The Arachne is an enemy in Final Fantasy IV. She does nothing but use Earthquake repeatedly. Since she appears in the Passage of the Eidolons, and likely the player should be under the Float status, her Earthquake ability will be harmless. Spears and bow/arrows are helpful in finishing the battle quickly.
Arachne is the name of a mythological Greek weaver who boasted she had skills surpassing that of Athena, the goddess of weaving. For this hubris, Athena transformed her and her descendants into spiders.