Bamidbar: How the sum of two censuses can be identical

(All of the following expanding upon Shadal.)

The count in parshas Bemidbar lists each tribe and the amount of people in that tribe. Then, in Bemidbar 1:45-47:

מה וַיִּהְיוּ כָּל-פְּקוּדֵי בְנֵי-יִשְׂרָאֵל, לְבֵית אֲבֹתָם, מִבֶּן עֶשְׂרִים שָׁנָה וָמַעְלָה, כָּל-יֹצֵא צָבָא בְּיִשְׂרָאֵל.	45 And all those that were numbered of the children of Israel by their fathers' houses, from twenty years old and upward, all that were able to go forth to war in Israel;
מו וַיִּהְיוּ, כָּל-הַפְּקֻדִים--שֵׁשׁ-מֵאוֹת אֶלֶף, וּשְׁלֹשֶׁת אֲלָפִים; וַחֲמֵשׁ מֵאוֹת, וַחֲמִשִּׁים.	46 even all those that were numbered were six hundred thousand and three thousand and five hundred and fifty.
מז וְהַלְוִיִּם, לְמַטֵּה אֲבֹתָם--לֹא הָתְפָּקְדוּ, בְּתוֹכָם. {פ}	47 But the Levites after the tribe of their fathers were not numbered among them. {P}

Thus, not including the Levviim, there were 605,350.

The Leviim are counted in the third perek, from the age of one month and up:

לט כָּל-פְּקוּדֵי הַלְוִיִּם אֲשֶׁר פָּקַד מֹשֶׁה וְאַהֲרֹן, עַל-פִּי ה--לְמִשְׁפְּחֹתָם: כָּל-זָכָר מִבֶּן-חֹדֶשׁ וָמַעְלָה, שְׁנַיִם וְעֶשְׂרִים אָלֶף. {ס}

39 All that were numbered of the Levites, whom Moses and Aaron numbered at the commandment of the LORD, by their families, all the males from a month old and upward, were twenty and two thousand. {S}

and number 22,000, though when summing up the numbers for Gershon (7500), Kehat (8600) and Merari (62000) earlier in the perek, there are an additional 300. (Rashi cites Bechorot 5a to explain that these additional 300 were themselves firstborn and thus

These were in place of the firstborn, who were approximately the same amount -- there were 273 more Israelite firstborn than the count of Levites.

מה קַח אֶת-הַלְוִיִּם, תַּחַת כָּל-בְּכוֹר בִּבְנֵי יִשְׂרָאֵל, וְאֶת-בֶּהֱמַת הַלְוִיִּם, תַּחַת בְּהֶמְתָּם; וְהָיוּ-לִי הַלְוִיִּם, אֲנִי יְהוָה.	45 'Take the Levites instead of all the first-born among the children of Israel, and the cattle of the Levites instead of their cattle; and the Levites shall be Mine, even the LORD'S.
מו וְאֵת פְּדוּיֵי הַשְּׁלֹשָׁה, וְהַשִּׁבְעִים וְהַמָּאתָיִם--הָעֹדְפִים, עַל-הַלְוִיִּם, מִבְּכוֹר, בְּנֵי יִשְׂרָאֵל.	46 And as for the redemption of the two hundred and three score and thirteen of the first-born of the children of Israel, that are over and above the number of the Levites,
מז וְלָקַחְתָּ, חֲמֵשֶׁת חֲמֵשֶׁת שְׁקָלִים--לַגֻּלְגֹּלֶת: בְּשֶׁקֶל הַקֹּדֶשׁ תִּקָּח, עֶשְׂרִים גֵּרָה הַשָּׁקֶל.	47 thou shalt take five shekels apiece by the poll; after the shekel of the sanctuary shalt thou take them--the shekel is twenty gerahs.

Now for the difficulty. In Shemot 38:26, by the donations to the Mishkan at its construction:

כה וְכֶסֶף פְּקוּדֵי הָעֵדָה, מְאַת כִּכָּר; וְאֶלֶף וּשְׁבַע מֵאוֹת וַחֲמִשָּׁה וְשִׁבְעִים, שֶׁקֶל--בְּשֶׁקֶל הַקֹּדֶשׁ.	25 And the silver of them that were numbered of the congregation was a hundred talents, and a thousand seven hundred and three-score and fifteen shekels, after the shekel of the sanctuary:
כו בֶּקַע, לַגֻּלְגֹּלֶת, מַחֲצִית הַשֶּׁקֶל, בְּשֶׁקֶל הַקֹּדֶשׁ--לְכֹל הָעֹבֵר עַל-הַפְּקֻדִים, מִבֶּן עֶשְׂרִים שָׁנָה וָמַעְלָה, לְשֵׁשׁ-מֵאוֹת אֶלֶף וּשְׁלֹשֶׁת אֲלָפִים, וַחֲמֵשׁ מֵאוֹת וַחֲמִשִּׁים.	26 a beka a head, that is, half a shekel, after the shekel of the sanctuary, for every one that passed over to them that are numbered, from twenty years old and upward, for six hundred thousand and three thousand and five hundred and fifty men.

So here there was also a count and donation, and the sum total of the people was once 603,550. This makes sense, since these two counts were in close. As Rashi writes there:

for six hundred three thousand, etc. This is how many the Israelites were. Their number equaled this [too] after the Mishkan was erected, [as appears] in the Book of Numbers [Num. 1:4]. Now too, when they donated to the Mishkan, this is how many they were. The number of the half-shekels of 600,000 [people] equals one hundred talents, each one the equivalent of three thousand shekels. How so? Six hundred thousand halves [of a shekel] equal three hundred thousand wholes, which equal one hundred talents. The [additional] 3,550 halves equal 1,775 shekels.

Shadal is bothered by this, for even though these two counts were in close proximity, they could not be the same census. For here the Levites were not counted, while there there is no reason to say they were not counted, for the Levites were not at that point separated off, nor were they mentioned at all in all the Torah. And furthermore, this count is to families, to beit avot (larger groups headed by a paterfamilias) in order to know how many are in each tribe, to make the encampments by flags, while there (in Shemos) each person gave a beka of silver, and the silver was counted without paying heed to the specifics of each and every tribe.

And, if in the count in Shemos the Levites were counted and in the one in the beginning of sefer Bemidbar Sinai they were not counted, how could both sums be 603,550?

My {=Josh} own answer would be: nu, nu -- Hashem knew what would be in the future, and so there as well He said not to take from the Leviim, and perhaps people even wondered about this until the matter was clarified later.

Shadal's answer is that we see that the Levites were basically the same number as the firstborn Israelites -- there were only 273 more firstborns. And those firstborns at that point had the status that the Levites would later take, and since they served bodily in the Mishkan they would not donate their money. And it is not so farfetched that 273 people from all of Israel died in the period between the two counts.

If so, the first (Shemos) count is the entire nation, including the Levites but not including the firstborn.
And the second (Bemidbar) count is the nation excluding the Israelite firstborn and excluding the 273 who died in the meantime.

Thus, the counts end up identical.

(I {=Josh} would suggest that the round numbers for the various tribes suggests that these were approximates, such that deaths and people entering their 20th year in the meantime can vary from tribe to tribe and still end up with the same total approximate.)

Now there are actually two opinions about how age is determined. According to Rashi on Shemot 30:16, age in years was determined on Rosh haShana, in Tishri. Thus, everyone becomes one year older on the same day. If so, there is no problem of someone all of a sudden becoming 20 in between the first and the second count.

Rashi writes:

and use it for the work of the Tent of Meeting [From this] you learn that they were commanded to count them at the beginning of the donation for the Mishkan after the incident of the calf. [They were commanded then] because a plague had befallen them, as it is said: “And the Lord plagued the people” (Exod. 32:35). This can be compared to a flock of sheep, treasured by its owner, which was stricken with pestilence. When it [the pestilence] was over, he [the owner] said to the shepherd, “Please count my sheep to know how many are left,” in order to make it known that he treasured it [the flock] (Tanchuma, Ki Thissa 9). It is, however, impossible to say that this counting [mentioned here] was the [same] one mentioned in the Book of Numbers, for in that one [counting] it says: “on the first of the second month” (Num. 1:1), and the Mishkan was erected on the first [day] of the first month, as it is said: On the day of the first month, on the first of the month, you shall erect, etc. (Exod. 40:2). The sockets were made from shekels realized from that counting, as it is said: “One hundred talents of the silver were used to cast, etc.” (Exod. 38:27). Thus you learn that they [the countings] were two-one at the beginning of their donation [to the Mishkan] after Yom Kippur in the first year [after the Exodus], and one in the second year in Iyar after the Mishkan had been erected. Now if you ask, how is it possible that in both of these countings the Israelites equaled six hundred three thousand, five hundred fifty? In the case of the silver of the community census, it says this number, and also in the Book of Numbers it says the same: “And all the counted ones were six hundred three thousand, five hundred fifty” (Num. 1:46). Were they [the countings] not in two [separate] years? It is impossible that in the first census there were none who were nineteen years old and consequently not counted, and by the second counting became twenty years old [and were counted]. The answer to this matter is that in the context of the ages of people, they were counted in the same year, but in the context of the Exodus they [the two dates] were two [separate] years, since [to figure the time] from the Exodus, we count from [the month of] Nissan, as we learned in [tractate] Rosh Hashanah (2b). In this context, the Mishkan was built in the first year [after the Exodus] and erected in the second year, for the new year started on the first of Nissan. People’s ages, however, are counted according to the number of years of the world, beginning with [the month of] Tishri. Thus, the two countings were [taken] in the same year. The first counting was in Tishri after Yom Kippur, when the Omnipresent was placated toward Israel to forgive them, and they were commanded concerning [building] the Mishkan. The second one [counting] was on the first of Iyar. -[from Num. Rabbah 1:10]

But Shadal also offers an answer according to Ramban that age is determined from the day of one's birth, along similar lines. See inside -- there is more to it.

Here is the text of Shadal, taken from the website:

הסך הזה מסכים בצימצום עם הסך היוצא מכסף פקודי העדה בשנה ראשונה קודם שהוקם המשכן ( שמות ל"ח כ"ו ועיין שם בנתיבות השלום שהאריך), והנה אי אפשר לומר ששני המנינים מנין אחד הם, כי כאן לא נמנו הלויים, ושם אין שום טעם לומר שלא נמנו, כי עדיין לא נבחרו ולא נבדלו ולא נזכרו הלויים כלל בכל התורה, ועוד כי המנין הזה הוא למשפחותם לבית אבותם , לדעת מזה כמה בכל שבט לעשות הדגלים, ושם לא היה הדבר כן, רק נתן איש איש בקע כסף ונמנו השקלים בלא השקפה לפרטי כל שבט ושבט, כי זה היה המבוקש במנין השני הזה. אם כך ששני מנינים הם, ובראשון נמנה שבט לוי ובשני לא נמנה, איך הסכימו שני הסכים האלה זה עם זה? אומר אני כי כשם שלא נמנו הלויים במנין השני הזה, מפני שנבחרו לשירות המשכן תחת כל בכור בבני ישראל, כן במנין הראשון לא נמנו הבכורות, כי עדיין היו ניגשים אל ה' לשרת בקודש. והנה מספר הבכורות היה קרוב למספר הלויים ויותר עליהם רע"ג ( למטה, ג' מ"ו ), ובכן איננו רחוק שקרה מקרה שהמתים בין מנין למנין היו ג"כ רע"ג, והנה נשאר המנין השני שווה לראשון, כי המנין הראשון היה העם בזולת בכורות, והמנין השני היה העם בזולת הלויים וזולת המתים. ומספר הלויים עם המתים הוא (לפי ההנחה) שווה למספר הבכורות. וכל זה בשיטת רש"י ( שמות ל' ט"ז ) ובמדבר רבה (פ א ח) (עיין נתה"ש שם) שלמנין שנות בני אדם מונין מתשרי, ובכן לא נוספו במנין השני הנערים אשר השלימו שנת עשרים, אבל לשיטת רמב"ן ( שמות ל' י"ב ) שלמנין שנות בני אדם איש איש ראש השנה שלו ביום היוולדו, ואם כן בהכרח הרבה נוספו במנין השני שלא היו בני עשרים במנין הראשון, יש לומר, שאע"פ שהבכורות כשנמנו מבן חודש ומעלה היו יתרים על הלויים, מ"מ בהימנותם מבן עשרים היה מספרם פחות מהם, כמו שהיה העניין במשפחות הלויים, שכשנמנו מבן חודש ( למטה פרק ג') היו בני גרשון 7500 ובני קהת 8600 ובני מררי 6200, בני מררי פחות מכולם, וכשנמנו מבן שלושים עד חמישים שנה היו בני גרשון 2630 ובני קהת 2750 ובני מררי 3200, בני מררי מרובים מכולם; אף כאן אולי הבכורות שהיו מרובים מן הלויים, מבן חודש היו פחותים מהם מבן עשרים, ושנכנסו במנין שני במקום הלויים, יצא שכר המשלימים שנת עשרים בהפסד מה שהיו הבכורות פחותים מן הלויים ובהפסד המתים בין ספירה לספירה, ונשאר המספר כשהיה, בלא תוספת ובלא מגרעת, כזה:

מנין ראשון מנין שני

ישראל ישראל

לויים בכורות

X אותם שמתו בין מנין למנין
Z אותם שהשלימו שנת עשרים בין מנין למנין

Y יתרון הלויים על הבכורות ( Z=X+Y ).

ואם תרצה, אמור שהבכורות כשנמנו מבן עשרים היה מספרם שווה למספר הלויים מבן עשרים, ושהמתים בין מנין למנין היה מספרם שווה למספר הנערים המשלימים שנת עשרים בין מנין למנין . וזה אמנם במקרה לא בנס, כי יהיה נס ללא דבר וללא תועלת כלל. ואל תתמה איך לא נמנו הבכורות במנין ראשון מבלי שתזכיר התורה דבר מזה, כי הלא כן תצטרך לומר בלויים במנין שני, שהיו בהם ש' בכורות שלא היו מן המנין ( עיין למטה ג' ל"ט בפי' רש"י). ואמנם הטעם האמיתי למה לא נמנו הבכורות במנין הקודם ההוא, כי שם לא היתה הכוונה למנות את העם, אלא שייתן איש תרומת כסף לצורך המשכן, והנה הבכורות שהיו עובדים עבודת ה' לא היה ראוי שייתנו מכספם להקמת המשכן, כי כבר הם וגופם נתונים אל המשכן ולכל עבודתו.