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An interesting idea on how Rashi measures diagonals. Please add comments or other examples.
On 5a, Rashi explains the case of the corner בקרן זוית as follows:
טפח מן הפתח בדופן האמצעי, ושלשה טפחים בדופן המשך כו'
“The door of four tefachim is a tefach on the middle wall, and three tefachim on the side wall.”
Tosefos asks on Rashi
בא“ד ולא דק דא"כ אין הפתח רחב אלא אלכסון של ג' טפחים על טפח והנה עינינו רואות שאין האלכסון של שלשה על אחד מגע לאלכסון של טפח על טפח עכ“ל.
“Rashi was not precise, because if so, the door would only be as long as the diagonal of 3 tefachim by 1 tefach, and our eyes can see that the diagonal is not ...” See Maharsha and Maharshal for the explanation of the end of Tosefos.
Actually, I would have thought that Tosefos could have asked even a simpler question. It’s not just that “our eyes see” that the diagonal of the 3x1 triangle is shorter than 4. You don’t need your eyes: it’s the Triangle Inequality of high school geometry: Any side of a triangle is shorter than the sum of the other two. Or, to put it more simply, it is quicker to go straight across than to go around!
Rashi’s p’shat does have certain advantages in the sugya: If you need four full tefachim at each corner, and then one whole side of the mavuy open entirely, there is very little left of the walls.
The truth is that Rashi often explains in a similar way. He measures diagonals in a halachic fashion, not straight along the diagonal. Here are several examples.
8b: A mavuy with one long side and one short side, and a korah across the diagonal opening. Rashi says
שהמבוי קצר לרחבו הרבה, ואין ברוחבו ובארבע אמות הנמשכין להצטרף יותר מעשר עכ“ל.
“The mavuy is very shortened (thinned), and its width along with the four amos (of the angled קורה distance along the side of the mavuy) doesn’t come to more than ten.” That is, the diagonal korah, across the door of the mavuy, is within its maximum length of ten amos.
Now if the korah is measured along the diagonal, the width of the mavuy cannot be קצר לרחבו הרבה “very shortened”: square root of (10 squared - 4 squared) = square root of (84), about 9.16.
Rather, Rashi doesn’t measure like that, and his mavuy is only 6 wide (which is very shortened) and with the four in the other direction, his קורה comes out to be “10” amos by his measurement.
78a: a ladder of 14 tefachim. Rashi:
שצריך למשוך רגלי הסולם ארבעה מן הכותל, לפי שאין סולם זקוף נוח לעלות עכ“ל
and so too several other comments there. 14 = 10 high, and 4 over on the ground. And again Tosefos there says that Rashi is לא דק.
94b: Rashi explains a break in two walls and ceiling of a house as being ten amos, five in this direction and five in that direction. Once again Tosefos says, לא דק, such a break is only about seven amos long.
At the beginning of perek Chalon, the gemara needs to find a circular window, where if you inscribe a square inside it, the square will be within 10 tefachim of the ground. So then a quarter of the circle’s circumference also must be within 10.
And there the דייני קיסרי gave the measurement of the circle as 24 tefachim. Tosefos asks on them that the true circumference is only about 16 and 4/5, like the gemara in Sukkah. But whereas over there, the gemara pushes away the דייני קיסרי - that we see that the number is wrong - here the gemara doesn’t argue on them at all.
The Gra there in the gilyon explains that the דייני קיסרי are taking the measurement of a square inscribed outside the circle, which is why they get 24. That’s actually not exact, more like 22.6 (16*sqrt(2)), but it’s close.
This measurement of the circumscribed square is exactly the same type of diagonal measurement that Rashi has been using. Imagine each small arc of the circle, but instead of measuring along the arc, measure how far up + how far over. You can see that the net result is going to be the same as if you measure all the way up, and all the way over, for each of the four quadrants, for the square around the circle.
On the other hand, in the gemara in Sukkah we are discussing how many people fit in the edge of the circular sukkah, and that is just a measurement directly along the curve, and therefore the measurement of the דייני קיסרי cannot be used. And so too for any place where Chazal discuss the distance around a circle being three times its diameter, it is dealing with where you must measure around with a string or some such. Whereas Rashi and the דייני קיסרי are talking about the halachic measurement.
On 5a, Rashi explains the case of the corner בקרן זוית as follows:
טפח מן הפתח בדופן האמצעי, ושלשה טפחים בדופן המשך כו'
“The door of four tefachim is a tefach on the middle wall, and three tefachim on the side wall.”
Tosefos asks on Rashi
בא“ד ולא דק דא"כ אין הפתח רחב אלא אלכסון של ג' טפחים על טפח והנה עינינו רואות שאין האלכסון של שלשה על אחד מגע לאלכסון של טפח על טפח עכ“ל.
“Rashi was not precise, because if so, the door would only be as long as the diagonal of 3 tefachim by 1 tefach, and our eyes can see that the diagonal is not ...” See Maharsha and Maharshal for the explanation of the end of Tosefos.
Actually, I would have thought that Tosefos could have asked even a simpler question. It’s not just that “our eyes see” that the diagonal of the 3x1 triangle is shorter than 4. You don’t need your eyes: it’s the Triangle Inequality of high school geometry: Any side of a triangle is shorter than the sum of the other two. Or, to put it more simply, it is quicker to go straight across than to go around!
Rashi’s p’shat does have certain advantages in the sugya: If you need four full tefachim at each corner, and then one whole side of the mavuy open entirely, there is very little left of the walls.
The truth is that Rashi often explains in a similar way. He measures diagonals in a halachic fashion, not straight along the diagonal. Here are several examples.
8b: A mavuy with one long side and one short side, and a korah across the diagonal opening. Rashi says
שהמבוי קצר לרחבו הרבה, ואין ברוחבו ובארבע אמות הנמשכין להצטרף יותר מעשר עכ“ל.
“The mavuy is very shortened (thinned), and its width along with the four amos (of the angled קורה distance along the side of the mavuy) doesn’t come to more than ten.” That is, the diagonal korah, across the door of the mavuy, is within its maximum length of ten amos.
Now if the korah is measured along the diagonal, the width of the mavuy cannot be קצר לרחבו הרבה “very shortened”: square root of (10 squared - 4 squared) = square root of (84), about 9.16.
Rather, Rashi doesn’t measure like that, and his mavuy is only 6 wide (which is very shortened) and with the four in the other direction, his קורה comes out to be “10” amos by his measurement.
78a: a ladder of 14 tefachim. Rashi:
שצריך למשוך רגלי הסולם ארבעה מן הכותל, לפי שאין סולם זקוף נוח לעלות עכ“ל
and so too several other comments there. 14 = 10 high, and 4 over on the ground. And again Tosefos there says that Rashi is לא דק.
94b: Rashi explains a break in two walls and ceiling of a house as being ten amos, five in this direction and five in that direction. Once again Tosefos says, לא דק, such a break is only about seven amos long.
At the beginning of perek Chalon, the gemara needs to find a circular window, where if you inscribe a square inside it, the square will be within 10 tefachim of the ground. So then a quarter of the circle’s circumference also must be within 10.
And there the דייני קיסרי gave the measurement of the circle as 24 tefachim. Tosefos asks on them that the true circumference is only about 16 and 4/5, like the gemara in Sukkah. But whereas over there, the gemara pushes away the דייני קיסרי - that we see that the number is wrong - here the gemara doesn’t argue on them at all.
The Gra there in the gilyon explains that the דייני קיסרי are taking the measurement of a square inscribed outside the circle, which is why they get 24. That’s actually not exact, more like 22.6 (16*sqrt(2)), but it’s close.
This measurement of the circumscribed square is exactly the same type of diagonal measurement that Rashi has been using. Imagine each small arc of the circle, but instead of measuring along the arc, measure how far up + how far over. You can see that the net result is going to be the same as if you measure all the way up, and all the way over, for each of the four quadrants, for the square around the circle.
On the other hand, in the gemara in Sukkah we are discussing how many people fit in the edge of the circular sukkah, and that is just a measurement directly along the curve, and therefore the measurement of the דייני קיסרי cannot be used. And so too for any place where Chazal discuss the distance around a circle being three times its diameter, it is dealing with where you must measure around with a string or some such. Whereas Rashi and the דייני קיסרי are talking about the halachic measurement.
