Back in the very earliest Tertiary deposits, and in all certainty even earlier, a group of comparatively small mammals was extensively spread through America, and apparently less widely in Europe, characterized by a primitive form of foot structure, each of which had five complete digits, the whole sole being placed upon the ground, as in the animals we call plantigrade. The grinding surfaces of their molar teeth were also primitive, bearing none of the complicated, curved crests and ridges possessed by present ruminants, but instead they had conical cusps, usually not more than three to a tooth; this tritubercular style of molar crown being about the earliest known in true mammals. In the opinion of many palaeontologists, the ancestors of the present hoofed beasts, or ungulates, were contained among these _Condylarthra_, as they were named by Prof. Cope. Of course, these early mammals are known to us only by their fossil and mostly fragmentary skeletons, but it may be said that at least in the ungulate line, the successive geological periods show steady structural progression in certain directions. Of great importance are a decrease in the number of functional digits; a gradual elevation of the heel, so that their modern descendants walk on the tips of their toes, instead of on the whole sole; a constant tendency to the development of deeply grooved and interlocked joints in place of shallow bearing surfaces; and to a complex pattern of the molar crowns instead of the simple type mentioned. To this may be added as the most important factor of all in survival, that these changes have progressed together with an increase in the size of the brain and in the convolutions of its outer layer. The _Condylarthra_ seem to have gone out of existence before the time of the middle Eocene, but before this they had become separated into the two great divisions of odd-toed and even-toed ungulates, into which all truly hoofed beasts now living fall. The first group (_Perissodactyla_) has always one or three toes functionally developed, either the third, or third, second and fourth, the two others having entirely disappeared, except for a remnant of the fifth in the forefoot of tapirs. They have retained some at least of the upper incisor teeth, and, except in some rhinoceroses, the canines are also left; the molars and premolars are practically alike in all recent species, and in all of which we know the soft parts, the stomach has but one compartment, and there is an enormous caecum. It is probable that they took rise earlier than their split-footed relations, and their Tertiary remains are far more numerous, but their tendency is toward disappearance, and among existing mammals they are represented only by horses, asses, rhinoceroses, and tapirs.