The Mishnah (2:5) teaches:
A city that is populated by yisraelim and nochrim, and there is a bathhouse that was used on Shabbat, if the [population] is majority nochrim then a yisrael can use it immediately after Shabbat. If there is a majority yisraelim then one must wait for the time it would take to heat the bath [before using it]. If it is half-half, then one must wait for the time it would take to heat the bath. R’ Yehuda says, that if the bath is small and the authorities have rights [to use it as they wish], then one can use it immediately after Shabbat.96
How do we understand the Mishnah?
When there is a majority of yisrael, it is presumed that the water was heated for the yisraelim – consequently it cannot be used immediately after Shabbat, rather they must wait the time it would take to heat the bath. The Mishnah Berurah explains that this is so that no benefit is derived from the nochri performing the melacha for them on Shabbat (Rashi, Ran). Furthermore it is a gezeirah to prevent one from asking a nochri to perform melacha on Shabbat (which is forbidden) in order that he can use the product immediately after Shabbat (Tosfot, Rambam).
The Mishnah Achrona points out that there are times that the product can be assur indefinitely. Unlike this case, it is where the melacha is performed public. This law is learnt from the Gemara Shabbat (151) where a coffin or grave site was constructed in a public manner. The Mishnah Berurah however cites the Ran that explains that the indefinite issur is only in the case of a grave site where it would be a disgrace to the deceased to be buried in place where it is known publicly that Shabbat desecration occurred. (He rules that that opinion can be relied on in cases of great need.)
Returning to the Mishnah, when there is a majority of nochri citizens, it is presumed that the water was heated for the nochrim and it can be used immediately after Shabbat. The Mishnah Achrona adds that the water cannot however be used on Shabbat itself. He continues noting that we learn in Gemara Shabbat (122a) that if there is a group with a majority of nochrim and a nochri lights a candle, all can benefit from the light. Why is this case different? The Tosfot explains that the Chachamim understood that there is an extra concern by food that one might perform the melacha themselves, as one is naturally draw to food. Such a concern does not apply to candle light. The Mishnah Achrona explains that since bathing also benefits the body, it too shares the same concern.
One question that was asked is that even in a city where the majority of the population are nochrim, it is quite possible that the majority of the people that will use the bathhouse after Shabbat will be yisraelim. Consequently does that impact on our understanding of the Mishnah? The Mishnah Berurah (326:38) explains that the majority that is referred to in the Mishnah is not referring to the dwellers but rather the people that would use the bathhouse motzei Shabbat. The reason is that if the majority of the users at that time are yisraelim, even though they might be the minority of the population, it can be safely assumed that the heating performed towards the end of Shabbat was for them.
96 Whether R’ Yehuda is arguing or adding to the first opinion is debated in the Rishonim. The Rashi implies that R’ Yehuda is adding to the first opinion, while the Rambam understand that the point that R’ Yehuda raises is debated. A discussion of this debate is beyond the scope of this article.
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