Why did Avraham instruct Eliezer to place his hand underneath his thigh?
Rashi: Because an oath requires the person swearing to hold an object of Mitzvah (such as a Seifer-Torah or Tefilin), and the B?ris Milah, which was was particularly dear to him, since it was the first Mitzvah that he had been commanded, and it came through pain,.
Rashbam: It is an act of subordination which we only find performed by an Eved to a master, or a son to a father. 1
Ha'amek Davar: It was to make a covenant with him, in addition to the Shevu'ah.
See Bereishis 47:29. Others who take an oath, tend to clinch it with a handshake or cutting something into two and passing between the pieces (as we learned in connection with the B?ris bein ha'Besarim, in Bereishis 15.
What is "ha'Moshel b'Chol Asher Lo"?
Yoma 28b: 'He ruled over his master's Torah - This teaches us that the slave of Avraham was an elder who sat and studied Torah.' 1
Lev Eliyahu (Bereishis, p. 69): He ruled over his Yetzer like Avraham. 2
Peninim mi'Shulchan ha'Gra: The terms "Katzin, Shoter, u'Moshel" are three kinds of appointments. Katzin teaches Chochmah and gives rulings in Yisrael, Shoter enforces the Halachah in practice, and Moshel is appointed over Midos. Eliezer is called "Avdo" (Katzin) "Zekan Beiso" (Shoter) "ha'Moshel." Also, the word 'DaMeSeK' 3 alludes to Moshel, Shoter, Katzin.
Ramban, Ha'amek Davar (from Sechel Tov): He was an overseer over Avraham's property.
One may not touch the Ever, lest it cause emissions (Nidah 13b)!
Moshav Zekenim #1: This applies only to youths, but not to old men.
Moshav Zekenim #2: It is forbidden only past the crown. Eliezer held it before the crown.
Ibn Ezra, Radak, and Chizkuni: He put his hand under the thigh, not on the Ever.
QUESTIONS ON RASHI
Rashi writes: "'Under my thigh' - for one who swears must hold an object of a Mitzvah." Why didn't Eliezer swear by his own Milah (all male members of Avraham's household had been circumcised - Bereishis 17:23)?
Oznayim la'Torah: Because Avraham's own Mitzvah was dearer to him than Eliezer's.
Gur Aryeh: According to the Halachah , one who swears must take hold of an object (Nekitas Chefetz), to which end part of one's own body would not qualify.
Rashi writes: "The Milah was the first Mitzvah that Avraham had been commanded... and it was dear to him." But Milah was also the only Mitzvah Avraham had been commanded! Why does Rashi add this extra reason?
Gur Aryeh: Rashi wishes to explain why Avraham used a Mitzvah object, when it would have been preferable to use an object with sanctity (Cheftzei Kedushah), such as Tefilin. 1 Avraham preferred to swear by the Milah, because it was the first Mitzvah commanded to him (even prior to the Akeidah), and it came through pain.
See Shevuos 38b.
Rashi writes: "The Milah... came through pain, and it was dear to him." But this reason would not explain why Yaakov later made Yosef swear by the Milah (47:29)?
Gur Aryeh: Even among the nations, the party taking an oath would put his hand under the thigh of the one administering it, showing subservience (Ibn Ezra). Yaakov needed to do this so that Pharaoh would later accept Yosef's oath as binding. But Avraham had no reason to follow the custom of the nations; rather he did so because of the Mitzvah of Milah.