.ח Easy as Pi

אמר ליה רב אסי לרב אשי לעולם גברא באמתא יתיב ורבי יוחנן מקום גברי לא קחשיב כמה הוו להו תמני סרי בשיבסר נכי חומשא סגיא היינו דלא דק ולחומרא לא דק

A "mathy" guy like myself can't help but be curious to examine the precise math behind this גמרא. The results are quite fascinating.

Let's start with what the precise הלכה would be:

If we do, in fact, require a circle that encompasses a 4x4 square, the radius of that circle would be √(32) by the Pythagorean Theorem. That would make the circumference of that circle √(32)π which is approximately 17.7715 אמות circumference. 

Now, let's examine ר' יוחנן's statement according to רב אסי. The גמרא assumes based on its loose interpretation of π that if we made a circle of 24 men, the circle inside of that ring would be 18 אמות circumference. We could end right here and declare that the margin of לא דק is even more narrow than the גמרא suggested. But let's try to get a little more precise. A circle of 24 אמות circumference would have a diameter of 24/π. The men take up exactly one אמה so the inner circle would then have a radius of 24/π - 2. The circumference of that inner circle would then be (24/π -2) x π which comes out to 17.7168 !!! So ר' יוחנן's statement is off by only 0.0547 אמות !!! The only issue, of course is that it is לא דק לקולא but at that minuscule amount, that should be allowable. 

:ז Pi in the ... תורה

רש"י ד"ה באמתא יתיב. כל אחד מקומו אמה נמצא הקיפה כ"ד אמות וקיימא לן (עירובין דף יג:) כל עגול שיש בהקיפו ג' טפחים יש ברחבו טפח דכתיב (דה"ב ד) בים שעשה שלמה עשר באמה משפתו אל שפתו עגול סביב וקו שלשים באמה יסוב אותו סביב לעשר אמות רוחב שלשים אמה היקף אלמא הכא נמי בתריסר אמות היקף סגי דנהוי פותיא ארבעה:

Since רש"י references the ים של שלמה, it would seem appropriate to reference the גר"א's famous פשט on that.
Read more here.

:ב Cutting off the Angles

First off, מזל טוב to all who recently completed מסכת יומא.

At the very beginning of סוכה we have the familiar discussion surrounding the 20 אמה walls. A number of years ago there was an interesting discussion on two of my other blogs regarding the interpretation of ר' זירא's position that 20 אמה  walls are not a problem with a סוכה larger than 4x4 אמות.
Please see R' Ari Storch's post on AstroTorah and my follow up on AlPiCheshbon.

:ס"ב The ט"ו בשבט Connection

[Originally published in the vicinity of ט"ו בשבט תשע"ד]

Ok, this is a little bit of a stretch...

מיעוט צפרים שנים *שתים

I was initially intrigued by the הגהה. Really? ציפור is feminine? Certainly there are other words that don't give off any apparent evidence of being feminine but are, nevertheless. But I wanted to double-check it. As I was approaching the חומשים where I would confirm that it most definitely is feminine - החיה, השחוטה, אותה, etc. - it hit me that the ubiquitous ט"ו בשבט ditty as well indicates that it is so. Here are the words:

השקדיה פורחת
ושמש פז זורחת,
צפורים מראש כל גג
מבשרות את בוא החג.

ט"ו בשבט הגיע חג לאילנות.
ט"ו בשבט הגיע חג לאילנות.

.יב The Big City Synagogue

הא דכרכים והא דכפרים ודכרכים אין מטמא בנגעים והתניא אחוזתכם אחוזתכם מטמאה בנגעים ואין ירושלים מטמאה בנגעים אמר ר' יהודה אני לא שמעתי אלא מקום מקדש בלבד הא בתי כנסיות ובתי מדרשות מטמאין בנגעים ואע"ג דכרכים נינהו אימא אמר רבי יהודה אני לא שמעתי אלא מקום מקודש בלבד

I'm having difficulty understanding how the גמרא actually answers the question here. The subject here is the גמרא's suggestion that the rural shuls may be subject to נגעים whereas urban shuls may not. So we directly question the assertion regarding urban shuls. The implication of the ברייתא is that any edifice at least outside of ירושלים may be subject to נגעים. So in the end, we understand רבי יהודה to mean that any מקום מקודש would be exempt. But then doesn't that exempt all shuls? How does this solve our problem. Yes, we have found an exemption for the urban shuls but we have not found a source to differentiate between urban and rural shuls whatsoever. Now, in truth, the גמרא eventually discards this approach for a similar reason later on. But couldn't they have done so already at this point?

ירידת הדורות ט

The גמרא discusses the various reasons for the destruction of the משכן and בתי מקדש as well as the level of severity of each. It is interesting to note a nuance in this discussion which is not discussed explicitly. There is a definite progression in the "literal" nature of the critical offenses. 

Let me explain by starting with the בית שני. There we are told that שנאת חינם was the culprit. As we all know, it was pure, unadulterated שנאת חינם.

The בית ראשון was because of the ג' חמורות. But what exactly was the גילוי עריות? The גמרא explains that it refers to immodest conduct on the part of בנות ישראל. They conducted themselves in such a way as to incite the יצר הרע. It stops well short of stating that actual ג"ע was rampant.

The conduct of בני עלי in the time of the משכן is yet an even more far-fetched classification of גילוי עריות. What I believe this nuance is indicating is the degree to which הקב"ה is מדקדק throughout the generations. The generation of משכן שילה was on a much higher level and therefore judged with much greater scrutiny, ultimately being punished for something which only remotely resembled גילוי עריות. As the generations progressed, more explicit transgressions were needed to mete out justice. 

:ט Let the Building bear witness

שאלו את רבי אלעזר: ראשונים גדולים, או אחרונים גדולים? אמר להם: תנו עיניכם בבירה. איכא דאמרי, אמר להם: עידיכם בירה

I found this exchange most intriguing. The גמרא is comparing the merits of the generations that saw the destruction of each בית המקדש. The consensus seems to be that the fact the בית המקדש has not been rebuilt, in contrast to the relatively short initial exile of 70 years, is proof of the greater merit of the earlier generations.

We are taught (actually ירושלמי in this פרק)

כל דור שאינו נבנה בימיו מעלין עליו כאילו הוא החריבו

It would seem that the responsibility for bringing the בית המקדש back would lie in the hands of the subsequent generations in exile. Yet, the גמרא seems to tie it back to the generation in which it was destroyed. I suppose one support for this could be that the prophecy of the 70-year exile was already given to ירמיהו כ"ט:י. The גמרא must understand that it was due to the merit of the generation of the destruction that an expiry was put on the ensuing exile from the very beginning, whereas as no such favour was granted the second time.