(Beraisa): The Mizbe'ach must be square. This is Me'akev.

Chulin 17b (Rav Huna bar Rav Ketina): Three things have the same intolerance for nicks: the Isur to break a bone of the Korban Pesach, a nick in the ear of a firstborn animal or (the Ervah of) a female animal is a Mum (blemish).

(Rav Chisda): To this list we may add a nick on a knife for Shechitah.

The Shi'ur for all of these equals the size of a nick to disqualify the Mizbe'ach, i.e. b'Chagiras Tziporen (a fingernail will get caught if one runs his nail along the Mizbe'ach).

Question (Beraisa - R. Shimon): A nick the size of a Tefach disqualifies the Mizbe'ach;

R. Eliezer ben Yakov says, it must be the size of an olive.

Answer: A Tefach or olive sized nick disqualifies in the plaster. For stones of the Mizbe'ach, b'Chagiras Tziporen disqualifies.

Menachos 35a (Beraisa): A tradition from Sinai requires Tefilin to be square.

(Rav Papa): The stitching must be square, and also the diagonals.

Support (Mishnah): One does not fulfill the Mitzvah with round Tefilin.

Bechoros 17b: The Torah gave precise measures for the Mizbe'ach and Kelim (Aron, Menorah and Shulchan). This shows that people can be exact!

Rejection: The Torah commanded to make the Kelim. It accepts whatever people are able to make - "ha'Kol bi'Chsav mi'Yad Hash-m Alai Hiskil."

Sukah 8a: The diagonal of a square is seven fifths the length of a side.

Rif (Hilchos Tefilin (after Menachos) 7b): Tefilin must be square. Rava taught that this refers to the stitching and the diagonal.

Mordechai (Hilchos Tefilin p. 11b, brought in Beis Yosef OC 32 DH v'Yihyu): Hilchos Gedolos and the Rif say that the length of the Shel Yad is half the middle finger, and the width is a finger width. They are called Ribu'a (square) because they have rectangular corners. This is wrong. They must be totally square. Since we do not find explicitly, and it is a tradition from Sinai, one may not be lenient without a proof.

Note: This is not in our text of the Rif's Hilchos Tefilin.

Rambam (Hilchos Tefilin 3:1): Tefilin and the stitching must be square

Rambam (2): One takes a square piece of wood (to make the boxes). The height, length and width should be the same. We are not concerned if the height does not equal the width, but the length and width must be the same.

Rosh (Hilchos Tefilin 7): The Ma'avarta is Me'akev. It is a slot passing through the leather attached to the Batim. The strap passes through it. Some say that it is is a small strap that passes above the hand Tefilin. The Rosh says that this is the custom. Even if it is not, this does not disqualify the box because it does not see the air. This is a problem only if an extra Bayis was attached.

Divrei Chamudos (12): The Beis Yosef says that nowadays the custom is not to pass a strap; he gave a reason. I say that when they make the box for the Shel Yad, there are folds, and it is not beautiful. Therefore, the custom is to pass a strap to cover the folds. In the Beis Yosef's region in Eretz Yisrael, there were no folds, so they did not pass the strap.

Minchas Yitzchak (6:1): The Rosh connotes that he was unsure. Since it is a Safek, surely one should also fulfill the first Perush (since it is Me'akev). Granted, if the first Perush is correct, there is no problem having an extra small strap. However, if the second Perush is correct, the addition to the box causes that it is not square, for it sticks out! Tosfos (Chulin 83b DH Ka) says that earth attached to the Mizbe'ach ruins the squareness! A hair's breadth separated the ramps from the Mizbe'ach, lest they spoil the squareness! Rather, we must say that it is not Pasul if it is not square in the middle (i.e. it must be square only at the top - PF). Divrei Chamudos was not concerned that the folds ruin the squareness. The custom to seal the folds with glue is for beauty. Also Semag says that it must be square only at the top.

Rosh (Hilchos Tefilin 9): The four boxes of the Tefilin Shel Rosh, and the box of the Shel Yad, must be square. Rashi holds that the stitching must be square in order that the box and its base will be square. We make cuts in the Ma'avarta (where the starp goes through) on both sides so it will be evident that it is not part of the Titura (base), and that the base is square.

Tosfos (Sukah 8a DH Kol): The calculation is not exact. If we make a square whose side is the diagonal of a square of side 5, it fills exactly half of a square of side 10 (see the picture in Tosfos), so its area is 50, not 49!

Shulchan Aruch (OC 32:39): It is a tradition from Sinai that Tefilin must be square in the stiching and diagonal, i.e. the length equals the width, so that the diagonal is like Chachamim said, seven fifths of the side.

Beis Yosef (DH v'Yihyu): Rashi says that the diagonal of Tefilin must be like Chachamim said, i.e. seven fifths of the side. If the length exceeds the width it is Pasul, even if the corners are rectangular. Tosfos says that some make only the base square. Perhaps the boxes of the Shel Rosh must be square when all four are stuck together. The same applies to the Shel Yad. Sefer ha'Terumos and Semag say that there is no proof that the Shel Rosh must be square when its four boxes are joined. However, this is clear from Shimusha Rabah (an early collection of Hilchos Tefilin). We are not concerned for the height, but the stiching and leather of the base must be square. Hagahos Maimoniyos (3:3) says so.

Gra (91): We cannot say that Rav Papa teaches only about the stitching, for if so there is no support from the Mishnah!

Pri Megadim (Eshel Avraham 53): Even though one cannot make Tefilin perfectly square, he must be as precise as possible.

Mishnah Berurah (173): If the only Tefilin he has are not square, he should put them on without a Berachah, and when he gets other Tefilin, he puts them on.

Mishnah Berurah (177): If the holes for the stitches are big, the thread is drawn to the side and the stitching is not square. One must be careful about this!

Mishnah Berurah (178): The boxes must be square throughout the height (from the base to the top). The Bayis itself must be square. It does not suffice to put something square on top of it.

Aruch ha'Shulchan (68): Even though the calculation of the diagonal is not exact, the Torah commanded to measure according to it.

Halichos Shlomo (4:25(36)): Tefilin need not be perfectly square according to measuring tools. It suffices if they look square to the eye. Acharonim say that the Halachah was said that the diagonal is two fifths bigger than the side, even though this is not exact. Tosfos showed that it is one part in 50 more than this. If so, also the diagonal of Tefilin may one part in 50 bigger.

Note: Tosfos showed that the square of the diagonal is one part in 50 bigger (than a square of seven). The Gemara said that the length of the diagonal is seven. The exact value (the square root of 50, i.e. 7.07...) is about one part in 100 bigger! Perhaps Halichos Shlomo means that the square of the diagonal may be one part in 50 bigger.

Igros Moshe (YD 2:146): The questioner said that the Griz checked the squareness through a microscope. This is astounding! Surely we are not concerned for what a microscope can see regarding Tefilin (or Dam Nidah, or nicks in a Shechitah knife). Previous generations did not have microscopes. Surely they did not sin, even b'Ohnes! We require the diagonal to be seven fifths the side. This is not exact! Rather, Tefilin are Kosher even l'Chatchilah if the diagonal is this big. Rashi (Bechoros 17b DH She'ani) explains that Klei ha'Mishkan were not made in the exact dimensions; the Torah accepts this l'Chatchilah. The same applies to squareness of Tefilin. Whatever looks square is Kosher.

Shulchan Aruch (ibid): If they were square and became not square, some say that one must square them.

Mishnah Berurah (180,181): The Beis Yosef writes 'some say' when only one Posek brings a law, even though no one disagrees. One need not check every day that they are still square, for we rely on Chazakah, unless one sees that they are not square. If the base warped and does not look square, even if it is square, one should fix it.

Bi'ur Halachah (DH Niskalkel): Surely, non-squareness disqualifies even if it is not along the entire height of the box, and even if it is above the minimal Shi'ur for the height of Tefilin. Tosfos (Sukah 49a DH she'Kol) says that a nick in the Mizbe'ach disqualifies because it is not square. Even a small nick disqualifies, even above the required height of the Mizbe'ach. Since it was made taller, all of it must be square. A nick disqualifies anywhere, not only at the edge. The same applies to Tefilin. If clearly one side is too far in, it must be fixed. Surely, a nick b'Chagiras Tziporen, and perhaps even a bit bigger, does not disqualify Tefilin. Even if one must be careful about this initially, if they were square, a tiny nick does not disqualify. Stones of the Mizbe'ach must be "Shlemos", so a nick b'Chagiras Tziporen disqualifies. The plaster need not be Shalem, so a nick disqualifies only if it is a Tefach or k'Zayis. This shows that even when something must be square, a small nick does not mean that it is not called square. The Gemara did not distinguish a nick in the corner from anywhere else. Rav Chisda did not list (among things that must be as smooth as the Mizbe'ach) a nick to disqualify Tefilin! The Taz (645:4) uses this to prove that such a nick does not disqualify a Lulav. The Shi'ur for Tefilin requires investigation. Perhaps more than b'Chagiras Tziporen disqualifies.

Shulchan Aruch ha'Rav (32:59): Tefilin must be square, i.e. the length must equal the width. This is at the base, the stitching, and the box itself (along the entire height). However, the height need not equal the length and width. If they were square, and the squareness was ruined, they are Pesulim.

Minchas Yitzchak (ibid.): He requires that the length and width are the same, but we are not concerned for a Pegam (indentation or protrusion) in the middle.