On the fourth week of the year, which encompasses the end of Sukkot (including Hoshanah Rabbah), as well as Shemini Atzeret and Simchat Torah, the eagle sings. During this week, as soon as each community completes the annual reading of the entire Torah, we immediately start our studies anew, just like the eagle renews its feathers from year to year. It is also worth noting that during these days, both for hoshanot and hakafot, we spend a large portion of our service circling the bimah, just like the eagle.
The eagle is the greatest of birds, flying higher than the rest. It therefore has an extremely broad and potent view and perspective on all Creation. Unlike other birds, which carry their young between their talons, the eagle carries them on their wings because no other animal can reach that high. So is our relationship with God: "You have seen what I did unto the Egyptians, and how I took them on eagles' wings and brought them to Me."
The eagle requests that G-d remember the nations (Psalms 59:6). The word “remember” can have both a positive (remember for good) as well as a negative connotation (remember in order to punish). The continuation of the eagle’s song appears to be more connected to the latter, as it states, “do not be gracious to any wicked traitors, selah.” Throughout Sukkot, the Jewish people have been bringing sacrifices on behalf of all nations. However, on Shemini Atzeret, we stop bringing sacrifices for others, and place them aside for the time being, so that the Jewish people can be alone with G-d.
The number four represents stability and strength more than the number three, just as a table with four legs is firmer than a tripod. The number four also refers to the matriarchs of the Jewish People: Sarah, Rebecca, Rachel and Leah. The Torah itself is quite explicit about how the matriarchs were more firm than the patriarchs when it came to protecting their family and their lineage from bad influences and from veering off to wrong paths. Sarah made sure that Yishmael was sent away in order not to be a bad influence for Isaac. When Abraham became apprehensive about this, G-d told him to listen to Sarah. Similarly, Rivkah made sure that Jacob would receive the proper blessings from Isaac, instead of Esau. She also insisted that Jacob not intermarry with the local tribes.
The stability of the number four is reflected in various aspects of the world itself. There are four basic elements in the world: fire, water, air, and earth. There are also four spiritual worlds, or dimensions, mentioned in the Kabbalah: Atzilut, Beriah, Yetzirah, and Assiyah. There are also four rivers that flow from the Garden of Eden, and four levels of Torah knowledge, also known as Pardes. Pardes literally means “orchard,” and stands for: Peshat (simple/meaning), Remez (implied/hinted), Derush (interpreted), and Sod (secret). All of the above concepts are deeply related.