Arguing from general considerations, but especially from the theory which regards the chromosomes as the bearers of the hereditary substance, I had come to the conclusion, before there was any full knowledge of the phenomena of the maturation of the ovum, that a reduction of the chromosomes by half _must_ take place, and had postulated a similar 'reducing division' for the sperm-cell, and further, for plants as well as animals--indeed, for all sexually reproducing forms of life. The two divisions in the sperm-cell corresponding to the polar divisions of the ovum with their reduction of chromosomes were demonstrated by Oscar Hertwig in the case of the thread-worm of the horse (_Ascaris megalocephala_)--a form which has proved so very important in relation to the whole theory of fertilization. It is true that in this case the course of the phenomena of reduction is less convincing than in some other forms which have been investigated more recently, as, for instance, the mole-cricket and the bugs. In these instances, at any rate, a 'reducing division' in spermatogenesis, quite corresponding to that of the egg-cell, has been demonstrated, and this demonstration is of particular value owing to the fact that the development of the sperm-cell, as we shall presently see, throws an entirely new light on that of the ovum, and especially on the phyletic significance of the polar bodies. We began our consideration of the processes of reduction with the full-grown egg-cell, but now let us go back to the earliest rudiments of the ovary of the embryo, and we find that it consists of a single primitive egg-cell, from which, by division, all the other egg-cells arise. In the same way the first rudiment in the testis or spermary is formed by a primitive sperm-cell, which does not differ visibly from the primitive egg-cell. Both now multiply by division for a considerable time, and in the ovary this is followed by a period of growth, during which multiplication ceases, and each cell increases considerably in size and lays in a store of yolk. Each cell thus ultimately reaches the condition with which we started previously, that of the full-grown _mother-egg-cell_.