Mooky

From my understanding, the line of descent of the English language looks like this:
Modern English << Middle English << Old English << Anglo-Saxon << Old Saxon/other Old Germanic languages.

Also, from my understanding, somewhere between 50-70% of all modern English words have Latin and/or Greek origins or roots, even though neither language is in English's line of descent.

I understand that Latin and Greek, as well as French, old Celtic, and other languages, have contributed "horizontally", i.e. by co-existence and parallel use, but how could this influence have been so great without being "vertical"?