Komentarz do Ohalot 12:7

עמוד שהוא מוטל לאויר (a column which lay [on the ground] in the open air) – a round column that is cast on the ground and lies on its circle. In the open air, as for example, in the courtyard or in the garden. And from the middle of the circle it begins to overshadow on the ground and gradually becomes shortened from here and there until its bottom is next to/adjacent to the ground, and regarding the mater of bringing defilement we require a handbreadth by a handbreadth at the height of a handbreadth squared, but if there is here to the side that is outward from underneath the height of a handbreadth even though that which is below is adjacent/next to in the place of his lying down on the ground, the height is not a handbreadth completely which defiles in the tent, and it is judged like the sloping of tents. But the round pillar which has in its circumference twenty four handbreadths is found that its width is eight handbreadths, that all that has in its width a handbreadth there is in its circumference twenty-four handbreadths, and it is found that a person is able to make a quadrangle underneath his partition/wall that is from here to and from there a handbreadth by a handbreadth. At the height of a handbreadth, there is here a square of eight [handbreadths] by eight [handbreadths] if you make within it a circle of eight [handbreadths], it is found that the corners overlap eight-fifths [handbreadth] for each and every corner, for each cubit in a square is a cubit and two-fifths crosswise/in a diagonal line, it is found that the diagonal line is in excess of the width by sixteen-fifths, take from them half for this corner and half for that corner, there is for each one eight-fifths, and if you come to make in the corner that is outside of the circle a square of a handbreadth by a handbreadth, it is found that his diagonal line/crosswise is in excess over its width by two-fifths, and it is found that the measure of the diagonal line is seven-fifths, there would not remain for us more than the need other than one-fifth, but because the one-fifth is not exact. And furthermore, that each cubit in a square is a cubit and two-fifths in a diagonal line – which is not completely lying exactly in line, for it is missing a bit from it.

Introduction Today's mishnah deals with a pillar that has fallen onto its side. There is going to be a space on each side of the pillar under which something could lie, because the pillar is round. For this space to be an ohel, the space must be one handbreadth square. Our mishnah calculates how large the pillar must be for the space to be a handbreadth.

[With regard to] a pillar lying [on its side] in the open air, If its circumference is twenty-four handbreadths, it brings uncleanness to everything under its side; But if it is not, the uncleanness cleaves upwards and downwards. Okay, ready for some math? If the circumference of the circle is 24, then according to rabbinic math, the diameter is 8. You could then put a square around this diameter where each side is 8 handbreadths. The diagonal line drawn from one end of this square to another would be 8 + 16/5. This is according to the rule that every handbreadth on the side of a square is equivalent to 1 1/5 handbreadths (we know that the real length is the square root of 128, according to the formula a2 + b2=c2. Still, they were pretty close) At the corners of this square with the circle in it would be a smaller square whose diagonal is 8/5 of a handbreadth (half of the extra 16/5). This is the area which would form the ohel over the uncleanness. Since it is more than a handbreadth, it brings uncleanness to the entire pillar. If the pillar is smaller than that, then the overhanging part will be less than a handbreadth and it will not convey impurity.