3×40=120

This observation draws attention to a recurring pattern in Moshe's life, with Chazal identifying it elsewhere as well

According to an opinion in the midrash rabbah (shemos 2:11) Moshe was 40 years old when he left the palace to see how his brothers were doing. 

Upon saving an israelite slave that was being brutalized he was taken aback upon realizing that some of his own people were intent on turning him in to the authorities as well as finally understanding why it is that they were still enslaved and not yet worthy of redemption (rashi to ibid 2:14 from various midrashim). 

Ultimately he needed to flee, spending the next 40 years in midyan as a fugitive. 

At the age of 80, after the folks that had it out for him no longer posed a threat, he was instructed to return to Egypt and free the people. He proceeded to do just that, spending the last 40 years of his life shepherding the people through the wilderness and bringing them to the doorstep of the promised land. 

He died there at the age of 120, tragically barred from entering. 

One can view his life as comprised of three segments of 40 years each. 

At the end of the first period he went out to attempt to protect his brothers and potentially liberate them only to be given a rude awakening that they were most definitely not yet ready for that. 

The next 40 years can be viewed as the Israelites rehabilitation period, ending with no more would-be informers amongst the people, presumably speaking to a newly acquired spirit of unity and solidarity. That in turn unlocked the geulah from mitzrayim with the next forty spent traveling to the land of milk and honey. 

Theres another 120 unit period in Moshe's life that is also broken up into segments of 40 units each, the 120 days from kabalas hatorah until yom kipur. 

There too the first segment ended in an attempt at giving klal yisroel the luchos, ultimately aborted, due to their unworthiness. 

The next 40 days can also be conceived of as rehabilitatory in nature, with the Israelites purging themselves of the sinners and Moshe engaging in prayerful supplication, ending with HaShem acquiescing to dwell amongst them, as the yoshev hakeruvim, with a new pair of luchos beneath His presence, albeit only after Moshe spend an additional 40 days on the mount.

This time, too, it met with success. 

Seems like an interesting correlation. 

R Yosef bchor shor, commenting on (devarim 9:24) מַמְרִ֥ים הֱיִיתֶ֖ם עִם⁠־י"י֑ מִיּ֖וֹם דַּעְתִּ֥י אֶתְכֶֽם, which is ostensibly referring back to all the previously recorded sins from the eigel onward (ibid pesukim 7-23), writes: מיום דעתי אתכם – תחילה שכשאמרתי לרשע: למה תכה רעך (שמות ב':י"ג), עמד והלשין עלי, connecting this statement of Moshe's to that first time he witnessed their misdeeds. (See Radatz Hoffman ad loc מיום דעתי – מקביל למה שנאמר בפסוק ז למן היום אשר יצאת וגו'⁠, והרי משה רבינו יודע את העם עוד לפני יציאת מצרים, על כן אנחנו צריכים לפרש כאן "מיום דעתי" כמו "מן היום שהייתי הפרנס שלכם", השוה משלי כ"ז:כ"ג, with Ri bchor shor rendering it literally)

Chazal located other instances of this pattern of 3*40, see the following sifri (devarim 34:7)

ומשה בן מאה ועשרים שנה – זה אחד מארבעה שמתו בן מאה ועשרים, ואלו הם: משה, והלל הזקן, ורבן יוחנן בן זכאי, ורבי עקיבא. משה היה במצרים ארבעים שנה ובמדין ארבעים שנה ופירנס את ישראל ארבעים שנה. הלל הזקן עלה מבבל בן ארבעים שנה ושימש חכמים ארבעים שנה ופרנס את ישראל ארבעים שנה. רבן יוחנן בן זכאי עסק בפרגמטיא ארבעים שנה ושימש חכמים ארבעים שנה ופירנס ישראל ארבעים שנה. רבי עקיבא למד תורה ארבעים שנה ופירנס את ישראל ארבעים שנה

I dont know if theres a deeper significance and nexus to all this but I think that at the very least it is quite interesting

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