For an explanation of the methodology of this series, see the introduction.

"Please let a little water be taken and bathe your feet and recline under the tree." (Gen. 18:4)

Peshat (basic meaning):

G‑d rewarded his [Abraham’s] children through a messenger...

Rashi: "Please let be taken"
through a messenger, and G‑d rewarded his [Abraham’s] children through a messenger, as it is said: "And Moses raised his hand, and he struck the rock." (Num. 20:11)

Siftei Chachamim: Just as Abraham brought water by means of a messenger, so too did his progeny had water brought to them through a messenger. But Abraham himself brought the guests bread, just like G‑d did to his progeny, as it says "I will rain bread down upon you from Heaven." (Ex. 16:4)

"and bathe your feet"
He thought that they were Arabs, who prostrate themselves to the dust of their feet, and he was strict not to allow any idolatry into his house. But Lot, who was not strict, mentioned lodging before washing, as it is said: (below 19:2) "and lodge and bathe your feet."

Remez (hinted meaning):

Baal HaTurim: "Please let a little water be taken"
The gematria of this phrase is equivalent (after adding 1 for the phrase itself) to the phrase "Mikal Zaku LaBe'er". From here [i.e. because Abraham said this] they merited the well [that accompanied them in the wilderness]. See also Baba Metzia 86b: in reward for three things Abraham did for his guests, his descendants merited 3 things: for the cream and milk, they merited the Mannah, for "he stood over them (in Hebrew, 'Omaid')", they merited the pillar (in Hebrew, 'Amud') of clouds, and in reward for 'let some water be taken," they merited the well of Miriam.

Derash (interpretive meaning):

Targum Yonatan: that the glory of Your Shechinah may not now ascend from Your servant, until I have set forth provisions under the tree.

Ohr HaHayyim: Abraham was hinting to the angels that the water he was speaking of was a reference to the Torah. When an angel assumes human form, his "body" is compared to the "foot". Abraham hinted to the angels that they should take some of the Peshat/simple meaning of the Torah, i.e "water" seeing that they were presently in human form. They could then rest "under THE Tree" the metaphor for the Torah, which was the Eitz Hayyim/Tree of Life. He referred to it as "THE Tree", though it was not previously featured in our story. [See more:

Maggid Mesharim: "Please let a little water be taken"
to give some water of chesed/kindness to wash and to rid themselves of the unclean kelipot from their feet, meaning from Kenesset Yisrael, the Congregation of Israel.

Abraham's choice of words reflects great sensitivity.

Rav Shalom Shvadron: Like a sparkling diamond, Abraham's heart glistened with sublime loving-kindness for his fellow man. When three desert travelers neared his camp, Abraham ran across the blazing sands to invite them for a sumptuous feast. During the course of his welcoming words he said to his guests, "Please let a little water be brought to wash your feet." But in light of Abraham's love of kindness, a question arises: Why did he say "Let a little water be brought?" The connotation of "a little water" seems inconsistent with his unbounded generosity.

However, in truth Abraham's choice of words reflects great sensitivity. When washing the feet of a guest it would be discourteous to request "a large amount of water be brought." For a large amount of water would imply that his guests' feet were terribly dirty. Whereas "a little water" suggests that the feet of his guests were basically clean; and just a little water was required to remove the bits of dust on their feet. Abraham carefully chose his words to avoid even the slightest embarrassment. Even more, his words gladdened the hearts of the travelers, for Abraham's words implied that his guest were neat, clean, and well-mannered. (Lev Shalom)

Sod (esoteric, mystical meaning):

Zohar Vayera 102:
"Let a little water, I pray you, be fetched..." to strengthen himself with that level which contains water.
This is how he purified people from all sins, including those from the impure side and idol worshiping. And just as he purified the men, so did she purify the women. Therefore, all those who came to him were completely purified. this tree, he knew who was attached to G‑d...
Come and see: wherever Abraham lived, he planted a tree. But it did not grow properly in all the places; it only grew properly when he lived in Cana'an. And by this tree, he knew who was attached to G‑d and who worshiped idols. For if a person cleaved to G‑d, the tree spread out its branches, covered his head, and formed a pleasant shade for him. But if a person was attached to idolatry, the tree raised its branches high. Then Abraham knew. Abraham reprimanded him and did not let him go until he cleaved to faith in G‑d!

In addition, whoever was pure was accepted by the tree. But whoever was impure was not accepted. Abraham then knew. If this was the case, he purified him with water. And there was a spring of water beneath the tree. If a person needed an immersion, the water immediately rose and the branches of the tree ascended upward. Abraham thus knew that he was impure and had to be immersed in water immediately. But if a person did not need to be cleansed immediately, the spring dried up. Then Abraham knew that he was still impure and needed to wait for seven days.

Come and see: even when he invited the angels, he told them to rest "yourselves under the tree," in order to test them. In this way, he examined every person. And the secret is that he said this for the sake of the Holy One, blessed be He, who is the Tree of Life for everyone. This is why he said " and rest yourselves under the tree" and not under idol worshiping.

BeRahamim LeHayyim:
The numerical value of this word for tree, "Ilan," as we know from the intentions of Tu BeShevat, the New Year of THE Tree [Ilan-singular, not Ilanot/"Trees"] is 91, and 91 is a profound number according to the Kabbalah that hints to the unification of the Holy One blessed be He and His Divine Presence, the Shechinah.

So under THE Tree (representing the Tree of Life) is where Abraham did the mitzvah of hospitality. Even though we are told that the angels "only pretended to eat," nevertheless tremendous Kavanot/devotions are attributed to the "eating" done under the tree. I look forward every year to this portion to be able to renew my learning of them, many which relate to the letter Alef. For perhaps this eating under the tree related to the first, the eating under THE Tree, the sin of which we intend to fix with all of our eating at all times, not just on Shabbat and on Pesach when the aforementioned meditations are recommended.
If all our eating is done correctly, we really are eating under THE Tree...of Life...
If all our eating is done correctly, we really are eating under THE Tree. not a Tree of Knowledge of Good and Evil, but rather a Tree of Life for all who hold fast to it. Now couple this eating with some Torah study (She is a Tree of Life), and add to this the hosting of guests (mitzvah), and we are most certainly uniting the Holy One Blessed Be He and His Shechinah/Divine Presence in perfect unity, for all Israel. Amen.
Perhaps that is why the simple explainer—Rashi —intended when he mentioned the Aramaic direct translation of the Torah, hinting to THE Tree, and all its tremendous significance.

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