The sixth chapter of Massekhet Eiruvin dealt with situations where two courtyards joined together and became one unit for the rules of eiruv and carrying on Shabbat.
The seventh perek,which begins on our, opens by discussing cases where hatzeirot (courtyards) that are next to one another either
cannot join each other,
are obligated to join each other,
or are permitted to do so.
The first Mishna deals with courtyards that are divided by a wall which has a window in it. If the window is within ten tefahim (handbreadths) of the ground and is minimally four tefahim square in size, then the courtyards can choose whether or not to join as one. If the window is higher than ten tefahim or smaller than four by four, the hatzeirot are considered separate and need to make their own eiruvin.
Rabbi Yohanan in the Gemara introduces the possibility of a round window, arguing that it would need to be 24 tefahim in circumference in order to ensure that a square inscribed in that window would be at least four by four. In the ensuing discussion about the relationship between circles and squares, the Gemara explains that Rabbi Yohanan’s position is based on the rule taught by the judges of Caesarea that “a square inscribed within a circle is half of the square.”
Later on in the Gemara, Rabbi Yohanan’s position is rejected in its entirety as being based on an error. Rabbi Ya’akov Kahane in his Ge’on Ya’akov argues that Rabbi Yohanan knew that his figures were not accurate, but chose to present a larger than necessary rule so that, in case of a mistake, there would be well over the minimum four by four, leaving room for error. Nevertheless, many attempts have been made to try and explain the mathematical positions presented by these Sages.
In explaining the rule of the judges of Caesarea, Tosafot argue that they are discussing the case of a square inscribed in a circle, which, itself, is inscribed in a square.
By drawing lines that bisect the outer square, the circle and the inner square, it becomes clear that the inner square is half the size of the outer one. The inner square is made up of four triangles, each of which is half of the four smaller squares that together make up the outer square.