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2. Bathing in water that was not heated through fire:
A. Hot Springs:
All the above is with regards to water heated through fire, however the hot water [from the] springs of Tiberius [and all other springs] are permitted for one to even dip his entire body inside of. [See Halacha 3 regarding if this is allowed when done for healing purposes!]
Hot water of Tiberius that is in a vessel: However it was only permitted [to bathe] in the hot waters of Tiberius when [the water] is in the ground, however when it is in a vessel then it has the same laws as water heated over fire being that all water that is in a vessel is not recognizable if it was [heated] from a fire or if it is from the hot springs of Tiberius, and there is thus [reason] to decree on these [hot waters of Tiberius] because of the hot waters [heated through fire]. [See Q&A regarding water heated through the sun!]
B. A roofed spring:
First opinion: [However] there are opinions which say that [the Sages] only permitted [washing oneself in] the hot springs of Tiberius when the area is not roofed, however if the area is roofed then it is forbidden.
Their reason is: because [a roofed spring] causes one to sweat and sweating is forbidden on Shabbos as will be explained later on.
Second opinion: [However] there are opinions which say that it is permitted to sweat in the hot waters of Tiberius.
Their reasoning is: because [the Sages] only forbade one from [actively causing himself to] sweat due to the decree made against bathing, as will be explained later on, and thus since it is permitted to bathe in the hot springs of Tiberius then certainly it is permitted to sweat in them.
The Final Ruling: Regarding the final ruling, in [a controversy over] a Rabbinical prohibition one may follow the lenient opinion.
C. The custom today by all springs:
Due to that not everyone knows to beware in all the Halachic matters involved in bathing [as brought in 2B], therefore the custom spread in these provinces to not bathe at all on Shabbos, not even with cold water [and certainly not by hot springs of Tiberius] even though there is no prohibition involved from the letter of the law [in doing so].
From the letter of the law, one may bathe even his entire body in a natural hot spring, as long as the water is not in a vessel. One may be lenient to bathe in it even if the spring is under a roof. However the custom is to avoid bathing even in cold water.
May one bathe in water that was heated by sun, such as in one’s solar boiler?
Heated from before Shabbos: It is forbidden to bathe majority of one’s body in such water even if it was heated before Shabbos. However one may wash minority of one’s limbs in such water if the water became heated by the sun before Shabbos. However, one must beware not to transgress the cooking prohibition in doing so, as cold water enters into the boiler upon opening it. It is thus only allowed to be opened if either a) The water in the boiler is below Yad Soledes [110 F.] or b) One shut off the water pipe which adds more cold water to the boiler.
Heated on Shabbos: If the water was heated by the sun on Shabbos, even if it is merely warm, then it is implied from Admur that it is forbidden to wash even minority of one’s body with it. However, there are Poskim who rule that even when heated on Shabbos it is allowed to bathe in such water minority of one’s body. Practically, we are stringent like Admur. Thus, if one brought a bottle of water outside into the sun and it became heated, one may not bathe even minority of his limbs in this water on Shabbos. However, if a cold bottle became room temperature, seemingly there is no need to be stringent. Likewise, one may wash minority of his body with water from the solar boiler even if it became heated through the sun on Shabbos.
 Admur ibid; Michaber 326:1; Shabbos 40a
Previous times: In previous times the Sages decreed even against bathing in hot spring water, however they later retracted their decrees when they saw the people were unable to uphold it. [Admur 301:60 as edited by Kuntrus Hashulchan 58; Shabbos ibid]
 M”B 326:8
 Admur ibid; Michaber ibid; Sefer Hateruma 233
 Admur ibid; Michaber ibid; Tur; Sefer Hateruma ibid; Semag L.S. 65
 326:2; Michaber 326:2
 First opinion in Admur and Michaber ibid; Rif Shabbos 62b; Geonim brought in Maggid Mishneh 22:2; Rambam ibid
 Admur ibid; Michaber ibid
 second opinion in Admur and Michaber ibid; Tosafus 147a; Rosh 22:12; Ramban; Rashba; Ran
 Admur ibid
 Admur ibid; M”B 326:11 in name of Elya Raba
 Admur 326:6
 Admur 326:1
 Admur 326:2
 See Piskeiy Teshuvos 326:2; Shabbos Kehalacha 18:35
 Implication of Admur in Mahdurah Basra chapter 259 Page 884 in the new Shulchan Aruch; SSH”K 14:3 in name of RSZ”A based on the stringency of Admur regarding sun heated water.
The reason: Since the water is within the boiler it has the same status as spring water that is in a vessel and is hence forbidden. [so is implied from SSH”K ibid] However from Admur ibid it is implied that even if the water is not in a vessel, if it was heated from the sun it is forbidden, even if it was heated before Shabbos and it is hence much more stringent than spring water. This is implied from the fact that Admur ibid uses the allowance to bathe in warm river water as a proof that it is permitted to bathe in warm water. This implies that if the river water were to be hot it would be forbidden. Now, the river water of Friday night was heated by the sun before Shabbos, hence implying the prohibition applies even with water heated by the sun before Shabbos, even if it is not in a vessel.
Other Opinions: Some Poskim rule that water warmed through the sun is even more lenient than spring water that is within a vessel, and is permitted to be fully bathed in on Shabbos even if it is in a vessel. The reason for this is because people do not become confused between warm water in a vessel and hot water heated by fire. [Tehila Ledavid 326:3]
 So is implied from the next Halacha [E] regarding warm water in which Admur rules in the Mahdurah Basra that if the river water was warmed by the sun before Shabbos it is allowed, as proven from the fact that one may bathe in a warm river on Shabbos, thus implying that if the water got heated on Shabbos through the sun then this would not be allowed. So understands also Tehila Ledavid 326:3 and SSH”K 14 footnote 17 and so concludes Piskeiy Teshuvos 326:2
 Tehila Ledavid 326:3, thus arguing on his understanding of Admur;
 Piskeiy Teshuvos 326:2
 See Tehila Ledavid ibid
 SSH”K 14:3; Piskeiy teshuvos ibid; Vetzaruch Iyun from Admur ibid
The reason: As since one did no action to heat the water it is considered as if it were to be heated before Shabbos, similar to water placed on a Blech before Shabbos. [SSH”K 14:3]