Eiruvin 13b

They studied their own rulings and those of Beit Shammai - When Beit Shammai would bring proof to their rulings from the Torah and Beit Hillel would bring proof from another verse, and Beit Hillel would expound the verse of Beit Shammai to show its relevance, and it was not light in their eyes, as they studied in Tractate Berakhot (Berakhot 10b), "Beit Shammai say, In the morning, every man should lean and read [Kiryat Shema'] and in the morning he should stand as it says, (Devarim 6:7) 'When you lie down and when you arise'; and Beit Hillel say, Every man should read as he is, as it says, (Ibid.) 'When you are on the way'. If so, what does it say, 'When you lie down and when you arise'? - [Recite] at the time that people normally lie down and at the time that people normally arise [i.e., in the evening and in the morning]." They mention the actions of Beth Shammai before theirs - As they first said in the story that the "elders of Beth Shammai went" and subsequently, the "elders of Beth Hillel".

There was another incident involving Rabbi Eliezer, who descended to serve as prayer leader before the ark on a fast day. And he recited twenty-four blessings, but he was not answered. Rabbi Akiva descended before the ark after him and said: Our Father, our King, we have no king other than You. Our Father, our King, for Your sake, have mercy on us. And rain immediately fell. The Sages were whispering among themselves that Rabbi Akiva was answered while his teacher, Rabbi Eliezer, was not. A Divine Voice emerged and said: It is not because this Sage, Rabbi Akiva, is greater than that one, Rabbi Eliezer, but that this one is forgiving, and that one is not forgiving. God responded to Rabbi Akiva’s forgiving nature in kind by sending rain.

In the first chapter of Masechet Eiruvin (13b): for what reason did the Beit Hillel merit to establish the Halakha according to them? Because they were calm when they were insulted and they would teach both their teachings and the teachings of the Beit Shamai. Furthermore, they would place the teachings of Beit Shamai before their own, as we learn in our Mishnah (Sukkah 28b): “one whose head and most of him were in the Sukkah, and his table is in the house – Beit Shamai validate and Beit Hillel validate. Beit Hillel said to Beit Shamai: was there not such an incident that the elders of Beit Shamai and the elders of Beit Hillel went to visit Rabbi Yochanan ben Hachorani etc.” From this understand and know that this quality is one who is centered, for one who is centered does not stray from the straight path at all, and he maintains his equilibrium, the opposite of someone who is not centered and because of this strays from equilibrium. And because of this, Beit Hillel merited that the Halakha is established according to them, because they were masters of calmness and they were not like Beit Shamai who acted with strictness, and it is known that strictness because of great anger leads to straying from equilibrium completely, and Halakha is the path which does not stray to the right or left at all – only walking with equilibrium and not swerving to the left or the right, and therefore what we call ‘Halakha’ is called ‘Halakha,’ because one who walks walks in straightness and does not tend from the straight and equal path at all, not to the right and not to the left; and this is the quality of Beit Hillel, that they had this quality in all of their conduct – they wouldn’t stray from equilibrium because they were centered. And don’t say that Beit Shamai weren’t wise, for on the contrary they were very wise and intellectually sharp. Only in the area of Halakha which is the path of the straight it is according to Beit Hillel because they were masters of calmness without veering from the order. And so they were [calm] even when they were insulted, and they taught their words and the words of their friends, and all of this refers to being completely calm, for in the case of other people, when someone says something to him, he is quick to become angry and to insult him back, while in the case of Beit Hillel they had the quality of being calm and did not insult even someone who insulted them. And so too when they taught their teachings and the teachings of Beit Shamai. Furthermore, they would put Beit Shamai’s teachings before their own. In all of these matters they were organized in their actions, and they were not from the group of highly competitive against their friends, and they were from the group that succeeds. And even though Halakhic discussions are nevertheless good, they [Beit Shamai] are not organized, and for matters of Halakha it is required that they not be not organized and are completely balanced. And all of these words that are mentioned [about Beit Hillel], they were well-organized and did not become disorganized. Even though the Halakha is not according to Beit Shamai, nevertheless the words of Beit Shamai are also the words of the Living God, as it is there. And this idea that they were stringent is because the attribute of Beit Shamai is also the attribute of wisdom, as it says: “if you see a Talmud Chacham, he takes vengeance and guards his hatred like a snake, gird him upon your hips”, and this idea we explained in its place, but with regards to Halakha it is fitting to be according to Beit Hillel for Beit Hillel were organized in their actions and did not stray from equilibrium, and therefore it was appropriate for Halakha.