Stehlin in his classic monograph on the “Geschichte des Suiden-Gebisses” has left little room for further descriptive work on the fossil forms of this group of mammals. I have followed his footsteps around most of the great palæontological collections of Europe, and have found no specimen of interest which has not received his careful consideration. All that I am proposing to do in this paper is to enquire anew into the relationships of some of the early members of the group. Stehlin’s main interest, as the title of his work indicates, lay in the dentition. He dealt carefully with the skeletal characters, but he came to them last. By reversing the process, and making skull characters my starting point, I am confining myself to a very small corner of the field, because valuable skull fragments are extremely rare and very unevenly distributed among the known genera. The consideration of big, indisputable morphological differences is alone possible: differences such as those by which we are accustomed to separate into families and genera the animals living to-day. The fossil forms can by these be divided into several well-defined and more or less nearly related groups. Any attempt to trace the phylogenetic relationships of the species within these groups is impossible on skull characters alone. There are certain families of early Tertiary Mammals which have often been regarded as allied or even ancestral to the Suidse owing to the similarity of their dentition. In seeking what further evidence their skulls might afford, I have been led very far from the true Suidæ into a study of the skulls of all the early Tertiary Artiodactyls in which these are known. The inadequacy of dentitional characters alone to solve the confused problems of Artiodactyl relationships has become more and more evident to me. The basicranial characters of the skull on the other hand, as being the most complex and the most conservative, the least likely to respond quickly to environmental changes, appear to form one of the surest guides—just as they have been found to do in the lower vertebrates, where they have been so much more carefully studied.