MISHNA: Five items in the burnt offering and the accompanying meal offering and libation join together to constitute the one peruta measure with regard to liability for misuse, and the olive-bulk measure with regard to liability for piggul, notar, and partaking of sacrificial foods while ritually impure. They are: The flesh; the fat of the burnt offering that is sacrificed on the altar; the fine flour of the accompanying meal offering; the wine of the accompanying libation; and the oil of the accompanying meal offering. And there are six items in the thanks offering that join together: The flesh, the fat, the fine flour, the wine, the oil, and the loaves accompanying the thanks offering.

GEMARA: The mishna teaches that five items in the burnt offering [ba’ola] join together. Rav Huna taught Rava the mishna as follows: Five items in the world [ba’olam] join together. Rava said to Rav Huna: Did you say: In the world, i.e., that there are only five items in the world of offerings that join together? But isn’t it taught in the mishna with regard to a thanks offering that there are six items in the thanks offering that join together: The flesh, the fat, the fine flour, the wine, the oil, and the loaves accompanying the thanks offering? Rav Huna said to Rava: One should teach in the mishna: Five items in the burnt offering [ba’ola] join together, not: Five items in the world.

§ The mishna teaches that the flesh and the fat join together. The Gemara notes that we already learned this, as the Sages taught explicitly in a baraita (Tosefta 1:28): The meat of burnt offerings and their sacrificial portions join together to constitute the amount of an olive-bulk that renders one liable for sacrificing them outside the courtyard, and to render one liable for eating them due to piggul, notar, and partaking of them while ritually impure.

The Gemara notes that the baraita teaches that with regard to a burnt offering, yes, this halakha applies, whereas with regard to a peace offering, it does not apply. The Gemara raises a difficulty: Granted, with regard to sacrificing them outside the courtyard, it stands to reason that in the case of a burnt offering, which is entirely consumed upon the altar, everything joins together, whereas with regard to peace offerings, whose meat is not burned on the altar, the meat and sacrificial portions do not join together. But with regard to rendering one liable for piggul, and notar, and partaking of them while ritually impure, in the case of a peace offering as well, why is he not rendered liable?

The Gemara elaborates: But didn’t we learn in the mishna below: All the pieces of sacrificial meat that are piggul join together with one another to constitute the olive-bulk measure for liability, and all pieces of sacrificial meat that are notar join together with one another to constitute the olive-bulk measure for liability? The mishna indicates that this is the halakha with regard to all types of offerings.

The Gemara answers: Rather, say that the baraita should read as follows: The meat of a burnt offering and its sacrificial portions join together to constitute an olive-bulk in order to permit one to sprinkle the blood for them. If the meat of the burnt offering was lost after it was slaughtered and only an olive-bulk of the meat and sacrificial portions remain, they combine to permit the sprinkling of the blood.

The Gemara further explains: And since they join together to constitute an olive-bulk with regard to sprinkling the blood, once that blood has been sprinkled they join together to constitute the minimum amount of an olive-bulk to render one liable for eating them due to piggul, notar, and partaking of them while ritually impure. By contrast, in the case of a peace offering, the meat and the sacrificial portions do not join together for sprinkling, and as the sprinkling is not valid, they do not subsequently join together to constitute an olive-bulk to render one liable for eating them due to piggul, notar, and partaking of them while ritually impure.

The Gemara adds: And who is the tanna who teaches this baraita? It is Rabbi Yehoshua, as it is taught in a baraita that Rabbi Yehoshua says: With regard to all the offerings in the Torah from which there remains an olive-bulk of meat that is fit to be eaten, or an olive-bulk of fat that is fit to be sacrificed on the altar, one sprinkles the blood. If all that remains is half an olive-bulk of meat and half an olive-bulk of fat, one may not sprinkle the blood. Since the meat and fat serve different functions, as the fat is burned on the altar while the meat is eaten by the priests, they do not combine to form the minimum amount that must remain in order to sprinkle the blood.

But with regard to a burnt offering, even if all that was left was half an olive-bulk of meat and half an olive-bulk of fat, one sprinkles the blood, because it is all consumed upon the altar. Since both the meat and the fat are sacrificed on the altar, they combine. And in the case of a meal offering, even if all of it remains, one may not sprinkle the blood.

With regard to the last statement, the Gemara asks: What is the relevance of a meal offering to this baraita? A meal offering does not require blood to be sprinkled on the altar. Rav Pappa said: The meal offering under discussion is the meal offering brought with the libations that accompany animal offerings. The baraita is teaching that if none of the meat of the animal remains, even if all of the accompanying meal offering is intact, the blood of the animal may not be sprinkled.

MISHNA: Teruma, and teruma of the tithe, and teruma of the tithe of doubtfully tithed produce [demai], and ḥalla, and first fruits all join together with one another to constitute the requisite measure to prohibit a mixture with non-sacred produce, and to form the requisite measure of an olive-bulk that serves to render one obligated for their consumption in payment of an additional one-fifth over and above the principal. All the pieces of sacrificial meat that are piggul join together with one another to constitute the olive-bulk measure for liability, and all sacrificial meat that is notar joins together with one another to constitute the olive-bulk measure for liability.

GEMARA: The Gemara asks: What is the reason that all of the foods listed in the mishna join together? The Gemara answers: The reason is that all of them are called teruma in the Torah. The Gemara elaborates: With regard to ḥalla it is written: “Of the first of your dough you shall set apart ḥalla for a teruma; like teruma of the threshing floor, so shall you set it apart” (Numbers 15:20).

With regard to first fruits, they are also called teruma, as it is taught in a baraita that discusses foods that may not be eaten outside Jerusalem: The verse states: “You cannot eat within your gates the tithe of your grain or of your wine or of your oil or the firstlings of your herd or of your flock, or any of your vows that you vow, or your freewill offerings, or the offering of your hand” (Deuteronomy 12:17). “And the offering of your hand”; these are first fruits. But with regard to the other items listed in the mishna, i.e., teruma, teruma of the tithe, and teruma of the tithe of demai, it is not necessary to explain why they join together or to cite a verse, as it is clear that they are called teruma.

MISHNA: All animal carcasses, whose consumption is prohibited and which transmit impurity through contact with them and through carrying, join together with one another to constitute the requisite olive-bulk measure. And all repugnant creatures join together with one another to constitute the requisite olive-bulk measure to render one who consumes it liable to receive lashes. The eight creeping animals enumerated in the Torah join together to constitute the measure of a lentil-bulk, which transmits impurity through contact, and to render one who consumes it liable to receive lashes.

GEMARA: The mishna teaches that all animal carcasses, whose consumption is prohibited and which transmit impurity through contact and through carrying, join together to constitute the requisite olive-bulk measure. Rav says: