What is "Heilel ben Shachar"?
Rashi: It is Venus, the morning star. Radak - it gives much light. Heilel is like "b'Hilo Nero Alai Roshi" (Iyov 29:3).
Malbim: It is light that begins to shine on the horizon from the east.
Why does it say "Eich Nafalta mi'Shamayim Heilel ben Shachar"?
Rashi: This is a lamentation over the Sar of Bavel - he will fall from Shamayim.
Malbim: Ancient nations used to attribute success of the kingship to stars; its star is illuminating on it. A star that is ascending, normally it does not set - how did it fall?! The Nimshal is, you were at the epitome of success - how did you fall so suddenly?
Why does it say "Nigdata la'Aretz"?
Radak: You became weak.
What is the meaning of "Cholesh Al Goyim"?
Rashi: He used to cast lots on the kings [that he captured], who will serve him on this day, and who on this day.
Rashi (from Shabbos 149b): He used to cast lots on the kings, to decide whom to sodomize each day.
Radak: (He weakens nations.) This is like "va'Yachalosh Yehoshua" (Shemos 17:13). "Al" is in place of the prefix Beis, as if it says Cholesh b'Goyim. Similarly, "v'Al Charbecha Sichyeh" (Bereishis 27:14) is like b'Charbecha. "V'Nasata Osam Al Sal Echad" (Shemos 29:3) is like b'Sal.
Radak (from Shabbos 148b, 149b): This is [lots], like 'Matilim Chalashim Al ha'Kodoshim on Yom Tov, but not on portions.' We learn from our verse that this is lots. "Ki Amad Melech Bavel El Em ha'Derech
