Ohalos, Chapter 12, Mishnah 6. Our chapter has been dealing with boards and beams and the question of when a flat surface earns the status of an ohel. This Mishnah takes a beam stretched across a house and asks two things: what happens to the tumah beneath it, and how do we know whether the beam is actually wide enough to matter.
The Beam Across the Room
The case: "koirah shehi nesunah mikoisel lekoisel vetumah tachteah", a beam laid from one wall to the opposite wall, with tumah, say a piece of a meis, lying on the floor underneath it. If "yeish boh poseiach tefach", the beam measures a tefach in its width, then "meviah es hatumah tachas kuloh", the beam carries the tumah along its entire length, so that any keilim lying beneath the beam become tamei, even where they are nowhere near the tumah itself. But the tumah stays down below. It does not break upward through the beam, because the beam serves as a chatzitzah that seals it off.
This is a good place to review the yesoidos we have been working with. A flat surface that measures a tefach by a tefach has the din of an ohel, and an ohel spreads tumah along everything under it. And when that surface is not a keli, as is the case with a plain wooden beam, it does double duty: it also holds the tumah back. So with our beam a tefach wide, whatever lies underneath is tamei, while whatever sits on top of the beam remains tahor.
What Is the Chiddush?
At first glance this halacha seems entirely familiar; we have met a surface of a tefach functioning as an ohel many times already. According to the Rambam, however, the Mishnah is adding something new: even a beam that does not maintain its full tefach of width along its whole span, one that thins out in places, still brings the tumah under all of it. There are a number of different ways the mefarshim understand exactly what the Rambam intends, and we will leave that discussion aside here.
A Beam Narrower Than a Tefach
The next clause reads "ve'im lav", and if it does not: a beam narrower than a tefach is governed by "tumah boka'as ve'oleh, boka'as ve'yoredes", the tumah breaks through upward and breaks through downward. A narrow beam simply does not qualify as an ohel, and that status works to be machmir and to be meikil at one and the same time. Lacking the strength of an ohel, it cannot serve as a barrier: tumah lying on the ground travels straight up and makes tamei whatever is positioned above the beam, and tumah placed on the beam travels straight down and makes tamei whatever sits directly beneath it. Yet the very same reasoning produces a kula, for a surface with no din of ohel also lacks the power to carry tumah horizontally. The tumah moves only in a vertical line, above and below, and nothing along the beam's length is affected.
How Much Must You Measure Around?
From here the Mishnah moves to a question of practical measurement: "kama yehei beheikeifah", what must the measurement around the beam come to in order to establish "viyehei boh poseiach tefach", that there really is a tefach of width there? Two figures are given, one for a beam that is rounded and one for a beam that is squared off.
"Bizman shehi agulah, heikeifah shloysha tefachim", when the beam is round, its circumference must be three tefachim. That is the standard rule: run a measuring tape around a circle, and the distance across the circle is a third of what the tape shows. Three tefachim around therefore tells you that you have a full tefach across the middle.
"Bizman shehu meruba, arba'ah", when the beam is square, the circumference must be four tefachim: a tefach up one side, a tefach across the top, a tefach down the other side, and a tefach along the bottom. The Mishnah gives the reason: "shehameruba yoser al ha'igul revi'a", a square exceeds a circle by a quarter. So with a round beam the tape reads three tefachim and that is your siman that there is a tefach across, while with a square beam the tape must read four. For now that is simply the plain meaning of the words; be'ezras Hashem, when we work through the calculations in the next Mishnah, the phrase "shehameruba yoser al ha'igul revi'a" will come into sharper focus.
The Yam Shel Shloimeh and the Accuracy of Chazal's Numbers
Be'ezras Hashem, when you eventually learn Gemara in Meseches Eiruvin, you will come across an extended sugya anchored in the pesukim of Divrei Hayamim that describe the yam shel Shloimeh, the enormous basin which stood in the Beis Hamikdash and which was fashioned by Shloimeh Hamelech. Its purpose is something you will learn about there. In the course of that sugya the Gemara develops a highly detailed cheshbon setting the dimensions of a circle against those of a square, taking the basin as measuring ten amos in width and thirty amos around.
Now the mefarshim here, like the sugya there, point out that in reality the circumference comes out slightly more than three times the width. One approach is that Chazal deliberately gave us a round, uncomplicated shiur rather than burden us with fractions. Another approach, that of the She'eilos Uteshuvos Tashbetz, the Maharam me-Rothenburg, is that Chazal never meant to replace an exact measurement; the three to one figure is a working approximation, and where the halacha depends on it you must take a precise measurement. Many poskim disagree and hold that this ratio is a halacha le-Moshe mi-Sinai: al pi halacha you measure three around and take a third for the width, and you may rely on that figure even though it is a rounded one.
Why Measure Around a Square at All?
Many of the mefarshim raise a further question on the Mishnah's second clause. Why do I need to be told that a square beam requires four tefachim of circumference? If the beam is square, simply measure the top or the bottom and see whether the width is a tefach. Why should I go through the exercise of measuring up one side, across, down the other side and along the bottom, arriving at four tefachim, and then taking a quarter to discover that the width is a tefach? Various mahalachim are offered to explain what the Mishnah is teaching with this, and we will leave the matter here.
Summary
Our case is a beam spanning the room from one wall to the other. A width of one tefach gives it the din of an ohel, and because a wooden beam is not a keli it functions in both directions at once: tumah below it makes tamei everything under the beam's whole length, including keilim far from the tumah, while anything resting above remains tahor; and tumah on top makes tamei what lies beneath it, with nothing rising through in the reverse direction. Should the width fall short of a tefach, there is no ohel here: tumah below breaks upward to affect what is immediately above, tumah above breaks downward to affect what is immediately beneath, and nothing at all is carried sideways. To establish that the tefach of width is present, take the measurement around the beam: three tefachim for a round one, four tefachim for a square one.