More Discussions for this daf
1. Taurus and Scorpio 2. North-South Determination 3. The Vilna Gaon's calculation
4. Rav Ada 5. Tekufos - Halachic vs Secular 6. Vernal Equinox
7. Gra on Tosfos D"H Abaya 8. Rashi's Siman 9. Rashi's inexact calculation
10. Solistice? 11. Rashi, D"H v'Ein Tekufos Tamuz.. 12. Rashi DH Mipnei She'Mafsid - Where does the 28 come from?
13. Bach on amud beis 14. Constellations 15. Akrav is in the South
DAF DISCUSSIONS - ERUVIN 56

Marc Diamond asked:

Dear Rabbi Kornfeld,

The Bach quotes the Venice edition which has a more precise girsah in Rashi of 1428.8, which makes the remainder of our version of Rashi completely mathematically consistent. Unfortunately, the more exact number would seem to be 1428.6 (to the nearest fifth) because 2000 / 1.4 = 1428.571428571428...If Rashi is lav davka, why did he go through the entire calculation in such detail?

Marc Diamond

The Kollel replies:

Your question is very good and to the point.

The MAHARSHAL addresses your question, and says, b'Dochek, that Rashi was not being exact. The MAHARSHA, though, rightly challenges the Maharshal, and points out another question as well. Rashi mentions the remaining 4/5ths only when he discusses the "Tavlah Shel ha'Zaviyos" (the square tablet used to fill the corners of the Techum) -- why does he mention it earlier, when he discusses the measurement of the Techum in each direction?

It must be that Rashi's intention is to explain that we would like to arrive at the sum of 2000 Amos for the length of the Techum at the diagonal (see Rashi). If there are only 1428 Amos to the sides of the square [drawn around the city], the diagonal will fall short by 4/5 of an Amah. Therefore, Rashi writes that the sides of the square around the city are each 1428 Amos, and in addition it is necessary to add 4/5 of an Amah to the diagonal in order to get 2000 Amos there. Rashi's calculation, then, is quite accurate (1428 x 1.4 = 1999.2 + 4/5 = 2,000 Amos).

Best wishes,

-Mordecai

Michael Mosesson commented:

Dear Rabbi Kornfeld

Just for your information: If the diagonal of a square is 2000 amah then the side is

only 1414.21 amah. You have to devide with square root 2 = 1.41421 . So Rashi's calculation

is not so exact anyhow. Futher, a side of 1428 amah gives a diagonal of 2019.5 amah.

\ 2 |

\ 0 |

\ 0 |

\ 0 | 1414.21 amah

\ |

\ |

Michael

The Kollel replies:

You are correct, according to modern geometry your conclusions are true. Chazal, though, approximated certain values, and the ratio of the diagonal of a square is one of them. Chazal use a value of 1.4 ("one and two-fifths"), and not the actual value of the square root of two, when calculating the diagonal of a square, as the Gemara says in Eruvin 76b and many other places. The comment of Rashi that we discussed was using that approximation. We were trying to show that according to that approximation, on which the words of Rashi are clearly based, all of the figures expressed in Rashi are consistent.

As for why Chazal did not use the exact, irrational, number for the value of radical two, see Perush ha'Mishnah of the Rambam to Eruvin 1:5 (and 2:5)

Be well, Mordecai