The paper which I lay before the Society is an attempt to treat with sufficient osteological detail an extinct family of Ungulates which had an immense range of distribution and a great variety of forms in the two periods of the earth’s history which preceded our own. The fate this family has met with at the hands of palæontologists is a somewhat sad one, presenting a warning example of the unscientific method that was paramount in the palæontology of the Mammalia after the time of Cuvier. With the exception of England, where here the study of fossil Mammalia was founded on a sound basis, and some glorious exceptions on the continent, we have very few good palæontological memoirs in which the osteology of extinct mammals has been treated with sufficient detail and discrimination; and things have come to such a pass, that we know far better the osteology of South American, Australian, and Asiatic genera of fossil mammals than of those found in Europe. Nearly all fossil Mammalia which have been described in detail belong to genera that still exist on our globe, or whose differences from fossil forms are trifling. After the splendid osteological investigations of Cuvier had revealed to science a glimpse of a new mammalian world of wonderful richness, his successors have been bent rather on multiplying the diversity of this extinct creation, than on diligently studying the organization of the fossil forms that successively turned up under the zeal of amateurs and collectors. From the year 1828, and even before, when Laizer, Pomel, Croiset, and others began to give short notices on the Mammalia of Auvergne, mammalian genera and species from this locality have been multiplied at a prodigious rate, every private collector giving his own generic and specific names, with no better description than stating the real or supposed number of teeth, and some phrases as to the general resemblances of the fossil in question. Others substituted in their short notices other names, while the scientific work of description did not proceed further than the mere counting of the number of teeth. This process has given rise to such an utter confusion in the palæontology of the extinct Paridigitata, that even now (forty years after the date of the earliest notices) we are utterly ignorant of the true extent and organization of the Miocene mammalian fauna of Auvergne, for instance—though materials for a detailed study of the subject abound in all great public, and many private, collections, the fossils being very common. No palæontologist, even of the highest standing, could boast of knowing, in our own time what Dremotherium, Dorcatherium, Elaphotherium, Gelocus , and so on really are, what are the bones belonging to each set of teeth (as the names were mostly given to these last), whether they had horns or were hornless like the Tragulidæ , and so on. If we add that German authors described the genera of Paridigitates which were found and named in France under different names (as Palæomeryx , Microtherium , Hyotherium , and so on), when they came from German localities, the confusion may be guessed. Having no good descriptions and no figures of the genera noticed in France, the German authors almost necessarily fell into the mistake of renaming what was already named. Once named, the genus was allowed to go forth with the short and wholly insufficient characteristics given to it by the first describer, the impossibility of adding one’s name after the generic or specific designation seeming to take all interest from it. And this, moreover, is the best case; for frequently the same form was described by an other palæontologist under a different generic name, or, if this was Utterly impossible, a new species was made of it, founded on some difference in size or other trifling character. Happily, however, a reaction began to set in, one of the first to head it on the Continent being Rütimeyer, who did not confine his study merely to the teeth of fossil Mammalia, but aimed with brilliant success at a complete investigation of the osteology of the extinct genera and of their affinities with the living ones. Gaudry’s work on the fossils of Pikermi (the best palæontological work that has appeared in France since Cuvier’s 'Ossemens Fossiles’), Fraas’s 'Fauna von Steinheim,’ Alphonse Milne-Edwards’s 'Oiseaux Fossiles,’ and many others may be cited as examples to prove that the new tendency has fairly set in and will bear good fruit. The wide acceptance by thinking naturalists of Darwin’s theory has given a new life to palæontological research; the investigation of fossil forms has been elevated from a merely inquisitive study of what were deemed to be arbitrary acts of creation to a deep scientific investigation of forms allied naturally and in direct connexion with those now peopling the globe, and the knowledge of which will remain imperfect and incomplete without a thorough knowledge of all the forms that have preceded them in the past history of our globe.