Gemara

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

102a: If a cave is four by six Amos, the diagonal is eight Amos.

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."

Eruvin 14a: The diameter is a third of the circumference.

Question: What is the source of this?

Answer (R. Yochanan): We learn from the Yam Shel Shlomo (a Mikveh). It was "Eser ba'Amah mi'Sefaso Ad Sefaso Soviv v'Chamesh ba'Amah v'Kav Sheloshim ba'Amah Yasov Oso Soviv."

Question: The circumference should exceed 30, for it is measured "Soviv" (around the outside)!

Answer: The circumference was measured from the inside.

Rishonim

Perush ha'Mishnayos (Eruvin 1:8): It is impossible to know the exact ratio of the diameter to the circumference. Scientists use 7/22 for an approximation. Since in any case we must use an approximation, we rely on a whole number, i.e. one part in three, in the entire Torah.

Perush ha'Mishnayos (Eruvin 4:8): The Mishnah says that the Techum is 2000 square Amos, like a square tablet. This teaches that even though one cannot make it perfectly square, it suffices to be as precise as possible, just like one cannot make a perfectly square tablet.

Tosfos (Sukah 8a DH Kol): The calculation of the diagonal 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 Tosfos' picture). Its area is 50, not 49!

Question (Tosfos Eruvin 14a DH veha'Ika): The Gemara connotes that the diameter is exactly a third of the circumference. It is not exact!

Answer (Tashbatz 1:165): Tosfos explains that the Gemara brought a verse to teach that we are not concerned for the excess. (This is not in our text of Tosfos.) The Gemara says that if a rectangle is 4 by 6, the diagonal is 8. Really, it is the square root of 52! Perush ha'Mishnayos (Ohalos 15:4) says that the Gemara was stringent to call it 8, for it is more than 7. There are two possiblilities. One is that Chachamim had a tradition to rely on approximation. (Shi'urim are a tradition from Sinai. Sometimes we find that a tradition was only for certain matters.) The reason is because the Torah was not given to angels (who can be exact). The verse about the Yam Shel Shlomo supports this. The other possibility is that Chachamim merely taught concisely, but practically, we must be precise. They rely on Chachamim, who know the Shi'urim, to ensure that people be precise in practice. I prefer the latter explanation. The Gemara says 'we are imprecise to be stringent, but not to be lenient' only regarding big errors. The Gemara says 'we are not exact about a small amount.' In Eruvin (23a), we say that if the side of a square is 70 Amos and a remainder, its area is 5000 square Amos, like Chatzer ha'Mishkan (which was 50 by 100), We did not rely on the approximation to say that it is exactly 70 because the approximation was not said regarding a rectangle, only regarding a square.

Aruch ha'Shulchan (YD 30:13): The Beis Yosef says that the width of a Sela (coin) is a third of a Tefach, and the circumference is a Tefach, for a square is a quarter bigger than the circle inside (the excess is a quarter of the square's circumference, which is a third of the circle's circumference). Even though the calculation is not exact, the Torah decreed to measure this way. Why did the Gemara ask the source that the diameter is a third of the circumference, and learn from a verse? One can measure with a string! Rather, it asked for the source to rely on the approximation.

Poskim

Shulchan Aruch (OC 32:39): It is a tradition from Sinai that Tefilin must be square. The diagonal must be like Chachamim said, seven fifths of the side.

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

Aruch ha'Shulchan (32:68): 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 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 discussed the length of the diagonal. This exceeds the exact value (the square root of 2, i.e. 1.414...) by about one part in 100! Perhaps Halichos Shlomo means that the square of the diagonal may be one part in 50 bigger.

Aruch ha'Shulchan (YD 84:36): I heard from a researcher that through a microscope, one can see many creatures in water and the air! The Torah was not given to angels. The Torah did not forbid what the eye cannot see. What the eye can see is forbidden, even if it is tiny.

Shulchan Aruch (648:12): If any amount on an Esrog changed appearance, it is Pasul.

Mishnah Berurah (46): This is only is anyone can see it. If it is too small to be seen and one must look intently, it is Kosher.

Halichos Shlomo (4(78)): This is a general rule, that Torah was given to each generation according to people's senses The Torah was not given to angels! If one can see normally, he may rely on what he sees. If one cannot see a speck of (menstrual) blood in a garment, we need not be concerned. If one can see something, even if he does not recognize that it is a bug, he must be stringent. (If one saw through a microscope, he should ask a Chacham what to do.) If the eye cannot see bugs move, they cannot be called "Sheretz ha'Aretz". This leniency does not apply if one could detect movement before the bugs are covered with a shell. If an Esrog looks complete, and one sees through a magnifying glass that it is missing a tiny amount, it is Kosher and Mehudar l'Chatchilah. "Hadar" and "Tamah" (complete) are according to natural sight. We permit homeopathic medicines even though some are made from Tamei animals. We are not concerned for Bitul Isur l'Chatchilah, because it is such a tiny amount that it is not considered a mixture according to human senses.

Even ha'Ezel (Kuntres Bein ha'Shemashos 1::1:3, cited in Halichos Shlomo, ibid.): The Ran says that perhaps we cannot see three stars within the time to walk three quarters of a Mil after sundown, because our sight is weaker. Bechoros 54b says that a person can see (animals) 16 Mil away. Nowadays, no one can see close to this! If so, we cannot prove anything from our sight.

Chachmas Adam (38, Binas Adam 34): We need not be concerned for bugs in vinegar that can be seen only through a microscope. The Beis Yosef is concerned for what can be seen in the sun with one's eye.

Igros Moshe (YD 2:146): We may eat things fermented through worms visible through a microscope. R. Chayim of Brisk said that the Torah forbids only what eyes can see. We are not concerned for what a microscope can see regarding inhaling organisms, Dam Nidah, squareness of Tefilin or nicks in a Shechitah knife. Previous generations did not have microscopes. Surely they did not sin, even b'Ones! If one ceased to breathe, we are sure that he died, and we bury him. An exceptional case occurred in which one who 'died' was later found to be alive (Pischei Teshuvah YD 357:1, citing the Chasam Sofer). Normally, we need not be concerned for this. However, if a radiogram detects electric responses, there is no majority to say that he died! One must continue to try to cure him, even on Shabbos.

Note: If an animal ate Chametz, its meat, milk, or eggs are not Chametz. (In some cases, we say that one who ate them benefited from Chametz, but he did not eat Chametz.) However, some say that if bacteria (too small for the eye to see) ate Kitniyos (e.g. corn syrup) and produced sugar, it is as if the Kitniyos changed form by itself, and the sugar is considered Kitniyos, just like whiskey is absolute Chametz.

Halichos Shlomo (2:34): If an electric razor shaves so close that one cannot feel the hairs that remain, it is forbidden, even if one can see the hairs through a magnifying glass.

Halichos Shlomo (12:40): If in a Sefer Torah it looks like two letters touch, it is Pasul, even if it is seen through a magnifying glass that they are separate.