Ohalot, Chapter Twelve, Mishnah Six
Today’s mishnah discusses a beam that goes from one wall to another. It also deals with the rabbinic calculation of circumference, so prepare for some math!
1) [With regard to] a beam which is placed across from one wall to another and which has uncleanness beneath it:
a) If it is one handbreadth wide, it conveys uncleanness to everything beneath it;
b) If it is not [one handbreadth wide], the uncleanness cleaves upwards and downwards.
2) How much must its circumference be so that its width should be one handbreadth?
a) If it is round, its circumference must be three handbreadths;
b) If square, four handbreadths, since a square has a [circumference] one quarter greater than [that of] a circle.
Section one: If the beam is one handbreadth wide, it forms an ohel and the impurity spreads to everything below the beam.
However, if the beam is less than one handbreadth, then it is not sufficiently wide to form an ohel. The impurity goes up and down, but does not spread to the sides.
Section two: The general rule for determining the diameter of a circle for the rabbis is three times the circumference (we know that 3 should really be Pi, but they were close). So as long as the circumference is at least 3 handbreadths, the diameter is one handbreadth.
If the beam is square, then its circumference must be four handbreadths for it to have a diameter (in this case, side) that is one handbreadth. A circle whose diameter is one handbreadth can fit into a square whose sides are each a handbreadth. The circumference of this square will be four.