"The Academy," as a place where arts and sciences were taught, 1580s, from phrase groves of Academe (translating Horace's silvas Academi), the name of the public gymnasium and gardens near Athens where Plato taught, from Greek he Akademeia (see academy).
Latin academia also was used in reference to Plato's doctrines. Academe in a modern, general sense of "the world of universities and scholarship" is attested in English from 1849. (academia in the sense of "academic community" is from 1956.)
Academe properly means Academus (a Greek hero); & its use as a poetic variant for academy, though sanctioned by Shakespeare, Tennyson & Lowell, is a mistake; the grove of A., however, (Milton) means rightly The Academy. [Fowler]