The 8th Day beginning
In English the name “Eighth Day of Solemn Assembly” is often used for Sh’mini Atzeret.
Presumably this tells us the obvious, that the day is connected with Sukkot, but in halachah the eighth day is regel bif’nei atzmo, an independent festival. Though attached to Sukkot, it has a character all of its own.
After all those weeks of festivals, God says: “It’s hard for Me to part with you”.
It is like the birth of a baby; after completing a viable week of life, a boy has his b’rit milah on the eighth day and though he doesn’t yet realise it, the eighth day is the beginning of his lifetime of growth. He has completed a week of private family celebration and is now ready for the wider community. His eighth day is a beginning.
Likewise, Sh’mini Atzeret is a beginning.