As part of the order of the day, we learnt (7:1) that the Kohen Gadol read three sections from the Torah and only one sefer Torah was used. We learnt that he would first read the section of acharei mot that deals with the avodah of Yom Kippur. We would roll the Torah and then read the nearby section of ach be'asor. The final section of u'be'asor from parashat Pinchas was recited by heart and not read for the Torah. The Bartenura, citing the Gemara, explains that since the second and third parshiyot are far apart from one another, in interest of kavod ha'tzibur – the honour of those present – we do not delay by rolling the Torah. Similarly, a second sefer is not used out of concern that it might raise questions as to the validity of the first sefer. Such a concern would be present in this case as it is read by the same person.
The Tosfot Yeshanim (Yoma 70b) however asks. We learn in Gemara Gittin (60b) that one is not allowed to recite any parts for the written text (devarim she'bichtav) by heart. The practice of the kohen gadol appears to contradict that ruling.
The Tosfot Yeshanim initially suggests that it is not prohibited to recite devarim she'bichtav by heart, it is simply a mitzvah min ha'muvchar; far more preferable to read devarim she'bichtav from the text. In this case however, due to kavod ha'tzibur, it is preferable to recite it by heart.
The Melechet Shlomo cites R' Yona who explains that we find that the recitation of korbanot was permitted. Recall the midrash where Avraham was concerned how future generations would achieve atonement without the Beit HaMikdash. Hashem responded that he would accept the recitation of korbanot in their stead. Consequently, since the above section is dealing with korbanot, it was covered by this heter and could be recited by heart.
The Tosfot Yeshanim cites the Riva who explains that the nature of the reading at that point was different. It was not part of the sedre of the day like the two previous parshiyot, but was simply describing the avodah of the day. Consequently, it is more comparable to sections of mussaf that can be recited by heart. The Riva cites the Yerushalim in support of this explanation. The Tosfot Yeshahim however cites his teacher that notes that there are sections, e.g. korbanot that do not simply describe the order of the day, and yet we recite them by heart.
The Ritva however understands from the Yerushalim that only the sections that one is obligated to recite in the tzibur is covered by the prohibition. Pesukim however that are recited for learning or as part of tefillah and praise, one can recite by heart. According to the Ritva, this explains the practice of reciting parts of tefillah by heart (e.g. korbanot, shema, pesukei dezimra). Nevertheless, he concludes that even an individual should not recite those obligatory section of keri'ah and sections as part of learning learning by heart.
The Tosfot Yeshanim however cites his teacher who explains that the prohibition of reciting sections by heart is only when one is reading to fulfil the obligation of another. Korbanot and the recitation of the kohen gadol on Yom Kippur which was simply giving over the order of the day, are therefore not covered by the prohibition. What about Hallel? He explains that that fits into a separate category, which is matter of praise and also not covered. The Tosfot Yeshanim cites the R' Eliezer from Mitz, who brings a proof for this position from another Gemara (Taanit 28.) which described the ma'amadot. The Gemara asks how the individuals could recite the sections at Mincha by heart in public. The Gemara answers that in that context it is comparable to the recitation of Shema. He understands the answer to mean that since it is recited by each individual, and not one person for everyone, it is not covered by the prohibition.
The Biur HaGra appears to side with this position. The Magen Avraham however explains that due to the different explanations cited by the Beit Yosef, the only pesukim one should recite by heart are those listed in the Shulchan Aruch (see O.C 49:1). The Tiffert Yisrael writes that this Mishnah appears to support the Magen Avraham where he explains that when left with no other options, one can recite sections by heart. Based on the position of the Biur HaGra, the Mishnah Berura rules leniently in certain cases. If one is giving a shiur and he is finding it difficult to find the pasuk, due to kavod ha'tzibur one can recite it by heart. Similarly, he says that one can rely on the Chovat Yair and recite tehillim by heart since it is comparable to tefillah.
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