Just for the record (and so I don't have to go and look them up again), the Latin words are:
Annuit Coeptis and Novus Ordo Seclorum.
The former phrase is apparently derivative of Virgil's Georgics:
"Da facilem cursum, atque audacibus annue cœptis."
(Whatever form, O Caesar, you assume,)
Smooth my path, condone this enterprise
(Of bold experiment in verse, and share
Concern with me for uninstructed farmers:
Grow used to prayers appealing to your name.)
The latter phrase comes direct from Virgil's Eclogues and is straightforward: it means "the new order for the ages".
So - "he/she/it has favoured the/this endeavour" sounds like a reasonably precise translation of annuit coeptis, with a slightly looser and bolder implied meaning "Providence has favoured this American project". The Eye of Providence is the motif, so Providence. The new order of the ages proclaims the beginning of an American era.
Hence, God approves of the US, and the US approves of God ("In God we trust"). The Latin phrases and the English phrase are mirrors. The English phrase perhaps also loosely recalls the phrase in the Declaration of Independence regarding the endowment of folks by our creator with certain rights.
Finally, there is an obvious parallel, via Virgil's Latin, in the US proclaiming itself a new Rome for a new age.
This is the fun of a free ranging conversation. You learn new stuff as you go along. Who knew boring green bits of paper could have so much in them.
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