Eruvin 56b

עירובין נ״ו ע״ב

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Topic

Squaring a city for the תחום שבת, and the geometry of the מגרש around the ערי לויים

Overview

This amud deals with measuring areas around a city using squares and corners. First, a ברייתא teaches that when we measure the תחום שבת, we turn both the city and its 2000-אמה boundary into squares. At the corners, we add a full 2000×2000 square rather than measuring 2000 אמות on the diagonal. The ברייתא calculates that this gains 1200 אמות at each corner, and אביי says these numbers fit a round city 2000 אמות wide. The גמרא then brings a ברייתא about the ערי לויים: their area extends 2000 אמות, the first 1000 are a מגרש, and this מגרש is called 'a quarter' of the whole. רבא gives the פסוק as the source, and the גמרא asks how 1000 out of 2000 can be a quarter. Through a surveyor's calculation of areas, plus several objections, the גמרא shows that the quarter works out exactly for a round city: 9 out of 36. Finally, אביי begins a second calculation using a 1000×1000 city, which continues on the next amud.

Opening lines

The steps (15)

1. StatementSquare the city, then square the boundaries
2. StatementHow to measure at the corners
3. StatementHow much is gained at the corners
4. StatementWhich city these numbers fit
5. StatementThe boundary and מגרש of the ערי לויים
6. QuestionWhat is the source for this?
7. Responseרבא brings the verse
8. ChallengeA quarter? It looks like a half!
9. ResolutionThe surveyor's calculation with corners
10. ChallengeStill not a quarter: it is more than a third
11. ResolutionAdd the city's own area to the total
12. ChallengeEven with the city added, it is still a third
13. ResolutionThe rule is about a round city
14. ResolutionThe math for a round city gives exactly a quarter
15. Statementאביי: The quarter also works for a 1000×1000 city