Inference: Shmuel holds like R. Shimon (who permits Davar she'Eino Miskaven).

Question: But Shmuel holds that we may extinguish (on Shabbos) a metal coal in Reshus ha'Rabim, in order that people will not be harmed (normally, this is forbidden mid'Rabanan), but not a coal of wood (for this is forbidden mid'Oraisa);

If he held like R. Shimon, he should permit extinguishing even a wooden coal (this is Melachah she'Einah Tzerichah l'Gufah, it is forbidden only mid'Rabanan)!

Answer: He holds like R. Shimon regarding Davar she'Eino Miskaven, but he is Mechayev for Melachah she'Einah Tzerichah l'Gufah like R. Yehudah.

(Ravina): Shmuel teaches that Isurim mid'Rabanan are permitted to prevent damage to the public - therefore, if a thorn is in Reshus ha'Rabim, one may carry it less than four Amos at a time [to the side of Reshus ha'Rabim];

If it is in a Karmelis, one may carry it [to where people do not walk] without stopping.

(Mishnah): But one may put...

(Beraisa - Beis Shamai): One may pour hot water into cold water, but not vice-versa;

Beis Hillel permit both of these.

This refers to a cup (a Kli Sheni used for drinking, people do not make it so hot);

However, in a bath (Tosfos - a Kli Sheni; alternatively, a Kli Rishon - Tosfos ha'Rosh), one may put only hot into cold [for people make it very hot].

R. Shimon ben Menasiya forbids.

(Rav Nachman): The Halachah is like R. Shimon ben Menasiya.

(Rav Yosef): A bucket (used for bathing) is like a bath.

Rejection (Abaye - R. Chiya's Beraisa): A bucket is not like a bath.

Question: How could Rav Yosef have thought that a bucket is like a bath - Rav Nachman taught that the Halachah follows R. Shimon [ben Menasiyah, who forbids mixing bath water in every case] - is there no way to bathe in hot water on Shabbos?! (Water on the fire is too hot, one must mix it with cold water; it is unfeasible to wait until it cools down.)

Answer: [Tosfos - Rav Yosef thought that] R. Shimon does not discuss the Seifa (a bath), rather, the Reisha - Beis Hillel permit hot into cold and cold into hot;

R. Shimon forbids [cold into hot].

Question: Does R. Shimon hold like Beis Shamai?!

Answer: No, he says that there is no argument about this (all forbid cold into hot).

(Rav Huna brei d'Rav Yehoshua): I saw that Rava was not particular about the Kli (Rishon or Sheni) - he relied on R. Chiya's Beraisa:

(R. Chiya - Beraisa): One may put a flask of water into a bucket, whether hot into cold or vice-versa.

Question (Rav Huna): Perhaps this is permitted only because a Kli separates the hot and cold water!

Answer (Rav Ashi): The Beraisa permits pouring a flask of water into a bucket, whether hot into cold or vice-versa.

(Mishnah): If one removed a frying pan or pot from the fire, one may not put spices into it [on Shabbos]; (R. Akiva Eiger - Rashi suggests that Bein ha'Shemashos this is permitted - perhaps he holds [like Tosfos] that cooking spices is not forbidden mid'Oraisa, and our Mishnah is like Rebbi, who permits Shevus (Rabbinic decrees of Shabbos) Bein ha'Shemashos.)

One may put spices into a bowl (a Kli Sheni).

R. Yehudah says, one may put [spices] into any [food], except for one with vinegar or brine.

(Gemara) Question: What does R. Yehudah discuss?

Perhaps he discusses the Reisha (putting spices in a Kli Rishon off the fire, compared to the first Tana), he is lenient;

Perhaps he discusses the Seifa (putting in a Kli Sheni), he is stringent.

Answer (Beraisa - R. Yehudah): One may put [spices] into any simmering pot or pan, except for one with vinegar or brine.

(Rav Yosef): Salt is like spices, [according to the first Tana] it is cooked only in a Kli Rishon, not in a Kli Sheni.

Version #1 - Objection (Abaye - R. Chiya's Beraisa): Salt is unlike spices, it is cooked even in a Kli Sheni.

The Beraisa also argues with Rav Nachman:

(Rav Nachman bar Yitzchak): Salt needs much cooking, like ox meat (it gets cooked only in a Kli Rishon).

Version #2 - Objection (Abaye - R. Chiya's Beraisa): Salt is unlike spices, it is not cooked even in a Kli Rishon [off the fire].

This supports Rav Nachman:

(Rav Nachman bar Yitzchak): Salt needs much cooking, like ox meat

(Mishnah): One may not put a Kli under a Ner to catch dripping oil;

If one put a Kli there from before Shabbos, it is permitted;

One may not benefit from the oil, for it is not prepared [for use on Shabbos, it is Muktzeh].

(Gemara - Rav Chisda): Even though it was taught that one may not put a Kli under a hen to catch an egg it will lay (e.g. it is about to lay an egg on an incline, it is prone to roll and break), one may put a Kli over an egg to prevent it from getting broken.

Question: What is the reason?

Answer #1 (Rabah): He holds that hens often lay eggs in wasteheaps (where they are in danger of getting broken), they rarely lay eggs on inclines;

Version #1 (Rashi): Chachamim permitted moving a Kli to save [something Muktzeh, like an egg laid on Shabbos] in a common case, not in an unusual case.

Version #2 (Tosfos): Chachamim permitted exertion to prevent a loss in a common case, not in an unusual case.

Question #1 (Abaye - Beraisa): If a barrel [of Tevel - Tosfos deletes this from the text] broke on the roof, one may bring a Kli underneath it [to catch the contents].

Answer: The Beraisa discusses a new barrel [it is prone to break].

Question #2 (Mishnah): One may put a Kli under a Ner to catch sparks.

Answer: A Ner often emits sparks.