How many camps is the Parah Adumah taken out from?

The passuk in this week’s sedrah says that the Parah Adumah had to be burnt outside of the camp:
זֹאת חֻקַּת הַתּוֹרָה אֲשֶׁר צִוָּה ה' לֵאמֹר דַּבֵּר אֶל בְּנֵי יִשְׂרָאֵל וְיִקְחוּ אֵלֶיךָ פָרָה אֲדֻמָּה תְּמִימָה אֲשֶׁר אֵין בָּהּ מוּם אֲשֶׁר לֹא עָלָה עָלֶיהָ עֹל. וּנְתַתֶּם אֹתָהּ אֶל אֶלְעָזָר הַכֹּהֵן וְהוֹצִיא אֹתָהּ אֶל מִחוּץ לַמַּחֲנֶה וְשָׁחַט אֹתָהּ לְפָנָיו.
“This is the statute of the Torah that Hashem has commanded; Speak to the benei yisrael and they should bring to you a perfectly red cow that has no blemish and upon which no yoke has been placed. You should give it to Elazar the cohen and he should take it outside the camp and he shouldshecht it there before him.”

The mishna is Parah (3:6) explains that they used to take the Parah Adumah outside Yerusholayim in order to burn it:
וכבש היו עושים מהר הבית להר המשחה...שבו כהן השורף את הפרה, ופרה וכל מסעדיה יוצאין להר המשחה.
“They used to make a ramp from the Har ha’Bayis to Har ha’Mishchah (Har ha’Zeisim) …on which the cohen who burnt the Parah Adumah and the Parach Adumah and all its attendants would go to Har ha’Mishchah.”

The simple explanation of this mishna is that the Parah Adumah had to be taken out of all three camps; the machaneh shechina, the machaneh leviyah and the machaneh yisrael. Accordingly, Rashi and Tosafos (Zevachim 105a) say that if the parah adumah was not shechted outside all three camps then it is passul.

However, the Rambam says in Hilchos Parah Adumah (3:1);
אין שורפין את הפרה אלא חוץ להר הבית שנאמר והוציא אותה אלא מחוץ למחנה ובהר המשחה היו שורפין אותה
“You can only burn the Parah Adumah outside of the Har ha’Bayis as the passuk says “והוציא אותה אלא מחוץ למחנה”, and they used to burn it onHar ha’Mishchah.”

It seems that according to the Rambam, the Parah Adumah only needs to be taken outside of two camps (the beis ha'mikdash and the Har ha’Bayis) although the minhag was to burn it outside of all three camps, outside of Yerusholayim.

Why does the Rambam argue on Rashi and Tosafos?

The Chazon Ish explains as follows;
The mishna in Zevachim (104a) says;
פרים הנשרפים ושעירים הנשרפים ... סובלין אותו במוטות יצאו הראשונים חוץ לחומת העזרה והאחרונים לא יצאו הראשונים מטמאין בגדים והאחרונים אינן מטמאין בגדים עד שיצאו יצאו אלו ואלו מטמאין בגדים רבי שמעון אומר אינן מטמאין עד שיוצת האור ברובן...
“[Korbanos which have their blood sprinkled in the heichal are taken outside of Yerusholayim to be burnt once their avodah is complete. There is agezeras hakasuv that the people who carry these korbanos outside Yerusholayim become tameh.] They used to carry them on poles from the beis hamikdash to the place that they were burnt. Once the people carrying the front of the poles went out of the beis hamikdash even if the people at the back had not yet gone out, the people in the front became tameh and the people at the back remained tahor. Once they both went out they both became tameh. Rebbi Shimon says that they do not become tameh simply by carrying the korban outside of the beis hamikdash, they only becametameh once the fire had taken hold of the majority of the carcass of the korban.”

The gemara (105b) explains:
יצאו אלו ואלו: מה"מ דת"ר להלן הוא אומר חוץ לג' מחנות כאן למחנה אחת לומר לך כיון שיצא חוץ למחנה אחת מטמא בגדים ...ור"ש האי מחוץ למחנה מאי עביד ליה מיבעי ליה לכדתניא רבי אליעזר אומר נאמר כאן חוץ למחנה ונאמר להלן (במדבר יט, ג) חוץ למחנה מה להלן חוץ לג' מחנות אף כאן חוץ לג' מחנות ומה להלן למזרחה של ירושלים אף כאן למזרחה של ירושלים אלא לרבנן היכא שריף להו כדתניא היכן נשרפין לצפון ירושלים חוץ לשלש מחנות ...

“In the case of the par heelam davar shel tzibbur and the par cohen hamashiach (which are burnt) the passuk says that you have to take them outside of three camps, but in the case of the par ve’sair shel yom ha’kippurim the passuk only says that it has to be taken outside of one camp(“וְאֵת פַּר הַחַטָּאת וְאֵת שְׂעִיר הַחַטָּאת אֲשֶׁר הוּבָא אֶת דָּמָם לְכַפֵּר בַּקֹּדֶשׁ יוֹצִיא אֶל מִחוּץ לַמַּחֲנֶה וְשָׂרְפוּ בָאֵשׁ אֶת עֹרֹתָם וְאֶת בְּשָׂרָם וְאֶת פִּרְשָׁם”)? This teaches you that as soon as the korban goes outside of one camp it makes the person carrying it tameh.

Rebbi Shimon argues with this halacha, so what does he use the passuk of the par ve’sair shel yom ha’kippurim for?

…It says in the par ve’sair shel yom ha’kippurim חוץ למחנה and it says in the parah adumah חוץ למחנה, just as the par ve’sair shel yom ha’kippurim have to be burnt outside of three camps so too the parah adumah has to be burnt outside of three camps and just as the parah adumahhas to be burnt to the east of Yerusholayim so too the par vesair shel yom hakippurim have to be burnt to the east of Yerusholayim.”

But according to the chachamim (who do not make a gezerah shaveh between the parah adumah and the par ve’sair shel yom hakippurim) where are the par ve’sair shel yom ha’kippurim burnt? To the north of Yerusholayim outside of the three camps…”

We can see from this gemara that there is a machlokess whether or not we make a gezerah shaveh between the par ve’sair of yom kippur and theparah adumah. Rebbi Shimon says that we compare them to each other and therefore the parah adumah has to be burnt outside of the three camps whereas the chachamim say that we do not compare them.

It would seem from this gemara therefore that according to the chachamim the parah adumah only has to be burnt outside of one camp and we can explain that the Rambam paskens like the chachamim.

Nevertheless, the Rambam is still difficult  to understand. The Rambam says in Hilchos Beis haBechirah (perek 7);
שלש מחנות היו במדבר מחנה ישראל .. ומחנה לויה שנאמר בה וסביב למשכן יחנו ומחנה שכינה והוא מפתח חצר אהל מועד ולפנים. וכנגדן לדורות מפתח ירושלים עד הר הבית כמחנה ישראל ומפתח הר הבית עד פתח העזרה שהוא שער ניקנור כמחנה לויה ומפתח העזרה ולפנים מחנה שכינה...
“There were three camps in the midbar, the machaneh yisrael, the machaneh leviyah and the machaneh shechinah. Correspondingly to these for all generations, from the entrance of Yerusholayim till the Har ha’Bayis is like the machaneh yisrael, from the entrance of the Har ha’Bayis till the entrance to the azarah is like the machaneh leviyah, and from the entrance of the machaneh leviyah and within is the machaneh shechinah

If according to the Rambam the parah adumah only has to be taken outside one camp then why does the Rambam say that it has to be taken outside of the har habayis which is outside the machaneh leviyah? It should be sufficient to take it out of the beis hamikdash and burn it even on the har habayis?

Perhaps we can explain the Rambam as follows. The passuk says;
...ויקחו אליך פרה אדמה תמימה ...ונתתם אתה אל אלעזר הכהן והוציא אתה אל מחוץ למחנה...
“Hashem told Moshe, ‘Speak to the benei yisroel and they will bring you a parah adumah .. and you should give it to Elazar and he should take it outside of the camp…”

In all other cases where the Torah says that you have to take the korban מִחוּץ לַמַּחֲנֶה , the Torah is referring to a korban that has been brought to thebeis hamikdash. Therefore the starting point for 'going out' is the machaneh shechinah.

In the case of the Parah Adumah, the Torah says that the parah adumah should be given to Elazar and then should be taken out of one camp (according to the Rambam). Seeing as there is no reason for Elazar to be in the machaneh shechinah, we can assume that he was in the place that he lived which was the machaneh leviyah. Therefore if he takes it outside of one camp he will be in machaneh yisrael.

That is why the Rambam says that it is sufficient meikar hadin for the parah adumah to be taken outside the har habayis, because this is one camp outside of the machaneh leviyah.

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