Mikvaos, Chapter 6, Mishnah 11. Our chapter has been occupied with mikvah samuch l'mikvah, one pool of water resting beside another, and with the question of when a mikvah psulah, a gathering of water that is not valid, ruins the kosher mikvah next to it. This Mishnah presents another such arrangement, this time in a bathhouse.
"Hamtahares sheb'merchatz": the rinsing pool in the bathhouse. The practice was that bathing took place in two stages. A person would first enter one bath, which held the hotter water, and it was then customary to step out of it into the pool alongside and cool himself off in the second one.
The Mishnah pictures the two pools on a slope, one above the other. "Tachtonah meleiah she'uvin": the lower pool is full of mayim she'uvin, drawn water. "V'ha'elyonah meleiah kesheirim": the upper one contains forty se'ah of rainwater and is therefore a kosher mikvah. Between the two pools, however, there is a nekev, an opening.
The Question of the Opening
We would like to treat the invalid pool as a separate entity, an unconnected mikvah b'tzad mikvah, so that the water we are gathering remains valid. But the drawn pool holds three lugin or more of she'uvin, and perhaps it is entirely drawn water. Everything therefore turns on the width of the opening between them. If the opening is wide enough that three lugin of the drawn water stand exposed to the kosher mikvah through it, the mikvah being filled with rainwater becomes invalid.
"Im yesh k'neged hanekev shlosha lugin, posul": if there are three lugin of water facing the opening between the two pools, the mikvah is posul.
Working Out the Measurement
"Kama yehei banekev v'yehei bo shlosha lugin?" How large must the opening be for us to know that three lugin stand within it? Let us do the arithmetic. A shiur mikvah is forty se'ah. Every se'ah contains six kabbim, and every kav contains four lugin. Forty se'ah therefore amounts to nine hundred and sixty lugin, which means that three lugin are one three hundred and twentieth of a mikvah.
That is precisely what the Mishnah answers: "Echad mishlosh me'os v'esrim labreicha," one part in three hundred and twenty of the pool. Where the mikvah holds exactly forty se'ah, an opening of that fraction of its volume tells you that three lugin are present at the nekev.
The Position of Reb Eliezer
The Position of Reb Eliezer
Reb Eliezer holds otherwise, and he does so even in the mirror image of the case. Picture the pools switched around: the water below is the valid one, while the pool above it is the one holding she'uvin. Even when three lugin of that drawn water stand at the opening joining them, his ruling is "kasher." To his mind, a mikvah in the course of collecting its rainwater suffers no damage from three lugin of a mikvah psulah being exposed to it through a nekev.
His reasoning: "shelo amru ella shlosha lugin shenaflu." The entire halacha of three lugin disqualifying a mikvah before it has reached its forty se'ah was stated only regarding three lugin that actually fell in. Here nothing has fallen anywhere; the drawn water remains sitting in its own pool. It is therefore a straightforward case of mikvah b'tzad mikvah, two neighboring bodies of water, and neither one invalidates the other.