This mishnah continues to deal with the laws of the beam placed at the entrance of an alleyway in order to permit carrying within it. So far we have learned about the height of the beam and its width, and in the previous mishnah (mishnah 4) Rabbi Yehudah expressed his view that the beam need not be strong enough to bear a half brick, and it is enough that it be a handbreadth wide. Our mishnah continues his words and addresses an interesting question: what is the law of a beam that lacks the proper form - a beam of straw, a crooked beam, or a round beam?
Let us begin with the principle:
The beam must be wide enough to receive a half brick, that is, a brick whose width is a handbreadth. With a square beam the matter is simple: we measure its width, and if it has a handbreadth, it is valid.
The three rulings in the mishnah:
"Haysah shel kash o shel kanim - ro'in oso ke'ilu hu shel matteches" - a beam made of straw or of reeds is not strong enough to hold a half brick; nevertheless we view it as though it were made of metal and is strong enough to bear it.
"Akumah - ro'in oso ke'ilu hu pashut" - a beam that is bent and curved out of shape, upon which a half brick could not rest properly, is viewed as though it were straight and not bent.
"Agulah - ro'in oso ke'ilu hu meruba" - a round beam, upon which a half brick cannot stand but rather rolls off, is viewed, so long as it measures a handbreadth, as though it were square and fit for a half brick to rest upon it.
How do we measure a round beam?
A round beam has no "width" in the ordinary sense, so how do we determine that its diameter is a handbreadth? Beyond that, the beam lies above the entrance of the alleyway with its ends attached to the walls, so they cannot be measured at all. Here the mishnah teaches an important rule, and shares with us a piece of geometric and mathematical knowledge:
"Kol sheyesh b'hekefo sheloshah tefachim - yesh bo rochav tefach" - any cylinder whose circumference is three handbreadths has a diameter of one handbreadth. In other words, the ratio between the circumference and the diameter is three.
But the true ratio between circumference and diameter is not exactly three; it is pi, which is approximately three point one four one five. Did the Sages err in their calculation? Certainly not. The Sages knew this well, but they spoke in approximate terms and rounded the number for the sake of simplicity, as the Torah regularly speaks in approximations where absolute precision is not required.
And there is an interesting point here: if the true ratio is somewhat greater than three, then a circumference of exactly three handbreadths yields a diameter slightly less than a handbreadth, and it would seem that the Sages were stringent without need, since the beam will be slightly narrower than a handbreadth. Rather, the Sages adopted a measure that is convenient for measuring in practice, and for the beam of an alleyway this approximation is sufficient.
The source of the ratio in Scripture:
The Gemara in Eruvin derives this ratio of three to one from a verse dealing with the basin that stood in the Beis HaMikdash, which was made by King Shlomo, concerning which it is stated that its circumference was thirty amos and its diameter ten amos.
The word used there for the diameter is "kav," and it has a written form and a read form - as with many words in Tanach that are written one way and read another: it is read "kav" (kuf, vav), but written "kaveh" (kuf, vav, heh).
The numerical value of kuf and vav is one hundred six, and of kuf, vav and heh - one hundred eleven. If you divide one hundred eleven by one hundred six and multiply the quotient by three, you get three point one four one five - the first four decimal digits of pi. Astounding.
In summary: we have learned the three rulings of the mishnah - a beam of straw or reeds is viewed as though it were of metal, a crooked beam is viewed as though it were straight, and a round beam is viewed as though it were square - and the central principle:
A square beam requires a width of one handbreadth.
A round beam requires a circumference of three handbreadths, so that its diameter will be one handbreadth.
From here we learn a foundation for the future: wherever the Sages use the measure of the circumference in order to calculate the diameter, they employ a ratio of three. This rule is worth remembering, for it recurs in many places throughout Shas.