It was necessary for the mishna to mention the goats sacrificed on the New Moons. It could enter your mind to say that since the term appointed time is not written with regard to them, these offerings do not override Shabbat or ritual impurity as do other communal offerings during their appointed times. Therefore, it teaches us that even the New Moon is called an appointed time.

This is in accordance with what Abaye said in order to defend the tradition that the spies returned from Eretz Yisrael and the entire Jewish people cried unnecessarily on the Ninth of Av, which resulted in the Ninth of Av becoming a day of crying for future generations. The calculation of the days does not work out precisely, and therefore Abaye said: They filled out Tammuz of that year, meaning that it was a thirty-day month, rather than a twenty-nine-day month as it is nowadays. There is an allusion to this in a verse, as it is written: “He proclaimed an appointed time against me to crush my young men” (Lamentations 1:15), meaning that the New Moon was proclaimed in order to harm the Jewish people in the future. This proves that even the New Moon is called an appointed time.

The Gemara asks: Is that to say that all of them come from, i.e., are derived from, the term appointed time? From where are these matters derived? The Gemara answers: It is as the Sages taught based upon the verse: “And Moses declared the appointed times of the Lord to the children of Israel” (Leviticus 23:44). What is the meaning when the verse states this phrase? This phrase is necessary because we had learned only that the daily offering and the Paschal lamb override Shabbat and ritual impurity, as it is stated with regard to them: In its appointed time, from which it is derived that each of them must be sacrificed in its appointed time and even on Shabbat, in its appointed time and even in ritual impurity.

With regard to the rest of the communal offerings, from where is it derived that they also override Shabbat and ritual impurity? As it is stated with regard to additional offerings that are brought on the Festivals: “These you shall sacrifice to the Lord in your appointed times” (Numbers 29:39).

The baraita continues: From where is it derived to include the omer and the lambs that are sacrificed with it, the two loaves sacrificed on Shavuot, and the communal peace-offerings that are sacrificed with them? The verse states: “And Moses declared the appointed times of the Lord to the children of Israel” after it lists Shabbat and the Festivals. This indicates that the verse established one time for all of them. All of these days are considered appointed times, and their offerings are not deferred.

The Gemara asks: Why do I need all these derivations? It should have been sufficient to provide one derivation and use that as a model for all communal offerings. The Gemara answers: They are all necessary. As, if the Merciful One had written this halakha only with regard to the daily offering in the Torah, I would have said: The daily offering is unique in that it is frequent and it is consumed, as it is entirely consumed as a burnt-offering, and that is why it overrides Shabbat and ritual impurity; but the Paschal lamb, which does not have either of these characteristics, does not override Shabbat and ritual impurity. Therefore, it teaches us that the Paschal lamb also overrides Shabbat and ritual impurity.

And if the Merciful One had written that this halakha applies to the Paschal lamb, I would have said that the Paschal lamb, for which one is punished with karet if one neglects to sacrifice it, overrides Shabbat and ritual impurity; but with regard to the daily offering, for which one is not punished with karet for neglecting to sacrifice it, say that it does not override Shabbat and ritual impurity. Therefore, it comes to teach us that the daily offering also overrides Shabbat and ritual impurity.

And if the Merciful One had written this halakha only with regard to these two offerings, I would have said that it is only with regard to these offerings that the halakha applies, because they have a stringent aspect. The daily offering is frequent and entirely consumed on the altar, and one who neglects to bring the Paschal lamb is punished with karet. But with regard to the rest of the communal offerings, which do not have these stringencies, say that they do not override Shabbat and ritual impurity. Therefore, the Merciful One writes: “These you shall sacrifice to the Lord in your appointed times,” to teach that even these override Shabbat and ritual impurity.

And if the Merciful One had written “These you shall sacrifice to the Lord in your appointed times” and nothing else, I would have said that only the other communal offerings that come to atone for sins are included, such as sin-offerings and burnt-offerings. Burnt-offerings atone for the neglect of positive commandments and for the violation of negative commandments that can be rectified through positive commandments. But the omer and the two loaves, which do not come to atone for sin but merely come to permit, as the omer permits the consumption of the new crop of grain and the two loaves permit using the new crop of grain as offerings in the Temple, do not override Shabbat and ritual impurity. Therefore, it teaches us that even these override Shabbat and ritual impurity.

And if the Merciful One had written: The omer and the two loaves, by themselves, I would have said: On the contrary, the omer and the two loaves, which are important because they come to permit, override Shabbat and ritual impurity, but these other communal offerings do not. Therefore, it teaches us each of the derivations separately.

Since the Gemara has discussed communal offerings that are brought even in a state of ritual impurity, it addresses the basic halakhot relating to this area. The Gemara posits two assumptions and then compares the opinion of Rabbi Yehoshua to the mishna. It states as a preface that the Sages originally assumed that everyone agrees that ritual impurity is overridden in cases involving the public. In other words, the prohibition against sacrificing offerings in a state of ritual impurity applies to communal offerings, but it is superseded by the obligation to sacrifice the offering. Therefore, the frontplate of the High Priest is required to appease God for the sacrifice of the offering in a state of ritual impurity.

There is no tanna that you have heard of who said that ritual impurity is entirely permitted in cases involving the public, i.e., that with regard to the public there is no significance to ritual impurity in the Temple, except for Rabbi Yehuda. As it was taught in a baraita: The frontplate of the High Priest, whether it is on his forehead or whether it is not on his forehead, appeases God and thereby facilitates the acceptance of offerings sacrificed in a state of impurity; this is the statement of Rabbi Shimon. Rabbi Yehuda says: When it is still on his forehead it appeases God, but when it is no longer on his forehead it does not appease Him, as indicated in the verse: “And it shall be on Aaron’s forehead, that Aaron may bear the iniquity of the sacred things which the children of Israel shall hallow” (Exodus 28:38).

Rabbi Shimon said to Rabbi Yehuda: The halakha with regard to the High Priest on Yom Kippur shall prove it, as the frontplate is not on his forehead, and it nonetheless appeases God if communal offerings are brought in a state of ritual impurity. The High Priest spends part of Yom Kippur wearing only the four white garments instead of his usual golden vestments, which include the frontplate.

Rabbi Yehuda said to him: Set aside Yom Kippur, as ritual impurity is wholly permitted in cases involving the public. The frontplate is needed only to atone for individual offerings that are brought in a state of ritual impurity. This proves by inference that Rabbi Shimon holds that ritual impurity is overridden in cases involving the public, but it is not wholly permitted. Therefore, the frontplate is needed to appease God for the sacrifice of the offering in a state of ritual impurity.

The Sages also presumed that everyone agrees that the frontplate appeases God only for the sacrifice of the offering and the burning of the requisite parts on the altar in a state of ritual impurity, but it does not appease God for the impurity of the portions of offerings that are supposed to be eaten. Therefore, although the offering is valid, it may not be eaten. As, the only tanna you have heard say the frontplate appeases God for the impurity of the portions of offerings that are supposed to be eaten is Rabbi Eliezer, as it was taught in a baraita that Rabbi Eliezer says: The frontplate appeases God for the impurity of the portions of offerings that supposed to be eaten. And Rabbi Yosei says: The frontplate does not appease God for the impurity of portions of offerings that are supposed to be eaten.

On the basis of these two assumptions, let us say that the mishna is not in accordance with the opinion of Rabbi Yehoshua, as it was taught in a baraita with regard to the verse: “And you shall sacrifice your burnt-offerings, the flesh and the blood, upon the altar of the Lord your God, and the blood of your offerings shall be poured out against the altar of the Lord your God, and you shall eat the flesh” (Deuteronomy 12:27), that Rabbi Yehoshua says: If there is no blood that is fit to be sprinkled on the altar, due to the fact that it became ritually impure or was lost, there is no meat, as the meat is also disqualified. Similarly, if there is no meat that is fit for use, due to the fact that it became ritually impure or was lost, there is no blood sprinkled on the altar, and the offering does not bring atonement.

Rabbi Eliezer says: Blood brings atonement although there is no suitable meat, as it is stated: “And the blood of your offerings shall be poured out.” Blood is the aspect of the offering most essential for atonement. And how do I uphold the significance of the juxtaposition of flesh and blood in the verse: “And you shall sacrifice your burnt-offerings, the flesh and the blood”? I hold that it is there to tell you that just as the blood is presented upon the altar via sprinkling, so too, the meat is presented via throwing.

You must say, based upon this, that there is a small gap between the ramp and the altar. In order to fulfill the requirement to throw, the priest would proceed as follows: Rather than walking to the arrangement of wood and putting down the meat, he would stand on the ramp and throw the meat of the offering over the gap between the ramp and the altar, onto the arrangement of wood on the altar.

The Gemara asks: And according to the opinion of Rabbi Yehoshua as well, isn’t it written: “And the blood of your offerings shall be poured out,” which indicates that the blood is the essential part of the offering? He could have said to you that it is written right next to it: “And you shall eat the flesh,” which indicates that the meat is also essential and must still be suitable for eating.