by his son.. He disagrees with him since he says, on condition that the upper one not be higher than twenty cubits and the lower not be lower than ten hand-breadths. If you would say that he follows his father’s opinion it could be arbitrarily high¹⁸⁴Mishnah 1.. Rebbi Yose, Ḥinena bar Shelemiah, Rebbi Yudan; Ḥinena bar Shelemiah in the name of Rav: Only if the upper one be within three hand-breadths of the lower one¹⁸⁵He permits the argument of R. Yose ben R. Jehudah if the difference in level of the two beams is negligible, i. e., smaller than 3 hand-breadths..