Rif Brachot 38b

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38b

{Brachot 51b continued}
upon the day and afterwards he blesses on the wine - for the day causes the wine to come, and the day has already come, and the wine has not yet come.
And Bet Hillel says: he blesses on the wine first, and afterwards he blesses on the day, for the wine causes the sanctification {of the day} to be said.
Another explanation - the blessing on the wine frequent occurs, and the blessing on the day does not frequently occur, and in a case of frequent vs. infrequent, frequent comes first.

{The Mishna had said:} "BETH SHAMMAI SAY THAT WASHING THE HANDS PRECEDES THE FILLING OF THE CUP, WHILE BETH HILLEL SAY THAT THE FILLING OF THE CUP PRECEDES THE WASHING OF THE HANDS":
Which cup? It is the cup of sanctification of the day {that is, the first cup, which precedes the meal}

{skips Mishnaic dispute regarding wiping hand with napkin}

Mishna:
BETH SHAMMAI SAY THAT [AFTER THE MEAL] THE FLOOR IS SWEPT BEFORE THE WASHING OF THE HANDS,
WHILE BETH HILLEL SAY THAT [THE DINERS] WASH THEIR HANDS AND THEN THE FLOOR IS SWEPT.

{Brachot 52b}
Gemara:
What is the reason of Bet Shammai?
Because of pieces {crumbs of bread}.
And Bet Hillel holds: here we are dealing with an attendant who is a scholar, who leaves alone pieces less than an olive's measure, and takes away pieces which have in it an olive's measure.

This supports Rabbi Yochanan, who says that pieces that do not have in them an olive's measure, one is permitted to destroy them by hand.

In what do they argue?

Bet Shammai hold it is permitted to make use of an attendant who is an ignoramus, and Bet Hillel holds it is forbidden to make use of an attendant who is an ignoramus.

Rabbi Yitzchak son of Rabbi Chanina cited Rav Huna: In all chapters the halacha is like Bet Hillel, with the exception of this one, where the halacha is like Bet Shammai.

MISHNA:
BETH SHAMMAI SAY THAT [THE PROPER ORDER IS] LIGHT, GRACE {=Mazon}, SPICES, AND HABDALAH,
WHILE BETH HILLEL SAY: LIGHT, SPICES, GRACE {=Mazon}, AND HABDALAH.

Gemara:
There is among the Rabbis {contemporaries} who says that this reference to Mazon in our Mishna refers to the blessing of HaMotzi, and we would thus derive from this Mishna that we may make Havdalah on bread. And they establish our Mishna as referring to a case occuring on Motza`ei Shabbat {Saturday night} going in to Yom Tov, that since you need to say the kiddush on bread, we attach along with it havdalah after it. And the reason he is compelled to say this is that it is difficult for him this case, in which Amemar visited the house of Rav Ashi, and he had no wine to make havdalah, and he slept that night without eating, and as a result he felt compelled to establish the case in our Mishna as referring to Motza`ei Shabbat to Yom Tov, and he establishes this case of Amemar as occuring on one of the other days of the year {that did not go into Yom Tov}, such that you have no difficulty from this case of Amemar.

And this matter is not so, for we do not find in any case havdalah upon bread. And this that we learn in the Mishna "LIGHT, GRACE {=Mazon}, SPICES, AND HABDALAH," it is not going on the blessing of HaMotzi, but rather on Birchat HaMazon, that is the three blessings, and we learn this explicitly in the Tosefta: Rabbi Yehuda said: Bet Shammai and Bet Hillel did not argue regarding Birchat HaMazon