Ki Savo: Devarim 26:10. Bowing During Tefilla

I
Tanchuma at the beginning of our Parsha:

היום הזה ה' אלהיך מצוך לעשות. זה שאמר הכתוב, בואו נשתחוה ונכרעה נברכה לפני ה' עושנו (תה' צה ו). והלא כריעה בכלל השתחויה, והשתחויה בכלל כריעה, ומה תלמוד לומר, נשתחוה נכרעה נברכה. אלא צפה משה ברוח הקודש וראה שבית המקדש עתיד ליחרב והבכורים עתידין ליפסק, עמד והתקין לישראל שיהיו מתפללין שלשה פעמים בכל יום, לפי שחביב תפלה לפני הקדוש ברוך הוא מכל מעשים טובים ומכל הקרבנות

The Tanchuma says that the words היום הזה ה' אלהיך מצוך לעשות, This day Hashem commands you to do.... imply that the mitzva of Bikkurim is so fundamental that it should be available to Klal Yisrael forever, on every day, in all times.  Moshe Rabbeinu, foreseeing the destruction of the Bais Hamikdash and the interruption of the Bikkurim ceremony, instituted that we say Tefilla three times a day.

The Tanchuma focuses on the passuk in Tehillim that refers to bowing down to Hashem in two ways, hishtachava'a and kri'a, and sees in the repetition an allusion to an alternative to, a replacement for, the Bikkurim.  The way the Tanchuma is structured is as follows:  An allusion to implied eternity in the words Hayom Hazeh; A reference to the reiteration of the word Bowing in the passuk in Tehillim; And only then the Tanchuma explains that Tefilla reflects the Bikkurim and enables us to experience, even without a Beis Hamikdash, the experience of Bikkurim.  It is interesting to consider what it is about the hishtachava'a in Bikkurim that makes it so fundamental, and how that vital message is expressed in our Tefilla.  (This section was inspired by a Drasha by Rabbi Frand.)

 Where is there bowing in the Bikkurim ceremony?  In passuk 26:10, where it says

והנחתו לפני ה'  אלקיך והשתחוית לפני ה'  אלקיך

that after placing the basket of Bikkurim before Hashem, we are to bow down to Hashem.

There are two approaches as to when this bowing occurred.  The Gaon here in the Aderes Eliahu says:

והשתחוית - זה הכלל כל היוצא מבית המקדש צריך השתחואה

This is a rule: everyone that leaves the Beis Hamikdash needs to bow down.

On the other hand, Tosfos in Sukkah (47b, D'H Habikurim) says:
בספרי (פרשת ראה) משמע דאיכא שתי תנופות דקתני והנחתו לפני ה' אלהיך מלמד שטעונין תנופה ב' פעמים אחת בשעת קריאה ואחת בשעת השתחויה 
that the bowing was done during the second Tenufa, when the Kohen lofts the basket in the cardinal directions.  
We find that the Torah mentions bowing in two contexts.  One connotes respect, and the other communicates gratitude (e.g., Breishis 24:51-2, where Eliezer was told that his mission was successful and that he could bring Rivka back to Yitzchak, it says הנה רבקה לפניך קח ולך ותהי אשה לבן אדניך, כאשר דבר ה'.  ויהי כאשר שמע עבד אברהם את דבריהם וישתחו ארצה לה' לה).  
The halacha (see Brachos 34a) is that in Shmoneh Esrei, we bow at the beginning and the end of two brachos- Avos, the first bracha, which ends with Magen Avraham, and Hoda'ah, which is Modim.  These two bowings express both the Gaon and Tosfos' Sifri.  When we bow in Avos, as we begin Shmoneh Esrei and reflect upon Hashem's greatness, this reflects the Gaon's bowing, that one must bow in the Beis Hamikdash to  demonstrate one's awareness of and subjugation to Hashem.  When we bow in Modim, this reflects the bowing the Sifri refers to, a way to express our gratitude for Hashem's love and beneficence.  As Rebbi Chanina says in Brachos there, דא"ר חנינא ראשונות דומה לעבד שמסדר שבח לפני רבו..... אחרונות דומה לעבד שקבל פרס מרבו ונפטר והולך לו.

So the Tanchuma becomes crystal clear.  When we bring Bikkurim, we bow to express these two feelings: Awe/Humility and Gratitude.  Bikkurim is the highest opportunity to express these feelings of humility and gratitude, and Hashem intimated to Moshe Rabbeinu that this experience is vital to our spirituality and ought to be eternal.  Moshe Rabbeinu, foreseeing that the Bikkurim would not last forever, instituted the Shmoneh Esrei, which forever formalizes our ability to express these two feelings.


II

Chazal (Brachos 34b) tell us that bowing during Tefilla should be limited to those places where it is prescribed, and not done wherever one feels like it.  As mentioned above, in Shmoneh Esrei it is found only five times- the beginning and end of Avos and Hoda'ah, and when we step back at the end.  But most people have the minhag to bow when we say Barchu at an Aliyah to the Torah and when we say Barchu cafter Yishtabach at the beginning of Birkos Krias Shma.  Interestingly, when I spoke to my shiur about this, some said "Nobody bows down by the Borchu of the Aliya," and the others said "Everyone bows down the Borchu of an Aliya."  But I think bowing by the beginning of Birkos Krias Shma is pretty universal.  Where does this minhag come from?  Is it a minhag that is brought in the poskim?  Why the difference in minhagim?

Let's start with the Kri'a at Barchu after Yishtabach.  In Orach Chaim 113:3 the Mechaber says that one should not just bow down wherever he feels like it, and one who does so is repugnant: הכורע בוכל קומה לפניך תשתחוה או בולך לבדך אנחנו מודים או בהודאה דהלל וברכת המזון הרי זה מגונה.  The Rama adds פירוש שאין לכרוע אלא במקום שתקנו חכמים.  The Biur Halacha D"H הכורע says the מגן גבורים דוחק למצוא טעם במנהג, the Magen Giborim tries hard to find an explanation for the minhag to bow at Barchu after Yishtabach.  The Biur Halacha, uncharacteristically, says that he found a basis for the minhag in a passuk (Divrei Hayamim I 29:20) that says    ויאמר דויד לכל הקהל ברכו נא את ה' אלקיכם ויברכו כל הקהל לה' אלקי אבתיהם ויקדו וישתחוו  לה' ולמלך, that David Hamelech told the gathering to say Barchu, and they said Barchu, and they bowed down.  We see here an association between Barchu and bowing.  (It goes without saying that there is a linguistic relation between the word Barchu and Birkayim, the knees.)

The Aruch Hashulchan also mentions this, in OC 57:1, where he says:

ודע שנוהגין לכרוע מעט באמירת "ברכו", ולא ככריעות שבשמונה עשרה. ונכון הוא. ואפילו לפי מה שיתבאר בסימן קל"ט דלקריאת התורה אין לכרוע, מכל מקום בשעת התפילה כורע. וכן המנהג הפשוט.

Note that he says לכרוע מעט...ולא ככריעות שבשמונה עשרה, just bend down a little- NOT like when you bow during Shmoneh Esrei.

So we see that while the source and rationale is obscure, and one might think that it is contrary to the Gemara's disapproval of bowing in places other than the ones listed, the fact is that the minhag is well known and established.

As for bowing before an Aliya:  The immediately preceding Biur Halach, D"H בסוף, brings the לחם חמודות, who, after quoting the Gemara in Brachos that disapproves of bowing where not prescribed by Chazal, says that טועים כל השוחים בברכת התורה תחילה וסוף, it's simply wrong to bow down by the brachos of an Aliya, and all those who do it are making a mistake.  The Biur Halacha does not seem to disagree with this opinion.  Interestingly, though, in Siman 139 SK 19, the Mishna Berura quotes a Magen Avraham that says that our minhag to bow by Birkos Hatorah of an Aliya is to show honor to the Torah- יש נוהגים לשחות ולהשתחוות בעת אמירת הברכות משום כבוד התורה- but you shouldn't bow a second time at the end of the Bracha.

(Aleinu is entirely unrelated to this discussion, because there, the text states ואנחנו כורעים ומשתחוים, so it would be wrong to say it and not do it, like the עדות שקר of Krias Shma without Tefillin- Brachos 14b).

All of this should make no practical difference to the reader.  It is descriptive, not prescriptive.  Each of us should do exactly as our mesora dictates.  I personally do bow after Yishtabach but not at an Aliya, because this is what I saw my father and Rav Rudderman and Reb Moshe do.  Everyone needs to pattern himself after a particular spiritual exemplar, and to do what they did.  I only write this to illustrate how our various minhagim deal with the Gemara, and so that if you see someone doing different than you do, you will understand why, so we can better resist our unfortunate tendency to disparage minagim that are unfamiliar to us.

III
UPDATE:
I was directed to an article by Rabbi Daniel Rakkah in the publication Ohr Yisroel, in which he brings an abundance of sources for kriyos in davening.  The article is in the fifth volume of Ohr Yisroel, which as of this writing is not available on Hebrewbooks.org.  The Editor/Publisher was kind enough to send me a pdf, which can be accessed here.
Thanks to the wonders of the internet, several of Rabbi Rakkah's marei mekomos are easily accessible.  Here are the ones that are currently online:
מחזור ויטרי
כלבו
אורחות חיים
The one that he brings that I cannot get online is the פסקי מהרי"ץ  from  רבי יחיא צאלח.  The Piskei Maharitz quotes a well known Medrash but with the addition of a few words that are unique to his version of the Medrash and which make it relevant to this discussion.
The usual version of the Medrash is

א"ר עקיבא בכל יום שחרית מלאך א׳ פותח פיו ואומר ה׳ מלך הי׳ מלך ה׳ ימלוך לעולם ועד עד שמגיע לברכו וכשמגיע  לברכו חיה אחת עומדת  ברקיע ושמה ישראל וחקוק עד מצחה ישראל עומדת באמצע הרקיע ואומרת ברכו את  ה׳ המבורך וכל גדודי מעלה עונים ואומרים ברוך ה׳ המבורך לעולם ועד.

In Rav Tzalach's version, the sentence is

.....וכשמגעת לברכו כורעת על רגליה ואומרת בקול ברכו את ה' המבורך וכל גדודי מעלה....

This Girsa is not found anywhere but here.  In any case, it does provide an early source for the minhag among the Teimanim.

By the way, I saw the Yalkut Yosef (3:421, 57:9) cited as saying that Rav Ovadiah Yosef does not bend at all by the Barchu of Birkos Krias Shma. 

UPDATE:

I just (2024) saw a Shoshan Kodesh from a talmid of the Ramban (brought to my attention by the Rivevos Ephraim) who says you should bow by Borchu.

https://hebrewbooks.org/pdfpager.aspx?req=19655&hilite=db76f1eb-5f60-488e-8e22-d072120cdcdf&st=%d7%a8%d7%90%d7%a9%d7%95&pgnum=14