I. IF no certain consensus has yet been arrived at as to what palæontology teaches in reference to the derivative hypothesis, the chief reason is our very imperfect knowledge of palæontology, arising partly from the necessary imperfection of the geological record caused by the very small chance of the remains of any creature living upon the earth being preserved in a perfect state; partly from the very minute portion of the record which is actually preserved in the rocks having as yet been rendered accessible to investigation; partly from the defective knowledge of the structure and relationship of those, documents, so to speak, which have already been brought to light, and of their existing representatives. The first cause must always remain a stumbling-block to these investigations. The second is gradually being removed by fresh explorations in many parts of the world, notably those now carried on with so much energy and success in North America. The third is one which only needs more numerous and more earnest workers to remove, and especially those who have the power and will to see the continuity of the manifestation of life upon the earth, and will abandon the old practice of studying the fauna of a particular epoch apart from that which preceded or succeeded it, and especially that of studying extinct forms without a thorough mastery of the key to the solution of the difficulties of their structure afforded by the more accessible existing species. Palæontology is no science apart—it can scarcely even be called a branch of zoology; it is simply the application of that science to elucidating the structure of beings now extinct. The thoroughly unscientific and mischievous system of arrangement of nearly ail our great public museums, both at home and abroad, where two distinct collections are kept up, under distinct custodians—one for animals existing at the present moment upon the earth, and the other for animals that have existed at all other periods put together—lias much to answer for in impeding the progress of sound zoological knowledge. Granted that our information is of a very limited nature, it still seems worth while occasionally to gather together the fragments of which it consists: and as it would be impossible in the time allotted to this course to do justice to more than a limited portion of the whole animal kingdom, it is proposed to take the class of mammals, as in many ways well suited for testing whether such facts as are known of their ancient history throw any light upon their mode of origin, and to point out, with impartiality, the results of the investigation. The poverty of the materials in some quarters, as well as their abundance in others, will thus be made manifest, and some useful landmarks afforded which may direct and stimulate future research.