1. (_a_) There is in the mythology a group of brothers, the Ivalde sons, from whose hands the most wonderful works proceeded, works which were presented to the gods, and by the latter were compared with those of the primeval artist Sindre.
(_b_) In the heroic saga there is a group of brothers, to whom Volund belongs, the most celebrated of the smiths handed down from the mythology.
2. (_a_) Ivalde is an elf and his sons elves.
(_b_) Volund, Egil, and Slagfin are elves (Volundarkvida, 32).
3. (_a_) Ivalde's sons are brothers or half-brothers of the goddess of vegetation, Idun.
(_b_) Volund, Egil, and Slagfin are brothers or half-brothers of swan-maids and dises of vegetation.
4. (_a_) Of Idun, the sister of Ivalde's sons, it is stated that she was for a time absent from the gods, and dwelt with the primeval artists in a winter-cold land, near Nat, the daughter of _Narfi_-Mimer.
(_b_) Volund and his brothers' swan-maids dwell for a time in a winter-cold land, which, as my researches have already shown, is situated _fyr nágrindr nedan_, consequently in the lower world, near the realm of Nat.
5. (_a_) Ivalde's sons were intimately associated with Frey and gave him precious treasures.
(_b_) Volund and Egil were intimately associated with Frey, and were his fosterers and wards.
6. (_a_) Ivalde's sons were most deeply insulted by the gods.
(_b_) Volund has been most deeply insulted by the Asas. He and Egil become their foes, and ally themselves with the powers of frost.
7. (_a_) The insult given to Ivalde's sons consisted in the fact that their works were judged inferior as compared with the hammer Mjolner made by Sindre.
(_b_) The best smith-work produced by Volund is a sword of such a quality that it is to prove itself superior to Mjolner in battle.
These circumstances alone force us to assume the identity of Ivalde's sons with Volund and his brothers. We must either admit the identity, or we are obliged to assume that the epic of the mythology contained two such groups of brothers, and made them identical in descent, functions, and fortunes. Besides, it must then have made the one group avenge not an insult offered to itself, but an insult to the other. I have abstained from the latter assumption, because it is in conflict with the best rules for a logical investigation--_causæ non sunt præter necessitatem multiplicandæ_. And the identity gains confirmation from all sides as the investigation progresses.