What is the meaning of "b'Yasdu"?
Rashi: This is a noun - in its founding. This is like from Chadash - b'Chadsho.
What did they see with their eyes?
Ibn Ezra: Bayis Rishon. When this Bayis (Sheni) was being founded, they cried.
Malbim: Both Bayis Rishon and the founding of this Bayis.
Why were they crying?
Rashi: When they saw this Bayis, they remembered the great Binyan of Bayis Rishon. Malbim - Chagai (2:3) said, "whoever saw the Bayis in its first honor, what you see now is like nothing in your eyes."
Malbim: They were building with Koresh's permission. He limited the size of the Binyan. Had they merited, a miracle would have been done in Ezra's days like in Yehoshua's days (Berachos 4). Therefore, they cried.
Pesikta Rabasi 31: They saw the Shechinah in Bayis Rishon, and there was no Shechinah in Bayis Sheni 1 .
How did they know that the Shechinah will not dwell in it even after it is fully built? Perhaps the Shechinah comes from when the begin building. We count the years of the Bayis from when they began building! (PF)
Why were many blowing Terumah amidst Simchah?
Rashi: They had great Simchah that they left their Galus 1 .
Malbim: The Rambam (Hilchos Terumos Perek 1) says that Yehoshua's Kedushah was Batel, for it was via conquest. When enemies conquered Yisrael, the Kedushah ceased. Ezra's Kedushah was via Chazakah; it will never be Batel. They rejoiced over this.
Above (11), Rashi said that the Simchah was due to making the Yesod of Beis Hashem. Why did he give a different reason? (PF)


