Gemara

39b (Beraisa #1): On Shabbos, one may wash his face, hands and feet in water heated from before Shabbos, but not his whole body.

(Beraisa #2 - R. Meir): One may not shower his entire body in hot or cold water;

R. Shimon permits;

R. Yehudah forbids in hot water, and permits in cold water.

(Rav Chisda): They argue about bathing in the ground, but all forbid bathing in a Kli.

(R. Yochanan): The Halachah follows R. Yehudah.

40a - Version #1 (Rav): If water was heated from before Shabbos, one may bathe his whole body in it on Shabbos;

(Shmuel): He may wash only his hands, feet and face.

Support (for Shmuel - Beraisa): One may wash his face, hands and feet in water heated from before Shabbos, but not his whole body one limb at a time, and all the more so regarding water heated on Yom Tov.

Version #2 (Rabah citing Rav): One may bathe his whole body in water heated from before Shabbos, except for one limb.

The Beraisa refutes Rav. (end of Version #2)

(Beraisa): If the holes of a bathhouse were plugged up from before Shabbos, one may bathe in it on Motzei Shabbos immediately;

After many people transgressed, Chachamim began to forbid (bathing).

(R. Shimon ben Pazi): In the end, Chachamim permitted bathing in hot springs and cold water, but Zei'ah (a steambath) is forbidden.

(Beraisa): One may walk through a city bathhouse.

(Rava): Bathhouses of villages are forbidden.

Since they are small, they get very hot (he will be Mezi'a).

(Beraisa): [On Shabbos] one may heat himself in front of a fire and shower in cold water. One may not shower in cold water and then heat himself in front of a fire, for this makes the water lukewarm

109a (Beraisa #1): One may bathe in water of Gerar, Chamsan, Asya, or Tiverya, but not in the Yam ha'Gadol (Mediterranean Sea), in Mei Mishre (water in which flax was soaked) or in the Dead Sea.

Contradiction (Beraisa #2): One may bathe in water of Tiverya or Yam ha'Gadol, but not in Mei Mishre or the Dead Sea.

Resolution #1 (Rav Nachman bar Yitzchak): One may not stay in the water for a long time (it is evidently to heal). One may bathe quickly.

Objection: A Beraisa permits bathing quickly in Mei Mishre!

Resolution #2: One may bathe in nice parts of Yam ha'Gadol, but not where the water is bad. One may bathe quickly in Mei Mishre and the Dead Sea, but not for a long time.

147a (Mishnah): If one bathed in a cave or in Chamei (natural he is particular spring water of) Tiverya and wiped himself, even if he used 10 towels, he may not take them;

Inference: Just like Chamei Tiverya are hot, also the cave water.

'If one bathed' connotes b'Di'eved. However, showering is l'Chatchilah! This is like R. Shimon (39b).

Rishonim

Rif and Rosh (18b and 3:6,7): The Halachah is, one may not bathe in hot water heated on a fire, in Kelim or in the ground. One may bathe in Chamei Tiverya. One may wash his face, hands and feet in water heated before Shabbos, but not the entire body, (even) one limb at a time, and all the more not in water heated on Shabbos.

Ran (DH Aval): R. Yehonason asked, since the Rif rules like Shmuel, why did he need to forbid the entire body in hot water? He answered that he rules like R. Yehudah to permit in cold water. I say that he needed to forbid [hot water] even in the ground.

Ran (DH Amar Rebbi): The Yerushalmi says that they left water over a fire to get hot on Shabbos. Yisraelim are not suspected of transgressing Shabbos mid'Oraisa

Rif and Rosh (62b and 22:12): Here we permit bathing only b'Di'eved. Above, we permitted in Mei Geder and Chamei Tiverya. Just Zei'ah is forbidden! Rather, the Mishnah says 'one who bathed' (b'Di'eved) due to cave water. Chamei Tiverya was taught with it to teach that the cave water is hot, just like Chamei Tiverya.

Rif (ibid.): Cave water is forbidden l'Chatchilah because the cave is covered, so people will come to sweat.

Rebuttal #1 (Rosh ibid.): If cave water becomes hot due to the cave, we would not forbid l'Chatchilah due to this! Rather, the cave water was heated on a fire. We infer this, for we say that the Mishnah is like R. Shimon, and unlike R. Yehudah. They argue about water heated on a fire.

Rebuttal #2 (Ran DH Gemara): They forbade Zei'ah only due to bathing. Since one may bathe in Chamei Tiverya, surely Zei'ah is permitted! Rather, the cave water was heated on a fire.

Rambam (Hilchos Shabbos 22:2): Chachamim forbade entering a bathhouse on Shabbos due to bathhouse attendants who would heat water on Shabbos, and say that it was heated from Erev Shabbos. They forbade entering a bathhouse even to sweat. They decreed not to shower the entire body even in water heated from Erev Shabbos, but the hands, feet and face are permitted. This refers to water heated on a fire. We decree due to bathhouses. One may shower the entire body in Chamei Tiverya and similar [hot springs]. One may not bathe in hot water in caves, for there is hot air and one will come to sweat, so it is like a bathhouse.

Mordechai (303): One may shower (pour water on the limbs) in cold water, but not bathe, even in the ground. The Ri says that this refers to ground like a Kli, e.g. a pond, which is cut off from the river, just like a Kli is. All permit in the river. Also Avi ha'Ezri says so. The Beraisa on 109a proves so. We permit Chamei Tiverya, and [all the more so] one may enter cold water to cool off, without rubbing [to clean himself].

Tosfos (39b DH v'Ha): Here, R. Meir forbids showering the entire body even in cold water, even in the ground. A Beraisa (109a) permits bathing in water of Tiverya or Yam ha'Gadol, and we establish it like R. Meir! Perhaps here we forbid in the ground like a Kli, and there he permits in a river. In a river, it looks like he entered to cool off. Even in Chamei Tiverya, he does not look like one who bathes in hot water.

Tosfos (109a DH Rochtzin): On 147a, we say that it is b'Di'eved to bathe in Chamei Tiverya. There we discuss bathing in the hole (the source), similar to a bathhouse. We permit in a river l'Chatchilah.

Poskim

Shulchan Aruch (OC 326:1): One may not bathe the entire body, even each limb individually, even in water heated from before Shabbos, whether in a Kli or in the ground. One may not even pour the water on himself.

Mishnah Berurah (2): Also the majority of the body is forbidden. The majority is like the entire body.

Kaf ha'Chayim (6): Women who immerse in the river and afterwards pour hot water on themselves may pour only on half the body. The Ran says that people used to pour water on themselves after Zei'ah. Therefore, there is concern lest people suspect that the water was heated on Shabbos.

Gra (DH Bein): It is clear from 147a that even R. Shimon argues only about showering [but bathing is forbidden in a Kli or in the ground].

Bi'ur Halachah (DH b'Mayim): R. Akiva Eiger permits one in pain to bathe in water heated from Erev Shabbos, even if the entire body is not sick.

Kaf ha'Chayim (5): It is a bigger Chidush to forbid in the ground. Water in a Kli does not retain its heat very long, so surely it looks like the water was heated on Shabbos. We forbid even in the ground, even though it can hold the heat for an entire day.

Shulchan Aruch (ibid.): However, one may wash his hands, feet and face with it.

Beis Yosef (DH Asur): Rashi explains that "Lehishtatef" is pouring on the body. This is unlike bathing, which is entering his body or limbs into water. Semag and Sefer Terumah say so.

Beis Yosef (DH v'Havah): It seems that we permit not only the hands, feet and face. The same applies to other limbs, as long as he does not bathe the entire body. However, Rashi explains that the Isur to put a kettle on the stomach is lest it spill, and he will bathe in hot water on Shabbos, even though it is only part of his body. However, the Rosh (Hilchos Mikva'os 37) permits heating a flask to rinse Beis ha'Setarim (covered places, e.g. the underarms), like Beis Hillel, who permit heating water on Yom Tov to wash the feet, but not the entire body. This implies that the Heter is not limited to the hands, feet and face. Even though there he discusses Yom Tov and here we discuss Shabbos, this is not a difference between Shabbos and Yom Tov. Perhaps Rashi agrees. He wrote 'lest the water spill on him.' He did not say 'lest it spill on his stomach.'

Taz (1): The Beraisa permits the hands, feet and face, but not the entire body one limb at a time. This implies that any one limb is permitted. Why did the Beis Yosef need to derive this from the Rosh?

Mishnah Berurah (5): If the water was heated on Shabbos, one may not wash even his hands in it.

Rema: The same applies to other limbs, as long as he does not bath his entire body.

Magen Avraham (3): Mas'as Binyamin (5) says that the custom is that women do not begin to wear white [to count seven clean days] on Shabbos or Yom Tov. Perhaps this is because they do not know to be careful about bathing. (They wash the area first.) Where there is no custom we permit, but she may not wash with a cloth, lest she come to squeeze.

Rebuttal (Shach YD 199:12, brought in Machatzis ha'Shekel): This is a flimsy reason. Our custom is that women begin on Yom Tov, but not on Shabbos. According to Mas'as Binyamin, there is no reason to distinguish! Rather, if she begins counting on Shabbos, (since seven full days are required) she will immerse on Motza'ei Shabbos. Chachamim tried to avoid this, for then she immerses a long time after shampooing [on Erev Shabbos].

Shulchan Aruch (ibid.): This refers to water heated through fire. One may bathe even the entire body at once in hot [spring] water of Tiverya. We need not say that it is permitted with cold water.

Beis Yosef (DH v'Chasuv): Sefer Terumah (233) says that Chachamim permitted bathing the entire body at once in Chamei Tiverya. The Gemara says 'Rochtzin', which connotes the entire body. I say that all the more so one may bathe in cold water. R. Yehudah says 'Lehishtatef is Asur in hot water, and permitted in cold water.' Rather, he permits even bathing in cold water. He mentioned Lehishtatef because R. Meir forbids even this in cold water. However, the Mordechai permits only showering in cold water.

Mishnah Berurah (7): In a place where they heat the Mikveh, one must ensure that it is only lukewarm. If not, one may not immerse in it.

Shulchan Aruch (ibid.): Chamei Tiverya are permitted only in the ground, but not in a Kli, lest it be confused with water heated on a fire.

Beis Yosef (DH v'Chasuv): Semag and Sefer ha'Terumah permit Chamei Tiverya only in the ground. This is reasonable, for if it is in a Kli, it is not evident if it is Chamei Tiverya or water heated on a fire. The Tur says that the Rosh permits in every case. The Rosh did not explicitly say so. He Stam permitted Chamei Tiverya without distinguishing between the ground and a Kli. This connotes that he permits in every case.

Gra (DH v'Ha): The Beraisa that permits Chamei Tiverya initially discussed a bathhouse in which the holes were plugged up. It is heated from underneath. The Beraisa on 147a forbids even Chamei Tiverya l'Chatchilah, i.e. in a Kli. Even R. Shimon agrees.

Mishnah Berurah (8): Even hot springs are forbidden where this is done for Refu'ah.

Mishnah Berurah (9): The custom is not to bathe in a river.