Hoofed animals are in many respects among the most important and the most interesting Mammals. They distinctly show that a true understanding of the natural relationship of animals can never be revealed to us merely by the study of living forms, but in all cases only by an equal consideration of their extinct and fossil blood-relations and ancestors. If, as is usually done, only the living Hoofed animals are taken into consideration, it seems quite natural to divide them into three entirely distinct orders, namely: (1) Horses, or _Single-hoofed animals_ (Solidungula, or Equina); (2) Ruminating animals, or _Double-hoofed_ (Bisulca, or Ruminantia); and (3) Thick-skinned, or _Many-hoofed_ (Multungula, or Pachyderma). But as soon as the extinct Hoofed animals of the tertiary period are taken into consideration—of which animals we possess very numerous and important remains—it is seen that this division, but more especially the limitation of the Thick-skinned animals, is completely artificial, and that these three groups are merely top branches lopped from the pedigree of Hoofed animals, which are most closely connected by extinct intermediate forms. The one half of the Thick-skinned animals—rhinoceroses, tapirs, and palæotheria—manifest the closest relationships to horses, and have like them odd-toed feet; whereas the other half of the Thick-skinned animals—pigs, hippopotami, and anoplotheria—on account of their double-toed feet are much more closely allied to ruminating animals than to the former. Hence we must, in the first place, among Hoofed animals distinguish the two orders of Paired-hoofs and Odd-hoofs, as two natural groups, which developed as diverging branches out of the old tertiary primary group of Primary Hoofed animals, or Prochela. The order of _Odd-hoofed animals_ (Perissodactyla) comprises those Ungulata in which the middle (or third) toe of the foot is much more strongly developed than the others, so that it forms the actual centre of the hoof. This order includes the very ancient, common, primary group of all Hoofed animals, that is, the _Primary-hoofed animals_ (Prochela), which are found in a fossil state in the oldest Eocene strata (Lophiodon, Coryphodon, Pliolophus). Directly allied to this group is that branch which is the actual primary form of the Odd-hoofed animals, namely, the _Palæotheria_, fossils of which occur in the upper Eocene and lower Miocene. Out of the Palæotheria, at a later period, the rhinoceroses (Nasicornia) and rhinoceros-horses (Elasmotherida) on the one hand, and the tapirs, lama-tapirs, and primæval horses, on the other, developed as two diverging branches. The long since extinct primæval horses, or Anchitheria, formed the transition from the Palæotheria and tapirs to the Miocene horses, or hipparions, which are closely allied to the genuine living horses.