The case of the square inside the circle

Eruvin 76 presents the geometry of a square inside a circle. The math seems badly wrong at the commentators’ first glance, leading to lots and lots of interpretation.

The Mishnah on Eruvin 76a says that for a window to possibly bridge two courtyards, it needs to be at least 4 by 4 tefahim and within 10 tefahim of the ground.

משנה. חלון שבין שתי חצירות, ארבעה על ארבעה בתוך עשרה - מערבין שנים. ואם רצו - מערבין אחד. פחות מארבעה על ארבעה, או למעלה מעשרה - מערבין שנים ואין מערבין אחד.

The Gemara then gives us the ruling of Rabbi Yohanan about a circular window:

אמר רבי יוחנן: חלון עגול צריך שיהא בהיקפו עשרים וארבעה טפחים, ושנים ומשהו מהן בתוך עשרה, שאם ירבענו נמצא משהו בתוך עשרה.

Rabbi Yohanan said: A circular window needs 24 tefahim in its circumference. And two plus a bit from those need to be within ten [tefahim from the ground], so that if you make a square, some of it will be within ten.

This is difficult to understand.

Making a keli unusable on Shabbat

The mishnah on Shabbat 42b states:

אין נותנין כלי תחת הנר לקבל בו את השמן, ואם נתנוה מבעוד יום - מותר. ואין ניאותין ממנו, לפי שאינו מן המוכן.

One may not place a keli under a lamp, on Shabbat, in order to catch the oil that may drip.

Rav Hisda says the principle also applies to a placing a keli under a hen to catch a newly hatched egg:

אמר רב חסדא: אף על פי שאמרו אין נותנין כלי תחת תרנגולת לקבל ביצתה - אבל כופה עליה כלי שלא תשבר.

Rav Yosef explains Rav Hisda on 43a:

רב יוסף אמר: היינו טעמא דרב חסדא - משום דקא מבטל כלי מהיכנו.

By causing a new egg, which is muktsah, to rest on the keli, you are mevattel keli me-heikhano, “nullifying the readiness of the keli.”

Abaye raises a number of cases against Rav Yosef that also involve bittul keli me-heikhano at first glance, including supporting a broken beam in the ceiling with a bench, placing a keli under a leaking roof, and using an overturned basket as a step for baby chicks to enter and exit their coop.

What is this prohibition of bittul keli me-heikhano?

The safek of bein ha-shemashot

Shabbat 34–35 discusses the laws of bein ha-shemashot, the period of twilight between sunset and full darkness. This time has a status of safek, uncertainty, when it comes to laws that depend on the time being day or night. There was no way I was going to skip a sugya on uncertainty.

A berayta on 34b states:

תנו רבנן: בין השמשות ספק מן היום ומן הלילה, ספק כולו מן היום, ספק כולו מן הלילה - מטילין אותו לחומר שני ימים.

Bein ha-shemashot is a three-sided safek: it might be fully day, it might be fully night, and it might be some combination of the two. In any question of halakhah we treat it stringently, as either one, or the other, or both of the days that it bridges.

There are two main shittot on understanding the side of the safek that bein ha-shemashot could be both days. Rabbeinu Tam understands the safek to be that any moment between sunset and nightfall could be the dividing line for when the previous day ends and the new day begins. The Ritva (as understood by some aharonim based on a comment on Yoma 47b) understands the safek to be that the entire period of bein ha-shemashot might be ruled as both days intertwined, just as an adroginus might be ruled both male and female.

In this post I discuss Rashi's shittah on how this safek works. It's a shittah whose meaning is, well...uncertain.

Ge’ullah and tefillah

The first half of Masekhet Berakhot covers two main topics, the Shema with its blessings and the Amidah. One subject arises multiple times as a glue that joins the two topics together (five times, to be exact, according to the Bar Ilan Responsa Project: on 4b, 9b, 10b, 26a, and 30a). This is the requirement to be somekh ge’ullah li-tfillah—that we connect the redemption at the end of the Shema’s blessings, “ge’ullah,” to the start of the Amidah, “tefillah.” The Halakhah gives this requirement high priority.

What is the reason for semikhat ge’ullah li-tfillah? And why is it so important?