From time to time, when a land-connection between North America and Eurasia was established, some of the successive equine genera migrated to the Old World, but they do not seem to have gained a permanent footing there until the end of the Miocene or beginning of the Pliocene, eventually diversifying into the horses, asses, and zebras of Africa, Asia and Europe. At about the same period, the family extended its range to South America and there gave rise to a number of species and genera, some of them extremely peculiar. For some unknown reason, all the horse tribe had become extinct in the western hemisphere before the European discovery, but not until after the native race of man had peopled the continents. In addition to the main stem of equine descent, briefly considered in the foregoing paragraphs, several side-branches were given off at successive levels of the stem. Most of these branches were short-lived, but some of them flourished for a considerable period and ramified into many species. Apparently related to the horses and derived from the same root-stock is the family of the Palaeotheres, confined to the Eocene and Oligocene of Europe, dying out without descendants. In the earlier attempts to work out the history of the horses, as in the famous essay of Kowalevsky ("Sur l'Anchitherium aurelianense Cuv. et sur l'histoire paleontologique des Chevaux", "Mem. de l'Acad. Imp. des Sc. de St Petersbourg", XX. no. 5, 1873.), the Palaeotheres were placed in the direct line, because the number of adequately known Eocene mammals was then so small, that Cuvier's types were forced into various incongruous positions, to serve as ancestors for unrelated series. The American family of the Titanotheres may also be distantly related to the horses, but passed through an entirely different course of development. From the lower Eocene to the lower sub-stage of the middle Oligocene the series is complete, beginning with small and rather lightly built animals. Gradually the stature and massiveness increase, a transverse pair of nasal horns make their appearance and, as these increase in size, the canine tusks and incisors diminish correspondingly. Already in the oldest known genus the number of digits had been reduced to four in the fore-foot and three in the hind, but there the reduction stops, for the increasing body-weight made necessary the development of broad and heavy feet. The final members of the series comprise only large, almost elephantine animals, with immensely developed and very various nasal horns, huge and massive heads, and altogether a grotesque appearance. The growth of the brain did not at all keep pace with the increase of the head and body, and the ludicrously small brain may will have been one of the factors which determined the startlingly sudden disappearance and extinction of the group.