of expressing these formulae, and that all animals have, within the limits of their own branch of the Animal Kingdom, the same structural elements, though each branch is entirely distinct. If this be true, and if these organic formulae have the precision of mathematical formulae, with which I have compared them, they should be susceptible of the same tests. The mathematician proves the identity of propositions that have the same mathematical value and significance by their convertibility. If they have the same mathematical quantities, it must be possible to transform them, one into another, without changing anything that is essential in either. The problem before us is of the same character. If, for instance, all Radiates, be they Sea-Anemones, Jelly-Fishes, Star-Fishes, or Sea-Urchins, are only various modes of expressing the same organic formula, each having the sum of all its structural elements, it should be possible to demonstrate that they are reciprocally convertible. This is actually the case, and I hope to be able to convince my readers that it is no fanciful theory, but may be demonstrated as clearly as the problems of the geometer. The naturalist has his mathematics, as well as the geometer and the astronomer; and if the mathematics of the Animal Kingdom have a greater flexibility than those of the positive sciences, and are therefore not so easily resolved into their invariable elements, it is because they have the freedom and pliability of life, and evade our efforts to bring all their external variety within the limits of the same structural law which nevertheless controls and includes them all. I wish that I could take as the illustration of this statement animals with whose structure the least scientific of my readers might be presumed to be familiar; but such a comparison of the Vertebrates, showing the identity and relation of structural elements throughout the Branch, or even in any one of its Classes, would be too extensive and complicated, and I must resort to the Radiates,--that branch of the Animal Kingdom which, though less generally known, has the simplest structural elements. I will take, then, for the further illustration of my subject, the Radiates, and especially the class of Echinoderms, Star-Fishes, Sea-Urchins, and the like, both in the fossil and the living types; and though some special description of these animals is absolutely essential, I will beg my readers to remember that the general idea, and not its special manifestations, is the thing I am aiming at, and that, if we analyze the special parts characteristic of these different groups, it is only that we may resolve them back again into the structural plan that includes them all.