Succah 8a

סוכה ח׳ ע״א

Sukkah 8a

Learn Succah 8a step by step: Aramaic beside English, every question, answer and proof colour-coded.

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Topic

Explaining רבי יוחנן's 24-person round סוכה: the geometry of fitting a 4x4 אמה square in a circle (true minimum 16⅘), and the rule that a rounded number may be stricter but not more lenient.

Overview

On the previous amud, רבי יוחנן ruled that a round סוכה must be big enough to seat 24 people around its edge, and the גמרא asked why so much is needed. Here the גמרא works through the geometry. The סוכה must hold a 4x4 אמה square, and the square's corners stick out, so the circle's width must equal the square's diagonal of 5⅗ אמות. That makes the true minimum circumference 16⅘ אמות (17 minus a fifth). Saying רבי יוחנן simply 'was not exact' (לא דק) fails, because 24 is too far from 16⅘. מר קשישא suggests three people sit in two אמות, which gives 16, but that is rejected because rounding may not make things more lenient. רב אסי explains that the 24 אמות ring includes the seated people, and the empty circle inside them measures 18. Since 18 is a little more than 16⅘, this is rounding toward strictness (לחומרא), which is allowed. Finally, רבנן דקיסרי begin a different calculation based on a circle inside a square, which continues on the next amud.

Opening lines

The steps (14)

1. ResolutionTwelve is enough only for a circle, not for a square
2. ChallengeA square is only a quarter larger, so sixteen should suffice
3. ResolutionA square inside a circle needs more because of its corners
4. ChallengeThe diagonal calculation gives only seventeen minus a fifth
5. Resolutionרבי יוחנן was not being precise
6. ChallengeRounding works for a small gap, not a big one
7. ResolutionThree people sit in two אמות
8. ChallengeSixteen is still a bit too small
9. ResolutionHe rounded down
10. ChallengeNo rounding toward leniency
11. ResolutionThe people's own space is not counted
12. ChallengeThe inner circle comes to 18, but only 16⅘ is needed
13. ResolutionHe rounded up, and rounding toward strictness is allowed
14. Resolutionרבנן דקיסרי begin a new calculation: a circle inside a square is a quarter smaller