Commentary for Chagigah 2:2

יוסי בן יועזר אומר שלא לסמוך – on the Festival day, because when one uses living animals, with all of one’s strength one presses hands [on the head of the sacrifice, to demonstrate ownership]. For this was the dispute that was between the patriarchs and the presidents of the Court, generation after generation.

Yose ben Yoezer says that [on a festival] the laying of the hands [on the head of a sacrifice] may not be performed. Yosef ben Johanan says that it may be performed.
Joshua ben Perahia says that it may not be performed. Nittai the Arbelite says that it may be performed.
Judah ben Tabai says that it may not be performed. Shimon ben Shetah says that it may be performed.
Shamayah says that it may be performed. Avtalyon says that it may not be performed.
Hillel and Menahem did not dispute. Menahem went out, Shammai entered.
Shammai says that it may not be performed. Hillel says that it may be performed.
The former [of each] pair were patriarchs and the latter were heads of the court.
Our mishnah is possibly the most unique mishnah in the entire Mishnah. It contains a debate about whether or not a person can lay their hands on a sacrifice on Yom Tov, the first and last days of a festival. According to Leviticus 1:4 and other parallel verses, when a person brings a sacrifice he lays his hands, or leans, on the sacrifice before it is slaughtered. On Yom Tov it is forbidden to use an animal and leaning on an animal is considered to be use of an animal. Therefore, the question could be asked, can one lean on the sacrificial animal on Yom Tov or must he do so the day before?
What is unique about our mishnah is that there are five pairs (zugot) of sages, each from a different generation, who debated this issue. The sages here are early sages who lived during the Second Temple period, from the beginning of the Hasmonean period until close to the beginning of the millennium. They might even be called proto-sages, or proto-rabbis. With one exception, all of these sages are found in Avot 1:4-10, where each transmits a moral exhortation. The final clause of the mishnah determines that of each pair the first was the patriarch and the second was the head of the court. As far as I know, there are no other mishnayot that have this structure in which the same debate being repeated generation after generation.
The fact that they debate specifically this issue is significant. We know that ancient Jews argued a lot about the laws of Yom Tov and Shabbat and that the Pharisees tended to be more lenient than the other two main sects, the Essenes and the Sadducees. This seems to be the trend in this mishnah as well Hillel allows one to lean on the animal, whereas Shammai does not.
Section four: Note that the mishnah switches order here. In all three previous pairs, the first said that it may not be performed, whereas the second said that it should be performed. It seems that there may have been some historical switch at this period, where the position of the patriarch changed.
Section five: Little is known about Menahem from rabbinic literature, except that he seems to have separated from the fold. Josephus, Antiquities 15:10, 5 relates a story about a certain Menahem who was an Essene. This Menahem receives a divine revelation that Herod will become king and he relates it to him. According to some scholars this is the same Menahem referred to in our mishnah. In any case, it is interesting that he is replaced by Shammai. It is as if the mishnah is letting us know that Shammai is not like Menahem. While Shammai’s opinions are usually not the accepted halakhah, he is still legitimate.

יצא מנחם – to the service of the king, and he withdrew being the colleague of Hillel, Therefore, it was not made known from his relation if he disputed with Hillel or not, and Shammai entered in his place to be the president of the Court in his place, and disputed with Hillel.

הראשונים – that were mentioned first in each pair were the patriarchs.