Gemara

(Beraisa): If a hot food fell onto a cold food, one rinses them off.

Objection: Surely, before the cold cooled off the hot, there was some absorption. One should have to peel off a layer!

Correction: Rather, one peels off a layer.

Shabbos 40b (Beraisa - R. Shimon ben Gamliel): One may anoint her hand with oil, heat it in front of a fire, and anoint her son without any concern.

He holds that cooking applies to oil, and warming it is cooking it. He permits only through a Shinuy (e.g. on the hand).

(Rav Yehudah): The law is the same for water and oil. Yad Soledes Bo is forbidden. Less than this is permitted.

Question: How hot is Yad Soledes Bo?

Answer (Rachbah): It is hot enough to scald a baby's stomach.

134b - Question (Chachamim of Eretz Yisrael): May one wash the entire body (on the third day from Bris Milah), or just the wound?

Answer (R. Yakov): One may put hot water on any wound. The Heter for Milah must be to wash the entire body.

(Rav): We do not refrain from putting hot water and oil on a wound on Shabbos.

Objection (Rav Yosef): For a regular wound we may only use water heated from before Shabbos. For Milah, we may use water heated on Shabbos! (Perhaps this is the leniency of Milah, but not to wash the entire body.)

Chulin 105b (Rav Yitzchak bar Yosef): One may not use for Netilas Yadayim water that is Yad Soledes Bo.

Rishonim

Rambam (Hilchos Shabbos 22:3): One may not rinse his entire body in water and warm himself near a fire, for he warms the water on him. This is like bathing the entire body in hot water.

Magid Mishneh: Warming water is not intrinsically forbidden. It is forbidden only due to bathing. The Ramban says so.

Rosh (Shabbos 3:9): One may not rinse his entire body in water and warm himself near a fire, for he warms the water on him. Therefore one must be careful not to warm his hands near a fire after washing them, unless he dries them first.

Rosh (10): R. Shimon ben Gamliel permits one to anoint her hand with oil, heat it in front of a fire, and anoint her son without any concern.

Ran (Shabbos 19a DH Tanu and DH Soledes): The Gemara permits to put water near the fire to get warm, but not hot. Rashi explains that one may put it near the fire where it could cook, as long as he removes it before it gets Yad Soledes Bo. The Ramban disagrees, for the text of the Ge'onim and Yerushalmi permits only in a place where Yad Sholetes Bo (the hand can stay there), but not where Yad Soledes Bo. Soledes means 'drawn back.' The Gemara asked about what is Yad Soledes Bo, for it is different for different people.

Me'iri (Shabbos 40b DH Mevi'ah): One may warm oil, for this is not cooking, and all the more so one may anoint her hand or child and warm it near a fire. However, it may not get so hot that if it were elsewhere and she touched it, her hand would recoil due to the heat. Regarding this we asked what is Yad Soledes Bo, and not regarding putting water near a fire. How can she gauge if the heat would make her hand recoil? (Surely, it is different when her hand is anointed. If it means that her hand would recoil now, there is no need to forbid! - PF. It seems that Mekor Chayim asked similarly, in what case do we forbid putting the body or hand near fire to get warm?)

Rosh (Avodah Zarah 2:13): After wine was boiled, it is called cooked. In Shabbos (40b), Amora'im argue about whether warming is called cooking. All agree that boiling is cooking. The concern for intermarriage applies also to wine that was boiled! Perhaps because it is not common, Chachamim did not decree about it.

Ran (Avodah Zarah 10a DH Yayin): The Ge'onim say that once wine was boiled, it is called cooked. The Ramban holds that wine is considered cooked after its volume diminished on the fire. The Yerushalmi forbids cooking Terumah wine, for this diminishes its volume. Perhaps both of these are the same. Through boiling, it diminishes.

Me'iri (76a DH mi'Mah): If hot meat fell onto cold milk, the milk does not receive heat in order to absorb anything. However, until the meat cools off, it absorbs a little of what is below, so one must peel off a layer. This is only for a heat that can cause emission [of absorptions], i.e. Yad Soledes Bo.

Rashi (Chulin 105b DH Soledes): If water is Yad Soledes Bo, it changed and lost the status of water.

Poskim

Shulchan Aruch (OC 326:4): One may not rinse his entire body in water and warm himself near a fire, for he warms the water on him. This is like bathing the entire body in hot water.

Magen Avraham (5): This is from the Rambam, who holds that this is not total bathing. Therefore, one may anoint his hands with water and warm them near a fire where it is not Yad Soledes Bo. This is why the Mechaber (Sa'if 5) brings another opinion.

Mishnah Berurah (17): Warming is not cooking. Since it is not total bathing, it is considered like water warmed from before Shabbos, and only the entire body is forbidden. The stringent opinion in Sa'if 5 considers it like bathing in water warmed on Shabbos. This is forbidden even for the hands, feet and face. Even though here, the water was never hot, we do not distinguish. Even if it was only warmed, and it is not Yad Soledes Bo, one may not bathe even one limb in them.

Shulchan Aruch (5): Some say that one must be careful not to warm his hands near a fire after washing them, unless he dries them well.

Beis Yosef (DH u'Mah): The Mordechai agrees with the Rosh, and it seems that also Tosfos and Rashi do. According to the Rambam, one may warm his hands while they are wet. The Rosh holds that one may warm water for the sake of drinking, but here this is like bathing in hot water. One might come to heat water for bathing the body. Alternatively, it is forbidden only where it could reach Yad Soledes Bo. The Yerushalmi says so.

Magen Avraham (6): The Mechaber forbids only where it is Yad Soledes Bo. The Rosh forbids due to bathing. We find that on Yom Tov, one may heat water to drink, but not for bathing!

Bi'ur Halachah (DH Yesh): The Magen Avraham forbids even where the water cannot become Yad Soledes Bo. If it could become Yad Soledes Bo, all would forbid. The Gemara permits to anoint the hand and warm it near a fire, because the decree of bathing did not apply to oil. Surely it is not permitted where it could become Yad Soledes Bo. According to the Beis Yosef's latter Perush, that the Rosh is stringent due to heating the water, the Rosh contradicts what he brings afterwards from R. Shimon ben Gamliel! At least he should have explained that R. Shimon ben Gamliel does not discuss this!

Bi'ur Halachah (DH Etzel): One should be stringent also about the wall of the oven, if it is very hot.

Chelkas Yakov (YD 44:3, 124:4): If water gets between 50 and 70 degrees (Celsius), this is Yad Soledes Bo. A woman should not immerse in water that is Yad Soledes Bo. Rashi (105b) says that it changes, and ceases to be water. If so, it is a problem even if it cooled off afterwards. Here, they heat the water during the day to 45 degrees, so it will cool off a little and be proper for Tevilah at night. This is a Safek Yad Soledes Bo.

Shulchan Aruch (YD 123:3): Our cooked wine is permitted if a Nochri touched it. It is called cooked after boiling it on a fire.

Beis Yosef (DH v'Chen, citing the Rashba): Any wine that was boiled is considered cooked. The Ge'onim had such a tradition. Some say that it is not called cooked until its volume diminished through cooking. It seems that one cannot boil it without diminishing its volume. We rely on the Ge'onim's tradition.

Shach (7): It must be boiled on a fire, i.e. until its volume diminishes.

Igros Moshe (YD 2:52): Surely, if wine was heated to Yad Soledes Bo, it is considered cooked, even if it does not bubble. Yad Soledes Bo is the Shi'ur for cooking for all laws regarding liquids. The Shach says that the volume of the wine must decrease due to the cooking. Surely, if it is heated to Yad Soledes Bo, it decreases somewhat. This is about 79 degrees, to be stringent.

Igros Moshe (YD 3:31): In wine factories, they cook the wine to 74 degrees. This is surely Yad Soledes Bo, so there is no problem if Nochrim touch it (afterwards). Surely this decreases the wine. I saw testimony from chemists that this decreases the volume more than 10%. One may rely on this in practice.

Igros Moshe (OC 4:74): It is proper to be stringent to be concerned lest 43 degrees is Yad Soledes Bo. There is a Safek until 71 degrees. That is surely Yad Soledes Bo, even to be lenient.

Minchas Yitzchak (7:61): A concentrated wine is made through heating it to between 60 and 82 degrees under a vacuum. Through this, it cooks and diminishes without getting it to 100 degrees, for such heat burns the sugar, which is detrimental for the wine. Sixty degrees is surely Yad Soledes Bo. Under a vacuum, it bubbles. Really, a stronger vacuum could cause it to bubble and diminish at a lower temperature. Perhaps even when it is heated to 60 degrees, it is not considered cooked, and Nochrim can forbid it after water is added to return it to the concentration of grape juice. Since heating it did not change the taste, it is not considered cooked. Or, perhaps since it was Yad Soledes Bo, and bubbled and diminished, it is called cooked. Also if one cooks on a high mountain, he need not heat it to 100 degrees. It cooks (boils) at much less than this.

Yachel Yisrael (20:8): The Chacham Tzvi forbids Tevilah (on Shabbos) in what people call 'hot water', even if it is not Yad Soledes Bo. The Mishnah Berurah permits as long as it is not Yad Soledes Bo, like Shulchan Aruch ha'Rav. Ketzos ha'Shulchan and Tefilah l'David permit only if one feels some coldness. They forbid anything more than the temperature of saliva, like the Chavos Ya'ir (101). Darchei Teshuvah proved from Bahag that it (Yad Soledes Bo) is no less than 40 degrees. Minchas Shlomo said that it is at least 48, and in any case one may be lenient up to 45 degrees. Many Poskim are stringent for anything more than 40 degrees. They say that the Chazon Ish held like this. Igros Moshe says to be stringent for 43 degrees. Everyone should follow his Rebbi.