출애굽기 27:18의 탈무드

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..