Beard, Heredity and the epicycle of the germ-cells. 3()9 



four grandparents contribute between them one-quarter, or each of them 

 one-sixteenth ; and so on, the sum of the series, V 2 + 1 U + V 8 + Vie + etc., 

 being- equal to 1, as it should be. It is a property of this infinite 

 series that each term is equal to the sum of all those that follow: 

 thus V, = '/4 + V 8 + Vio + etc., V* + V 8 + Vw + etc., and so on. The pre- 

 potencies or subpotencies of particular ancestors, in any given pedigree, 

 are eliminated by a law that deals only with average contributions, 

 and the varying prepotencies of sex in respect to different qualities 

 are also presumably eliminated". 



Assuming for the moment the correctness of this, its embryological 

 basis is furnished by the formation etc. of the primary germ-cells. 

 The germ-cells in any embryo, possessing from their mode of formation 

 like qualities and having these and the like ancestry with that, which 

 formed the embryo, these qualities are necessarily halved at the follow- 

 ing determination of sex and reduction. At the close "of this halving 

 the ,,parental" qualities can embryologically under Gal ton's law only 

 be represented by at most V 2 , or V* for each ,,parent", and so on 

 for each preceding generation; for in these also primary germ-cells 

 of like characters were formed, of which one gave rise to an embryo 

 in every case. 



The line of ancestry is, of course, from and through these germ- 

 cells, and never from the embryo or sexual generation of a preceding 

 generation. 



But, as the germ-cells, associated withany given embryo, are all 

 of like characters among themselves (including that from which the 

 embryo arises), on the production of eggs and sperms, and the sub- 

 sequent union of these with other sexual products, the result is the 

 same, as if the line of ancestry had been through the embryo ; so far 

 at any rate as the ancestral characters are concerned. According to 

 Gal ton the parental qualities are at most represented in their pro- 

 geny by '/, ('/4 + V 4 ). 



In the same way, and because between offspring and grandparent 

 there are two sets of germ-cells (in addition to those still immature 

 in the offspring) and two reductions, the graudparental portions taken 

 together can only be half of the parental portions taken together, that 

 is to say, V*, and so on through any number of generations. It will 

 be quite unnecessary to carry out the examination further; for study 

 of the diagram will make evident the light it throws from the em- 

 bryological side on Gal ton's law, and how it furnishes this law with 

 its basis in the facts of development. 



In his book on the ,,Germplasm" (English edition p. 257) Weis- 

 mann has adversely criticised Gal ton's law. His objections would 

 be valid in cases where in-breeding had taken place. For, Gal ton's 

 law can only hold good, if no in-breeding occur, and if none have 



