MISHNA I.: R. Jehudah attested six cases where the decisions of the Beth Shamai are lenient, and those of Beth Hillel rigorous. The blood of a carcass is, according to the former, clean, but unclean, according to Beth Hillel. The egg of a fowl carcass, if it looks like the ordinary egg sold in market, is allowed by Beth Shamai, but not otherwise, while the Beth Hillel prohibit it in all cases. However, both prohibit the egg of an internally injured, for it was formed in a prohibited stage. The menses of a heathen woman as well as the clean blood of a leprous woman in confinement, Beth Shamai declare clean and Beth Hillel consider it to be like her spittle or urine. The fruit of the Sabbathic year one may enjoy with or without reward, according to Beth Shamai, the Beth Hillel hold that one may eat it and reward somehow the owner. A leather bag is subject to defilement, according to Beth Shamai, if it is bound and fastened, and the Beth Hillel hold so even when it is not bound. (Shebieth, IV.)
MISHNA II.: R. Jose quotes also similar decisions of six cases. Beth Shamai allow to serve on the table, but not to eat, poultry together with cheese, while Beth Hillel forbid the one as well as the other. Likewise allow the former to separate Terumah from olives for their oil and for the wine from grapes, and the latter prohibit it. According to Beth Shamai he who sows seeds within four ells in the vineyard has thereby sanctified one row, while according to Beth Hillel, two rows (i.e., the rows in question must not be sown). Flour put into boiling water is, Beth Shamai say, free from Chalah, and the Beth Hillel say it is not. The Beth Shamai allow to use rain-water (running down hill) as a legal bath, the Beth Hillel do not. Finally, Beth Shamai allow a proselyte, who underwent circumcision on the eve of Passover, to immerse himself and then partake in the Passover-offering, while Beth Hillel declare that he who parts with his prepuce is like one returning from the grave.
MISHNA III.: R. Ismael cites to the same effect the decisions of the following three cases: The book Ecclesiastæ does not, according to Beth Shamai, render unclean the hands, while it does so according to Beth Hillel. Sin-cleansing water that has already performed its destination is declared clean by former and unclean by latter. The same divergence of opinion the two schools show with regard to the cleanness and uncleanness of black cumin and its tithe. (Negaim, Mishna III. Chapter V.)
MISHNA IV.: R. Elazar quotes two cases to the same effect. The blood of a woman lying-in, who has not bathed (as prescribed) is considered by Beth Shamai as her spittle and urine, while Beth Hillel declare it defiling always, moist or dry. The former agree, however, with the latter view when the woman in question bore in a state of running issue, then the issue defiles immaterial whether dry or moist. (Tract Nidah II., Mishna VI.)
MISHNA V.: If of four brothers two who are married to two sisters die, the latter perform Chalitzah but cannot enter the levirate marriage; and if such marriage has been hastily concluded, divorce must follow. R. Eliezar quotes the Beth Shamai as declaring this marriage to remain, and Beth Hillel as requiring divorce.
MISHNA VI.: Aqavia b. Mehallalel testified four things, which the sages persuaded him to retract, promising him therefor the chair of presiding justice in Israel, to this he responded: I shall prefer to hear the name fool all my life to becoming a wicked even for one hour before the Omnipresent; but let nobody say "He retracted for the sake of an office!" Here are his rules: He declared unclean the white hair (left from a previous case of leprosy) as well as the yellow blood (of a woman), both which the sages declare clean; he allowed to make use of the faded hair of a blemished first-born cattle slaughtered immediately after the hair has been put into a (wall) niches, while the sages forbid it; finally, he prohibited to give the jealousy-water to a female proselyte or to a freed maid-slave, which the sages allow.
The following episode was thereupon presented to him: A certain Karkmith, a freed maid-slave in Jerusalem, was made to drink the aforesaid water by Shmaia and Ahtalion, to which he replied: They did it only in a "make-believe" way. (They, being themselves proselytes, did it.) And they placed him under ban, and when he died the court stoned his coffin. R. Jehudah remonstrated: That Aqavia b. Mehallalel, who among all Israel on whom the doors of the temple court-yard closed, was unequalled in both erudition and piety, should have been placed under ban? Impossible! It was Eliezar b. 'Hanoch that was excommunicated for his trifling with the rule concerning hand-cleaning; and when he died the court sent to put a stone on his coffin; whence it may be inferred that the coffin of him who dies while under ban is to be stoned.
MISHNA VII.: While on his death-bed he (Aqavia b. Mehallalel) thus spoke to his son: Reject the four rules I have been teaching; I adhered to them because I had received them from a majority, and the others likewise had them from a similar source; we both, therefore, remained true to our traditions; but you have learned them of an individual and then of a majority, now it is more advisable to abandon the opinion of the individual and to follow that of the majority. Then the son's request to commend him to his friends he refused, saying: It is not because I find fault with you, but let your own conduct be your recommendation. (Explained at length in Pessachim. V., Mishna IV.)
Sources: Sacred Texts