Who was Eve?
Two names are given to her, both bestowed by the man, her mate. The first, Heb: ishshah, "woman" (literally, "man-ess"), is not strictly a name but a generic designation, referring to her relation to the man; a relation she was created to fulfill in default of any true companionship between man and the beasts, and represented as intimate and sacred beyond that between child and parents (Gen 2:18-24).
The second, Eve, or "life," given after the transgression and its prophesied results, refers to her function and destiny in the spiritual history or evolution of which she is the beginning (Gen 3:16,20). While the names are represented as bestowed by the man, the remarks in Gen 2:24 and 3:20b may be read as the interpretative addition of the writer, suited to the exposition which it is the object of his story to make.
Eve and Her Relation to Adam
As mentioned in the article &ADAM, the distinction of male and female, which the human species has in common with the animals, is given in the general (or P) account of creation (Gen 1:27); and then, in the more particularized (or J) account of the creation of man, the human being is described at a point before the distinction of sex existed. This second account may have a different origin, but it has also a different object, which does not conflict with but rather supplements the other.
It aims to give the spiritual meanings that inhere in man's being; and in this the relation of sex plays an elemental part. As spiritually related to the man-nature, the woman-nature is described as derivative, the helper rather than the initiator, yet equal, and supplying perfectly the man's social and affectional needs. It is the writer's conception of the essential meaning of mating and marriage. To bring out its spiritual values more clearly he takes the pair before they are aware of the species meanings of sex or family, while they are "naked" yet "not ashamed" Gen 2:25), and portrays them purely as companions, individual in traits and tendencies, yet answering to each other. She is the helpmeet for him (Heb: ezer keneghdo, "a help answering to him").
The Change in the Condition of Eve
True to her nature as the being relatively acted upon rather than acting, she is quicker than the man to respond to the suggestion initiated by the serpent and to follow it out to its desirable results. There is eagerness of desire in her act of taking the fruit quite different from the quasi matter-of-course attitude of the man. To her the venture presents itself wholly from the alluring side, while to him it is more like taking a desperate risk, as he detaches himself even from the will of God in order to cleave to her.
All this is delicately true to the distinctive feminine and masculine natures. A part of her penalty is henceforth to be the subordinated one of the pair (Gen 3:16), as if for her the values of life were to be mediated through him. At the same time it is accorded to her seed to perpetuate the mortal antipathy to the serpent, and finally to bruise the serpent's head (Gen 3:15).
Eve in Later History
After these opening chapters of Gen, Eve is not once mentioned, nor even specifically alluded to, in the canonical books of the Old Testament. It was not in the natural scope of Old Testament history and doctrine, which were concerned with Abraham's descendants, to go back to so remote origins as are narrated in the story of the first pair. The name Eve occurs once in the Apocrypha, in the prayer of Tobit (Tobit 8:6): "Thou madest Adam, and gavest him Eve his wife for a helper and a stay; of them came the seed of men"; the text then going on to quote Gen 2:18. In 1 Esdras 4:20,21 there is a free quotation, or rather paraphrase, of Gen 2:24. But not even in the somber complaints of 2 Esdras concerning the woe that Adam's transgression brought upon the race is there any hint of Eve's part in the matter.
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IBSE, (in the public domain) with minor edits.