פירוש על חולין 1:3

השוחט מתוך הטבעת – In the large ring [of the trachea] which is above all of them, is [what the Mishnah] is speaking of.

Introduction Slaughtering must be performed on the animal’s neck below the larynx, preferably below the first hard ring of the trachea and up to the place where the bronchial tubes begin to branch. Cutting outside of this area is called “hagramah” and renders the animal invalid. Our mishnah deals with a case where a person cuts right below the first hard ring of the trachea.

ושייר בה מלא החוט של פני כולה – on the side of the head, for he did not incline the knife to exit from the ring to the side of the head until he completed [cutting] the entire ring, it is considered valid.

If one slaughtered [by cutting] at the [top] ring [of the trachea] and left a hair's breadth of its entire circumference [towards the head], the slaughtering is valid. As stated in the introduction, the cut should be made below the first hard ring of the animal’s trachea. If he makes the cut at this top ring, the slaughtering is valid as long as a hair’s breadth of the trachea on the side of the head remains. According to the first opinion, the hair’s breadth of the trachea must remain on the entire circumference of the highest ring.

מלוא החוט – that is to say, a bit, but if it was before he completed the entire [ring], he slanted the knife to the side of the head and completed the ritual slaughter above from the ring which was not the place of the ritual slaughter, which is הגרמה (cutting the animal’s throat in a slanting direction – letting the knife slide beyond the space ritually designed for cutting), even though most of the windpipe was cut at the place of the ritual slaughter, the Rabbis invalidated it, since it concluded with a disqualification.

Rabbi Yose son of Rabbi Judah says: if there was only left [towards the head] a hair's breadth of the greater part of its circumference, [the slaughtering is valid]. Rabbi Yose son of Rabbi Judah says that it is sufficient for the hair’s breadth to remain on the greater part of the circumference, and not necessarily the entire circumference.

רבי יוסי בר' יהודה אומר מלוא החוט על פני רובה – If he left in it as much as the breadth of a thread of the hair from the ring [of the trachea] at the side of the head over the majority of the ring, that is to say, he cut most of the windpipe within the ring, but the minority of it was cut in a slanted direction and it left its [body] at the side of its head, and he completed the slaughter above it, it is considered valid, as he slaughtered most it in a valid manner, but the other part, he slaughters by cutting mere skin. And the legal decision is that a person who slaughters above the large ring, where the thyroid cartilage (Adam’s apple) begins to protrude and above it is considered invalid, whereas from the thyroid cartilage and below is considered valid. And that means that he left glands, that he slaughtered these two glands of the flesh that are above the large ring but below the thyroid cartilage, and he left a bit of them at the side of the head.