Ohalos, Chapter 12, Mishnah 7. Our Mishnah continues to teach us how a small space, even a space we would hardly notice, can qualify as an ohel and change everything about how tumah travels. Here the Mishnah takes a case that asks us to picture something very ordinary and then look at it very carefully.
The Mishnah opens: "Amud shehu mutal la'avir", a pillar, a large round pole, that is lying out in the open air. It is not standing upright; it is lying on its side on the ground. And it is not under any roof: if you stand beside it and look up, you see the clouds, or on a clear day, the blue sky above you.
The space nobody thinks about
Close your eyes for a moment and picture yourself walking up to a very large round pole resting on the ground. Because it is round, the part that bulges out toward you is at its middle. From that middle point, the surface begins curving back inward, both upward and downward, until at the very bottom the pole touches the earth.
Now stand at the side of the pole and touch the point that sticks out the furthest. If you were to drop a straight line from that point all the way down to the ground, you would discover that there is a space behind your line: a wedge of air between the ground and the curving underside of the pole.
If you do not happen to have a large pillar lying around the house, take a round pen and lay it on the table. Look at the shadow underneath it. Imagine a line dropped from the widest point of the pen straight down to the tabletop. Something very thin could slide in behind that line, into the little sloping gap that runs from where the curve begins receding until the point where the pen rests on the table.
The question
The Mishnah's question is about exactly that wedge of space beneath the sloping side of the pole. Suppose tumah is lying there in one spot, and further along that same gap, underneath the pole, there is a keli. Is there a tefach by a tefach of space in there? If there is, that space is an ohel, and the keli becomes tamei. If there is not, it is not an ohel. Our Mishnah gives us a way to measure it without ever crawling underneath.
The measure: twenty four tefachim around
The measurement given is: "Im yesh behekeifo esrim ve'arba tefachim", if its circumference comes to twenty four tefachim. Wrap a measuring tape about the amud and follow it once fully around. Should the tape read twenty four tefachim, the ruling is "mevi es hatumah tachas dofno": the pole conveys tumah beneath its side. In other words, in the low region where the surface bends back toward the earth, you are guaranteed a square tefach of airspace, and such a space carries the status of an ohel.
From this two rulings follow. If tumah lies at one place within that wedge and a keli sits at another place within it, the keli contracts tumah, since an ohel carries tumah horizontally along its length. The reverse holds as well: should the tumah be lodged in the gap while a keli rests on the pole's upper surface, the ohel functions as a barrier against tumah rising, and that keli remains tahor.
"Ve'im lav", and if not: should the tape come back with a figure below twenty four tefachim, the sliver of air near the earth will not measure a full tefach in each direction once you square it off. Nothing there covers a square tefach, so no ohel exists, and tumah has no route by which to travel horizontally beneath the pole.
Consequently, tumah at one spot in that gap leaves a keli lying at another spot within it tahor. But the coin has two sides. The very absence of an ohel means that nothing is present to restrain the tumah or seal it in. It is bokaas ve'olah, bokaas veyoredes: it breaks upward and it breaks downward. Should part of a meis rest beneath the curving underside of the amud, the tumah ascends straight through the wood and defiles a keli positioned directly over it on the pole's rounded surface. Reverse the picture: put a piece of a meis on top, where the surface slopes outward toward the widest point, and the tumah travels in a straight line down through the pole and defiles a keli lying below it.
To sum up the principle: with a tefach by a tefach you have an ohel, which blocks tumah from passing through it but renders tamei whatever is under it. Without a tefach by a tefach there is no ohel, so nothing spreads sideways, but the tumah pierces upward and downward in a straight line.
For those who enjoy the mathematics
A reasonable question presents itself: how were Chazal able to establish that a circumference of twenty four tefachim yields precisely a square tefach down in that lower wedge? If the halachah as such satisfies you, feel free to continue on. For anyone who wishes to travel the road with us, here is the reasoning, one stage at a time.
Go around to the end of the pole and look at its circular face. Because the pole is lying down, you do not have to climb anywhere to see it. Take a marker and draw two lines across that circle: one from the top straight down to the bottom, and one from the right side straight across to the left. It will look like the view through a ship's periscope, or like the cross hairs on a radar screen. The circle is now divided into four quarters: upper right, upper left, lower right, lower left.
Chazal supply us with the practical ratio between a circle's perimeter and the line drawn through its center: that crossing line equals one third of the perimeter. This is why a circumference of three tefachim corresponds to a diameter of one tefach. Our circumference is twenty four tefachim, so the line across the face measures a third of that, eight tefachim. Facing the end of the pole, the span from the right edge to the left edge through the middle is eight tefachim, and the span from the upper edge down to the lower edge is eight tefachim as well.
A reminder before we continue: Chazal themselves tell us that three to one is not the precise ratio. It is an approximation, the closest simple figure to work with, and Chazal ruled that we may work with it.
The circle inside the square
Now picture a square frame built tightly around that circular face. Each end of my vertical line meets the frame, above and below; each end of the horizontal line meets it too, on the right and on the left. Yet in all four corners of the frame a bit of empty area remains, because the circle bends away while the frame's sides continue on straight.
The square's side from top to bottom, and from side to side, is the same eight tefachim as the circle's lines, since those are precisely the points where circle and square meet. And again, why eight? Because twenty four around gives you a third of that across.
Now, how long is each of the four lines forming the quarters? Measuring upward along the vertical, from where the rim sits at the lowest point until you reach the middle, gives half of eight, which is four tefachim. Traveling from the middle out to the left rim likewise gives four tefachim. Think of a round pie (pardon the mashal) cut into four portions: every portion has one curved edge and two straight edges, and each straight edge comes to four tefachim. Indeed, any straight line drawn from the middle point outward until it reaches the rim measures four tefachim.
The corner: the million dollar question
Let us focus on one quarter, say the lower left; all four corners are identical anyway. Draw a diagonal line inside the square, from the lower left corner of the square across to the upper right corner. Most of that line runs inside the circle, but the stretch nearest the corner lies outside the circle, in that leftover curved space.
Everything hinges on this question: what is the length of the diagonal's segment that runs from the square's corner until it arrives at the circle's rim? The response: roughly one tefach, and the meforshim explain that it is in fact slightly more than a tefach.
Where does that come from? Chazal give us the rule for a circle set inside a square: the diagonal reaching the corner is longer than the corresponding line inside the circle, by a factor of one and two fifths, 1.4. We established that the line from the edge of the circle to its center is four tefachim. Stretch that same line out to the corner of the square, and by Chazal's rule it grows to about a tefach and a fraction more. That extra stretch, the space in the corner of the square where the circle has already curved away, comes out to a little more than a tefach. And that is exactly what we were looking for: proof that there is a tefach by a tefach available in that corner space, which is the space beneath the sloping side of our pole.
One closing thought. The precise relationship between a circle and a square is among the most difficult questions in all of mathematics. When you look at how Chazal arrived at their figures, it is pilei pelaos, wonders upon wonders, coming as close as it is possible to come, because an exact answer is unattainable; only the Borei Olam knows it exactly. And with that approximation Chazal handed us, we can stand beside a pillar lying in the open air, run a tape measure around it, and know whether the sliver of space beneath its curve is an ohel or not.