Eruvin 76a-b

Dispute regarding geometry (just before the first two dots):

עירובין עו.-עו: (ש"ס ווילנא); ז"ל

א"ר יוחנן חלון עגול צריך שיהא בהיקפו עשרים וארבעה טפחים ושנים ומשהו מהן בתוך י' שאם ירבענו נמצא משהו בתוך י' מכדי כל שיש בהיקפו שלשה טפחים יש בו ברוחבו טפח בתריסר סגיא הני מילי בעיגולא אבל בריבועא בעינן טפי מכדי כמה מרובע יתר על העגול רביע בשיתסר סגיא ה"מ עיגולא דנפיק מגו ריבועא אבל ריבועא דנפיק מגו עיגולא בעינן טפי מ"ט משום מורשא דקרנתא מכדי כל אמתא בריבוע אמתא ותרי חומשי באלכסונא בשיבסר נכי חומשא סגיא רבי יוחנן אמר כי דייני דקיסרי ואמרי לה כרבנן דקיסרי דאמרי עיגולא מגו ריבועא ריבעא ריבועא מגו עיגולא פלגא: עכ"ל

R. Johanan ruled: A round window must have a circumference of twenty-four handbreadths, two and a fraction of which must be within ten handbreadths from the ground, so that, when it is squared, a fraction remains within the ten handbreadths from the ground. Consider: Any object that has a circumference of three handbreadths is approximately one handbreadth in diameter: should not then twelve handbreadths suffice? This applies only to a circle, but where a square is to be inscribed within it a greater circumference is required. But observe: By how much does the perimeter of a square exceed that of a circle? By a quarter approximately; should not then a circumference of sixteen handbreadths suffice? This applies only to a circle that is inscribed within the square, but where a square is to be inscribed within a circle it is necessary [for the circumference of the latter] to be much bigger. What is the reason? In order [to allow space for] the projections of the corners. Consider, however, this: Every cubit in [the side of] a square [corresponds to] one and two fifths cubits in its diagonal; [should not then a circumference] of sixteen and four fifths handbreadths suffice? – R. Johanan holds the same view as the judges of Caesarea or, as others say, as that of the Rabbis of Caesarea who maintain [that the area of] a circle that is inscribed within a square is [less than the latter by] a quarter [while that of] the square that is inscribed within that circle [is less than the outer square by] a half. [translation by Rev. Dr. Israel W. Slotki in The Babylonian Talmud, Rabbi Dr. I. Epstein, ed., Seder Mo'ed Vol. III (`Erubin), London: Soncino, 1938]