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The Mishna in Ohaloth 12:6 informs us that you need a Tefach of width to bring Tumah from one location to another:

קוֹרָה שֶׁהִיא נְתוּנָה מִכֹּתֶל לְכֹתֶל וְטֻמְאָה תַחְתֶּיהָ, אִם יֶשׁ בָּהּ פּוֹתֵחַ טֶפַח, מְבִיאָה אֶת הַטֻּמְאָה תַחַת כֻּלָּהּ. וְאִם לָאו, טֻמְאָה בוֹקַעַת וְעוֹלָה, בּוֹקַעַת וְיוֹרָדֶת. כַּמָּה יִהְיֶה בְהֶקֵּפָהּ וִיהֵא בָהּ פּוֹתֵחַ טֶפַח. בִּזְמַן שֶׁהִיא עֲגֻלָּה, הֶקֵּפָהּ שְׁלשָׁה טְפָחִים. בִּזְמַן שֶׁהִיא מְרֻבַּעַת, אַרְבָּעָה, שֶׁהַמְרֻבָּע יָתֵר עַל הֶעָגוֹל רְבִיעַ:‏

[With regard to] a beam which is placed across from one wall to another and which has uncleanness beneath it: If it is one handbreadth wide, it conveys uncleanness to everything beneath it; If it is not [one handbreadth wide], the uncleanness cleaves upwards and downwards. How much must its circumference be so that its width should be one handbreadth? If it is round, its circumference must be three handbreadths; If square, four handbreadths, since a square has a [circumference] one quarter greater than [that of] a circle.

Firstly, regarding the circle, as we all know, if the circumference is only 3 tefachim then we are missing about 3mm from the required Tefach.

But more peculiar is the "divide the circumference of the square by 4". It seems one could simply measure any one side.

However, even that isn't accurate, for if the square beam isn't sitting perfect horizontal - like this: ■ - but is (in the extreme case) sitting tilted - like this: ♦ - then the "divide the circumference by 4" is totally wrong! We now have 1.41 Tefachim. A circumference of 2.84 Tefachim would be sufficient to get us a 1 Tefach for the Tumah to travel underneath.

So: Why is the Mishna assuming the beam is sitting "square" (like this: ■) and why do the meforshim all seem perplexed by the unnecessary mathematics, but none of them seem to notice that we have a real question if the beam is sitting like this: ♦ ?

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Danny Schoemann
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  • It's simply דרוש וקבל שכר - I't not like we'd like to know the empiric truth, we love to argue. :) – Al Berko May 23 '19 at 08:27
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    Would one ever orient a ceiling beam like that? – Joel K May 23 '19 at 08:30
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    Why does the direction the beam is sitting make a difference? No matter what, if the perimeter is 4+, dividing it by four will give you at least one tefach, making it liable to pass tuma to anything under it. Whereas measuring only one side might not be effective if, for example, the beam is rectangular with the length being 1.5 Tefachim and the width only being half a tefach. – Lo ani May 23 '19 at 11:33
  • @AlBerko please take comments like that elsewhere. – Dr. Shmuel May 23 '19 at 23:23
  • @Dr.Shmuel Just speculation: if we have a beam of uneven form, we can approximate it either to a square or to a circle, like if it is rhombus or oval (remember 2000 years ago they didn't have wood processing machines). How do we measure its variable width? We measure the circumference with a rope and then divide it by 3 or 4.depending on the form. The result is the average width accepted for the Halachah. – Al Berko May 24 '19 at 08:48

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Firstly, regarding the circle, as we all know, if the circumference is only 3 tefachim then we are missing about 3mm from the required Tefach.

I have no idea where you get 3mm from. It's closer to five times that, because, assuming a tefach is 4in=10.16cm, the diameter is 10.16/π≈3.23cm, or ~16mm smaller than 10.16/3≈3.39cm. (Rav Moshe's 9cm tefach puts this at 9/π≈2.86cm, or ~14mm smaller than 9/3=3cm.)

So, if anything, the Mishnah is overestimating in this case. Nevertheless, this rule is applied in other cases to be more lenient, a problem which the Gemara already deals in Eruvin 14a and a topic I've addressed previously on this site. The Mishnah there (13b) says similarly that a beam with a circumference of three tefachim is wide enough to qualify as a Koreh to permit carrying in an alleyway, even though it ordinarily needs a full tefach of width. To that the Gemara asks for a source. According to the Tosfos HaRosh on the spot, the Gemara is asking how we know that we can assume π=3 halachically, even though that equality is false; according to the Aruch HaShulchan (OC 363:22, YD 30:13), it's asking how we know such a beam is permissible, even though we don't have a full tefach of width. According to either interpretation, the Gemara is directly addressing how we know such a beam is fine even though it doesn't actually measure up.

But more peculiar is the "divide the circumference of the square by 4". It seems one could simply measure any one side.

Sure, in practice you probably could. But the Mishnah seems to want to make the point that "A square has a [perimeter] one quarter greater than [that of] a circle," which has halachic ramifications elsewhere (ex. Sukkah 8a, Eruvin 76a). Similarly, the Mishnah I cited above in Eruvin doesn't need to discuss the case of a circular beam, as any novelties contained therein are addressed by the previous cases of the Mishnah. According to the Gemara, the case is only discussed just so the Tanna has an excuse to mention that a circle has thrice its diameter in its circumference.

Now for the main part of your question.

I don't have any sources on this, but it seems reasonable to assume one of the following:

  • The Mishnah is dealing with a flat square because of the case at hand: it's being rested on a tent on either end, which is simplest construed as lying flat on either end.
  • The Mishnah is dealing with the minimum case, that is to say that the square gives the minimum shadow beneath it; this occurs when the square is lying flat, as its diagonal will always be longer than its sides.
  • The Mishnah wants to make sure that the beam indeed has a shadow of a tefach, as while it looks square, it might actually be rectangular, rhombic, or parallelogrammatic. By measuring the perimeter and dividing by four, it would ensure that, regardless of the case, it will have a bottom side measuring at least a tefach.

From the fact that the Mishnah doesn't even present the rule which allows one to generalize to rotated squares – i.e. that any square with a side of 1 has a diagonal of 1.4 – that seems to eliminate the second theory.

DonielF
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  • (Or 2a) the beginning (circle) of the Mishna calculates l'Humra, the end (square) calculates L'Kula (or vice versa, depending on how you look at it). But since it's talking about moving the Tumah from one house to the other via the beam, in the circle we're very strict ("stealing 3-4mm) and for the square we're lenient, we assume that it's got 90° corners.
    • – Danny Schoemann May 26 '19 at 08:49
  • @DannySchoemann 2. Yes, but the Gemara brings a proof from a passuk. 3. I‘d think the opposite, actually - by assuming that it has 90 degree corners, we assume the most stringent case, where it has the minimum horizontal distance. – DonielF May 26 '19 at 11:38
  • Why? If it's this ♦ shaped (i.e. it's sitting slightly crooked) then I need a much smaller circumference and smaller side (any & all sides) for the Tumah to travel to the next house. Based on the Mishna as is, I can assume the next house is Tahor but in reality, I am mistaken. That sounds like a leniency to me. – Danny Schoemann May 26 '19 at 16:02
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    @Danny If a side of the beam measures one tefach, its diagonal will measure ~1.4 tefachim. You can check that by the Pythagorean Theorem. So to the contrary - you may have a case where the beam is in fact tahor but you think it's tamei, but you'll never have a case, if you follow this Mishnah, where the beam is tamei but you think it's tahor. – DonielF May 26 '19 at 16:03