We must now ask why some groups of variations are co-ordinated while others are not, and it is here that we encounter the most formidable of the difficulties of any hypothesis of transformism which depends on the concept of natural selection. If we assume that the environment induces the appearance of variations, it seems to follow that these variations are likely to be co-ordinated, but we then invoke the principle of the acquirement of characters and their transmission by heredity. If, on the other hand, we assume that variations appear spontaneously, and quite irresponsibly, so to speak, in the germ-plasm of the organism, the selection, or elimination, by the environment will not occur until the co-ordinated or un-co-ordinated variations appear. It is far more likely that a large number of simultaneously appearing variations will be un-co-ordinated than that they will be co-ordinated. Merely as a matter of probability the progressive modification of a species will take place slowly--too slowly to account for what we see. Two examples will make it easier to appreciate this difficulty. Evolution has undoubtedly proceeded in definite _directions_. There are two dominant groups of fishes, the Teleosts and the Elasmobranchs, and both must have originated from a common stock. All the characters in each kind of fish must have been useful (since they were selected), and all must have been modifications of the characters of the common stock. The latter became modified along two main lines, or directions, which are indicated by the characters of the existing Teleosts and Elasmobranchs. The whole skeleton, the gills, the circulatory system, and the brain differ in certain respects in these groups. Therefore a modification of the brain in the primitive Elasmobranchs was associated with a modification of the cranium, and therefore with the jaw-apparatus, and so with the branchial skeleton and the gills, and therefore also with the heart, and so on. Suppose that the evolutionary process included ten useful and co-ordinated variations--not an unlikely hypothesis--and suppose that each of these ten useful variations was associated with nineteen useless ones. The chance that any one of them did occur was therefore one in twenty; and if they all occurred independently, that is, if the occurrence of any one of them was compatible with the occurrence of any other one, or of all the others, then the chance that all the ten variations occurred simultaneously was 20^{-10} that is, one in the number 20 followed by 10 cyphers, a rather great improbability.