But it is also intelligible that this twice repeated division of the mother-egg-cell is necessary if the reduction in the number of chromosomes to half is only possible in this way, since _this reduction is indispensable_. If each of the two conjugating germ-cells contained the full normal number of chromosomes, the segmentation-nucleus would contain a double number, and if that went on, the number of chromosomes would increase in arithmetical proportion from generation to generation, and would soon become enormous. Even though we were not otherwise certain that these chromosomes are units of a permanent nature, which only apparently break up in the nuclear reticulum, but in reality persist, the fact of reduction would point in this direction. For if they were not permanent structures and distinct from one another, and if their number depended solely on the quantity of chromatin which the nucleus contains, the reduction in number might be secured if the chromosomes in the growing egg and sperm-cells increased in size more slowly than the cell-body and the other parts of the cell. But from the fact that the reduction takes place not in this simple way, but, in sperm-cells and in ova which require to be fertilized, only through cell-division and a specific mode of nuclear division, we may conclude that it cannot happen otherwise, that chromosomes are not mere aggregates of organic substance, but organs whose number can only be reduced by the extrusion of some of them from the cell. It is true that there are ova in which the process of reduction does not follow the course we have described, but the exceptions only serve to confirm our view of the reducing significance of the polar divisions, and of their persistence because of the necessity for reduction. As far back as the middle of the nineteenth century it was known that in various animals the eggs develop without fertilization. This reproduction by 'parthenogenesis' was first established with certainty by the German bee-keeper Dzierzon in 1845, and then scientifically corroborated by Rudolph Leuckart and C. Th. von Siebold. Although parthenogenesis was at first observed only in a few groups of the animal kingdom, in bees and some nocturnal Lepidoptera (Psychidæ and Tineidæ), it has become more and more apparent in the course of years that this 'virgin reproduction' is by no means a rare form of reproduction, and that it occurs regularly and normally in many cases, especially in the very diverse groups of the great series of Arthropoda. Thus among insects it is found in certain saw-flies, gall-flies, ichneumon-flies, in the honey bee, and in common wasps, and it is particularly widespread among plant-lice (Aphides) such as the vine-aphis (_Phylloxera_), whose prodigious multiplication in a short time depends partly on the fact that all the generations, with the exception of one, consist only of females with a parthenogenetic mode of reproduction.