On Cycle-Accuracy

The Lunar Cycle

We have a tradition that the average Molad interval is 29 days, 12 hours, and 793 chalakim, where there are 1080 chalakim in an hour.  Written differently, 29 days, 12 hours, 40 minutes, 73 chalakim (this is the way it is written in Pirkei d'Rebbi Eliezer).  

In Pirkei d'Rebbi Eliezer chapter 7 it is written:

The total of the days of the lunar year is 354 days, 8 hours and 876 chalakim.  All the hours of the lunar month are 708 hours and two thirds of an hour.  All the hours of a lunar year are 8504 hours.

That is to say that it gives the precise length of the lunar year, and then goes on to get imprecise.  First it drops the 876 chalakim, and then it reworks the calculations without the "pesky" chalakim.  If all the hours of a lunar month are 708 hours and 2/3 of an hour, then that means that a lunar month is 29 days, 12 hours, and 40 minutes.  That means that the imprecision is 73/1080 of an hour, or approximately 4 minutes.

The Solar Cycle

The year that we use for Birkat HaChamah is 365 days and 6 hours.  The difference between this and the solar year of Rav Adda is 365 days, 5 hours, and approximately 997 chalakim, or 365 days, 5 hours, 2/3 of an hour, and 277 chalakim.  The difference between the two is about 83 chalakim per year, or about 4.6 minutes per year.

In both places, we have inaccuracies on the "small" scale of less than 5 minutes per unit (lunar month or solar year). 

Birkat HaChamah Online Book

I've been writing a (short) book on the math behind Birkat HaChamah for the last year or so, and I have decided to start to make it public online with Google's new web publishing app.  I have 8 chapters, an appendix, and I've translated Pirkei D'Rebbi Eliezer chapters 6 and 7.  Please leave comments on this post if you have any.

Here's the link.

Update:  All parts of this book have been put online.

The Three Weeks

America's celebrated date marking the declaration of its independence, July 4th, 1776, was the 17th of Tamuz, 5536.  If we count forward until the 9th of Av using 12 days per year, we arrive at 5788, which is towards the end of a "day" which spans 5777-5789.

Birkat HaChamah - Mezarim

From my translation of chapter 6 of Pirkei d'Rebbi Eliezer:

And in 366 degrees (not 360), the Sun goes up and down, 183 it goes up in the East, and 183 it goes down in the West, corresponding to the days of the solar year. And in 366 windows, the Sun goes out, and it comes in the East, 91 days on the southern end, and 91 days on the northern end, and 1 window in the middle called the Nogah window. In Tekufat Tishrei, it (the sun) starts from the Nogah window and goes around to the southern end, window after window, until it arrives in the Shabbtai window. In Tekufat Tevet, it starts from the Shabbtai window and goes around and returns back, window after window, until it arrives to the Taalumah window, from where the light goes out, as it says, "And Taalumah brings out light" (Iyov, 28:11). In Tekufat Nisan, it starts from the Taalumah window, from where the light goes out, and it goes to the northern side, window after window, until it arrives in the Naamon window. In Tekufat Tamuz it starts from the Naamon window and goes around and returns back, window by window, until it reaches the Cheder window, from where the storm wind (ruach sufah) goes, as it says (I'm not going to translate the pasuk, but its Iyov 37:9).

In those [windows] that are in the East, it would go out, and correspondingly in the West, it comes in, and the Shechinah is always in the West. It comes in and prostrates itself before HaKadosh Baruch Hu, and says before Him, "Master of the Universe, I did all that you commanded of me".

The window in the middle of the Firmament, its name is Mezarim, and it (the sun) doesn't go out and enter in it aside from one time in the great cycle, that through (the window), it goes out on the day that it was created, in the night, in the west.

החלון שבאמצע הרקיע, שמו מזרים, ואינו יוצא ונכנס בו אלא פעם אחת של מחזור גדול, שבו יוצא ביום שנברא בלילה במערב

On the Significance of Birkat HaChamah

My online book regarding Birkat HaChamah is mostly complete, one chapter of translation is in editing.  Again, you can find it here.

Therein, I describe how the midrashim related to Birkat HaChamah serve as a cover for other cycles underneath it that help in understanding the course of our history.

There is another great theory that has been published, by Moshe Lerman, which shows that using these midrashim, we can actually determine approximately when and how the Jewish calendar was created.  You can find the article here.

Kaftor VaFerach ch. 51 on Yovel - English

On the matter of Yovel: The beginning of the count was from 14 years after they entered the Land.  From Yetziat Mitzraim to the building of Bayit Rishon is 480 years.  In the desert, 40 years, 14 years of conquering and division, therefore 54 years (after Yetziat Mitzraim was the beginning of the counting).  Take these (54 years) from the 480 years, and you end up with 426 years, and then Bayit Rishon was built.  Add this to the 410 years that it stood, and you get 836, and then it was destroyed.  800 years is 16 Yovels.  If so, it was during the 36th year of the 17th Yovel that it was destroyed.  Take the remaining 14 years from the 70 years (of Galut Bavel) and you have 56 years.  And the 420 years that (the 2nd Beit HaMikdash) stood, behold that is 9 Yovels, and you are left with 26 years from the 10th Yovel cycle.  If so, the Yovel took place 24 years after the Churban.  Take 24 years away from 172 (which complete 4000 years), and you are left with 148 years.  If so, the count of Yovels is from 148 years after the Churban.  For example, for our date that we have mentioned, add 148 and the complete 5th millennium years and 82 years, and you will have 1230 years.  1200 years is 24 Yovels.  And we are in the 30th year of the 25h Yovel, and thus on this pattern.  And Chazal have already said: Just like the Shmitah makes rest one year for every seven, thus the world will rest for one thousand years for every seven thousand years, as is brought in Perek Chelek (Masechet Sanhedrin)

Kaftor VaFerach ch. 51 on Yovel - Hebrew

ביובל:  תחילת המנין מי"ד שנה אחר שנכנסו לארץ.  כבר קדם שמיציאת מצרים עד הבנין ת"פ.  במדבר היו מ' וי"ד כבוד וחלוק, הרי נ"ד.  ואז התחילו למנות.  תגרעם מת"פ וישאר תכ"ו ואז נבנה בית ראשון.  חבר על זה ת"י מהבית ויהיו תתל"ו ואז נחרבה הראשונה.  ת"ת הם י"ו יובלות.  אם כן אל ל"ו מהיובל הי"ז נחרבה.  ואל י"ד אחריו היה יובל.  ולזה מאותם ע' שנה של גלות בבל אל י"ד מהם היה יובל.  הסר י"ד משבעים נשאר נ"ו.  ות"כ שעמדה הרי ט' יובלות וישאר כ"ו מהיובל העשירי.  אם כן, אל כ"ד שנה אחר החרבן היה יובל.  נגרע כ"ד מקע"ב וישאר קמ"ח.  ואם כן תחלת מנין ליובלות הוא מן קמ"ח אל החרבן.  דמיון:  לתאריכנו הנזכר, חבר קמ"ח ואלף החמישי שלם ופ"ב ויהיו אלף ור"ל.  אלף ור' הם כ"ד יובלות.  ואנו בשנת השלשים מיובל כ"ה, וכל על זה הדרך.  וכבר אמרו חכמים ז"ל:  כשם שהשמטה משמטת שנה אחת לשבע שנים, כך העולם משמט אלף שנה לשבעת אלפים וכדאיתא פרק חלק.

Source: part 1, part 2.

Haftarat Parah

A few words of Torah before my brother's second yartzheit on yom shlishi.

The haftarah for Parshat Parah has a few points that we should ponder:

• Yehezkel 36:20 - The goyim see Jews living in Chutz La'Aretz and they say to themselves that this is the nation of Hashem, and they have left His land.  This is considered a Chillul Hashem.  It does not refer to the religious level of the Jew in Galus.  The very existence of Jews being physically out of Eretz Yisrael is a Chillul Hashem.
• After Kibbutz Galuyot comes divine assistance in Teshuvah.  Verse 24 states that Hashem will gather us in, verse 25 states He will then sprinkle upon us clean water.  This is of course analagous to what is stated in chapter 30 of Devarim.  Am Yisrael will do some level of Teshuvah, and after Kibbutz Galuyot, Hashem will help us achieve he rest.

I have been learning the Perek Yomi cycle of Mishnayot for a year and a half now.  This morning's chapter begins with the words:

ארץ ישראל טהורה ומקוואותיה טהורין

Eretz Yisrael is pure, and its mikvaot are pure.

For a person to purify himself in a mikveh, he must submerge completely.  For Am Yisrael to become pure, we must all come home.