A rhipidistian paired fin, made to assume the position of a primitive tetrapod limb, would suffer certain modifications which throw light on the origin of the latter and on the homology of its elements. The new theory here stated, and that of Gregory and Raven (from which it differs in several points), are radically different from all hitherto proposed. The proto-tetrapod pectoral and pelvic fins were of similar structure. Each consisted of five axial mesomeres, each bearing distally a simple preaxial radial and an actual or potential postaxial process; a rudimentary sixth mesomere may have been present. The downward flexure of the extended fin between 1st-2nd mesomeres, and forward torsion of the part distal to the 2nd mesomere, leads to conditions closely comparable with part of the limbs of primitive tetrapods. The digits (including their proximal podialia) are new formations, and the terms archepodium and neopodium are proposed for those parts of the tetrapod limb derived directly from the rhipidistian paddle skeleton and for new formations respectively. The prepollex (prehallux) is a digit, but postminimal and pisiform elements are less certainly so. The main axis of the rhipidistian paddle is directed between the 'first' and 'second' fingers. Several variants of this scheme are discussed. The structure of the stegocephalian limb is discussed; it corresponds well to this theoretical pattern, but there is some reduction of the distal part of the archepodium. The limb structure of urodeles and other tetrapods can easily be derived from this common ancestral pattern, contrary to the views of Holmgren. The development of limbs of living tetrapods is in general agreement with the main results of the new theory, which alone explains the relationships of the radiale (tibiale) in many forms. Many peculiarities of the tetrapod limb seem to be relics of characters of the rhipidistian fin. Each mesomere and preaxial radial of the rhipidistian paddle was probably related to one myotome. Relics of this condition seem to be found in the segmental motor innervation of the muscles of human extremities. The relationship between paired fins in Dipnoi, Coelacanthini and Rhipidistia is discussed.