Both Uses of

emulate
in
Frankenstein - 1831 version

This lady died, but her lessons were indelibly impressed on the mind of Safie, who sickened at the prospect of again returning to Asia and being immured within the walls of a harem, allowed only to occupy herself with infantile amusements, ill-suited to the temper of her soul, now accustomed to grand ideas and a noble emulation for virtue.†
p. 127.2 *

emulation = imitation

We passed rapidly along; the sun was hot, but we were sheltered from its rays by a kind of canopy while we enjoyed the beauty of the scene, sometimes on one side of the lake, where we saw Mont Saleve, the pleasant banks of Montalegre, and at a distance, surmounting all, the beautiful Mont Blanc and the assemblage of snowy mountains that in vain endeavour to emulate her; sometimes coasting the opposite banks, we saw the mighty Jura opposing its dark side to the ambition that would quit its native country, and an almost insurmountable barrier to the invader who should wish to enslave it.†
p. 196.9

Definitions:

(1) (emulate)
imitate (copy or match)
(2) (meaning too rare to warrant focus)
meaning too rare to warrant focus:

Much more rarely, emulate can mean to try to reach equality is some sense -- as in "This artists' drawings cannot emulate his water colors."