출애굽기 27:18 탈무드 참조: Jerusalem Talmud Peah & Jerusalem Talmud Eruvin

Rav said³⁰Here starts the discussion of how large the interruption between two crops of the same kind must be so that two separate peot are due.: Fallow land and ploughed land of a bet rova‘³¹In general, surface area measurements are given in the Talmudim by the amount of seed grain needed for the area in cultivation. The measurements are standardized by the tradition that the courtyard of the Tabernacle, which was 50 by 100 cubits (Ex. 27:18), defined the area covered by two seah. One seah are six qab. Hence, the area covered by a quarter qab, the bet rova‘, is an area of 5000:48 = 104 1/6 square cubits. A cubit was at least 45 and at most 61 cm, probably 54.6 cm., other produce³²For example, if two fields of wheat are separated by a strip of barley. even the tiniest amount. Rebbi Joḥanan said: Fallow land, ploughed land, and other produce at three preliminary furrows³³Three wide parallel furrows. According to Rashi (Is. 28:24), in ploughing an uncultivated field one first ploughs wide and coarse “starter” furrows; for sowing one then ploughs narrow ones adapted to the particular seed. According to Maimonides (Kilaim 3:2), the coarse furrows serve to cover breaks in the soil which developed during the heat of summer. There is no practical difference between the two explanations. In any case, the total width of the three furrows cannot be larger than two cubits.. Do they disagree³⁴Is it necessary to assume that Rav and R. Joḥanan present different traditions? One tries to avoid disagreements over basic measurements as much as possible.? What Rav said³⁵In matters of the separating produce only. refers to produce subject to peah, what Rebbi Joḥanan said refers to produce not subject to peah³⁶In the next paragraph it will be explained that, at least for Rebbis Meïr and Jehudah, plots subject to the obligation of peah are more powerful to interrupt between fields of the same kind than those not subject to it.. But did we not state that fallow or ploughed land is subject to peah³⁷At least for the Sages who disagree with R. Meïr in the Mishnah. Since here Rav and R. Joḥanan give different measurements for identical situations, they seem to disagree.? What Rav said refers to an average field, what Rebbi Joḥanan said refers to a plot 50 by 2³⁸If the entire field, in this example 100 square cubits, is smaller than a bet rova‘, the rule of Rav becomes inapplicable and must be replaced by a smaller limit..

HALAKHAH: Paragraph. “In addition, Rebbi Jehudah ben Bava said,” etc. Rebbi Samuel bar Naḥman in the name of Rebbi Jonathan, they inferred it from the courtyard of the Tabernacle: the length of the courtyard 100 cubits wide 50 by 50⁸¹Ex. 27:18. Babli 23b. The expression “50 by 50” is read as meaning that the standard measure of area shall be a square of 5’000 square cubits.. 50 times 100 are 5’000. 70 by 70 are 5’000 minus 100, and we have stated “seventy cubits and a remainder.” And Samuel stated, they stated cubits and two thirds of a cubit. Seventy times two thirds and seventy times two thirds make 140 thirds each; 140 thirds and 140 thirds are 93⅓. There is missing from there 4/9 for the corner, there remain nineteen thirds minus one ninth. As we have stated, there is a slight difference which the Sages could not compute⁸²Since 70² = 4900 and 71²= 5041 the square root of 5’000 is between 70 and 71. In order to compute 70⅔² one uses the binomial formula in its geometric form: A square of sides 70⅔ is composed of a square of sides 70 + two rectangles 70 by ⅔ + a square of area 4/9 for a total of 49937/9. The difference to 5000 is 62/9 = 19/3 -1/9. The additional correction was a problem. Not only is the square root of 5000 irrational, not expressible as a fraction, but of the two square root algorithms used in Antiquity, the Babylonian, using a kind of Newton approximation, is an approximation from above and the Hellenistic, using an equivalent of modern continued fractions, starts with an approximation from above, when the legal situation here demands an approximation from below. 70⅔, is a reasonable rational approximation from below to √5000 since 707/9² already is >5009. The second term of the continued fraction development is an approximation from below, 70140/197, not a practical expression. The systematic approximation of square roots from below which used to be taught in our schools is essentially dependent on the notion of place values underlying Indian- Arabic numerals. The first approximation of √5000 both in the Babylonian and the Hellenistic methods, 705/7, is an exceedingly good low-denominator approximation from above, (705/7)² = 500025/49, and therefore not usable in this context. The approximation of 70⅔ cubits allows a measuring error of 1 digit without exceeding √5000. Tosephta 4:9..