Berakhot 142

1:
This mishnah too is concerned with the question of ritual impurity. Bet Shammai hold that one may only dine off a table that is ritually pure. Since one may well need the towel, which doubled as a table napkin (like the French 'serviette') for further use during the meal it is essential that it be kept ritually pure. (Since the towel is now moist from drying the hands it is most susceptible to ritual impurity.) Therefore, maintain Bet Shammai, it should be placed on the ritually pure table so that it will stay ritually pure. Bet Hillel do not require a dining table to be ritually pure at all. That means that if you place the moist towel on a ritually impure table it will become ritually impure itself; that impurity could then be passed on to the food being served during the meal. Therefore, according to Bet Hillel, it would be best to place the napkin on the cushions of the dining couch; there it is most unlikely to come into contact with the food, and at the very most could pass on tertiary impurity to the hands of the diner. Tertiary impurity cannot be passed on.

1:
It was the custom in rabbinic times (a custom still observed in ultra-religious households) to wash the fingers in a minimal amount of water after the meal. (According to rabbinic tradition this second washing was in order to obviate the danger of very strong salt, used for flavouring during the meal, causing damage to the eyes if accidentally rubbed in.) This washing of the fingers in a few drops of water after the meal (and before reciting Grace) is called Mayim Aĥaronim. The tidying of the dining area referred to in our mishnah is, in fact, the sweeping away of crumbs and other morsels of food from on and under the table. This is not so much for reasons of tidiness as to make sure that there are no remains of food lying around which could come into contact with the Mayim Aĥaronim – which would then render them ritually impure. This is one of the few halakhot that were decided in accordance with the view of Bet Shammai – that the fingers are washed only after tidying the dining area. However, this is almost meaningless, since both the habit of clearing away the table and the custom of Mayim Aĥaronim are now defunct. (The clearing away of the table does not refer to clearing away dishes and the remains of the food, but the physical removal of the table – which was a low table next to the dining couches.) It seems as though Bet Shammai lose even when they win.