[2] In 1817, Vater was elected a member of the American Philosophical Society[3] Following Alexander Geddes, he applied the fragmentary hypothesis to the whole of the Pentateuch, treating it as an aggregate of numerous minor documents that had been compiled together.
[4] Vater's major work, Commentar über den Pentateuch was published in three volumes in Halle between 1802 and 1806.
[5] This work's primary purpose was to advance the Supplementary Hypothesis against the earlier Documentarian endeavors of Jean Astruc, Johann Gottfried Eichhorn, and Karl David Ilgen.
Many of Vater's conclusions – most prominently, his assertion of the late nature of the Pentateuch as compared to the historical books – mirror the independent work of Wilhelm Martin Leberecht de Wette, whose Beiträge zur Einleitung in das Alte Testament was published in 1806–7.
[6] Besides the Commentar, his works include: He also edited and continued Henke's Allgemeine Geschichte der christlichen Kirche (1818–23).