We have learnt that two courtyards sharing a common wall may make one eruv together, only if there is a suitable window or opening in the wall- otherwise, the wall serves as a closed מחיצה (partition) between the two courtyards and the eruv does not work.

The minimum size of a halachik opening is 4 tefachim by 4 tefachim, and at least some of it needs to be below 10 tefachim in height.

We also discussed the complex issue of how to ensure that a circular window fits the minimum size and concluded that it need to be large enough for a square of 4 by 4 tefachim to be inscribed inside it.

Although Rabbi Yochanan seemed to require a circle with a circumference of 24 tefachim, Assuming π equaling 3 and Ö2 equaling 1.4, the Gemara concludes that one with a circumference of 16.8 tefachim is sufficient, and that Rabbi Yochanan was relying on the judges (or Rabbis) of Caesaria’s different formula in his ruling, who believed that in order to circumscribe a square, the perimeter of a circle must be twice the perimeter of the square.

We saw different views as to how to understand what seems like such a large error on the part of the judges of Caesaria and Rabbi Yochanan, as well as how to understand what appears to be a lack of mathematical precision on behalf of both the Rishonim and Chazal.

Though the issue of why Chazal did not use more accurate measures for rational numbers like π and Ö2 is also essential to our discussion, even more difficult was the far greater “error” (at least according to the way Rashi explained it) of Rabbi Yochanan and the judges he relied on that the hypotenuse of an isosceles right-angled triangle is equal to the sum of the other two sides, whereas according to the ancient theorem of Pythagoras, it equals Ö2 times the width of one of the sides, less than 75% of what they claimed.

We noted that even if Rabbi Yochanan and/or the judges of Caesaria and/or Rashi were unaware of this theorem, it is so easy to see that a factor of 2 is completely off by the simplest of measuring, and that ascribing such an error to any of them is extremely problematic, even without dealing with the question of whether and what type of “ruach hakodesh” they might have had.

I suggested that it might make more sense to explain that everyone understood that the length of the hypotenuse is less than the sum of both sides, but because measuring the precise length was difficult and involved square roots which were often irrational, they preferred to use the highest possible value of the hypotenuse of a right-angled triangle, which approaches (but never reaches) the sum of its’ width and height as the ratio between them decreases. (see chart below plotting hypotenuse on the y access and width on the x axis) assuming a constant height of 10)

We also mentioned that the Gra has another approach, which “seems” not to fit the text so well, but interprets the view of the judges of Caesaria and Rabbi Yochanan in a way that fits the maths perfectly- He claims that when the judges said that the square inside the circle is half, it does not refer to the perimeter of the square but to its area, and is not comparing it to the area of the circle it is inscribed in, but the square which superscribes the circle it is inscribed in (something Tosfos suggested was the intention of the judges of Casearia but misunderstood by Rabbi Yochanan and the Gemara!) and that Rabbi Yochanan’s requirement for 24 tefachim was also referring to the perimeter of the outer square.

On our daf, we have yet another instance of this issue, and seeing how the Rishonim handle it here, might shed some light on the earlier discussion, as well as its parallel discussion in Sukkah 8a (regarding the minimum circumference of a circular sukkah.)

On Daf 77, we have been discussing other options for linking the two courtyards, and one of them is by using a ladder that allows easy access.

On daf 78, Rav Yehuda quotes Shmuel as ruling that if a wall is 10 tefachim in height, it requires a ladder of 14 tefachim to permit it (serve as the equivalent of an opening.)

Rashi explains that seeing as the ladder cannot be placed vertically and still provide easy access, its base needs to be at least 4 tefachim away from the wall, and that given its diagonal position (forming a hypotenuse,) it needs to be at least 14 tefachim (the sum of the two sides of the right-angled triangle formed) in order to reach the top of the wall.

Once again, we see that Rashi seems to assume that the hypotenuse of a right-angled triangle equals the sum of the other two sides, which we clearly know to be incorrect.

Once again, it is hard to believe that Rashi could make such a large error. It is also perplexing that what the earlier sugya and the sugya in Beitza seemed to view as an error of Rabbi Yochanan based on the judges of Caesaria, Rashi is now attributing to Shmuel, without even saying that it was an error.

This might strengthen what we suggested that at least some of Chazal preferred to use one general rule for all right-angled triangles, and err on the side of caution using the maximum the “hypotenuse” could ever be, assuming a collapsed triangle consisting of two lines at 180 degrees to each other.

Instead of having to constantly work out square roots that vary according to the ratio between the lengths of the other two sides, the simple measure of over-estimating the hypotenuse as the sum of the other two sides is then used across the board according to this view.

It would still be difficult why Rav Yehuda would be quoting Shmuel as expressing this stringent cautious view that was seemingly rejected in the earlier sugya and in Sukkah.

Further, the Gemara itself in Sukkah suggests that perhaps Rabbi Yochanan was simply לא דק (not being precise) and rejects that suggestion based on the size of the imprecision being way too much to contemplate.

Tosfos on our daf, however, claims that Rashi was indeed “לא דק” (not being precise,) given that the ladder would need to be placed at a distance of 10 tefachim from the wall (forming an isosceles right-angled triangle) in order for it to have to be 14 tefachim in length, given the ratio of 1.4:1 of the hypotenuse to the other side ( Tosfos on Eruvin 57a already offered his own proof that it is slightly more than 1.4, showing independent mathematical ability but perhaps at the same time unfamiliarity with the theorem itself.)

Though Tosfos’ understanding of Rashi might backup my suggestion that the inaccuracy in our case was intentional and not due to ignorance, it remains difficult why Rashi would assume that Shmuel was being imprecise to a level that the sugya in Sukkah dismissed as implausible (using Pythagoras, the required length of the wall would be Ö116 or about 10.77, far closer to the height of the wall itself!)

Either way, we seem to see that according to Rashi, using the sum of the two sides as an “estimation” of the length of the hypotenuse was not limited to Rabbi Yochanan and/or the judges of Caesaria regarding the hypotenuse of an isosceles right-angled triangle, but extends also to Shmuel’s treatment of the hypotenuse of any right-angled triangle!

The journey continues…

*These posts are intended to raise issues and stimulate further research and discussion on contemporary topics related to the daf. They are not intended as psak halacha*