BIOLOGICAL RELATIONSHIPS OF THE 1, 2, 4, SERIES 



373 



numbers not only necessarily and definitely, but perhaps somewhat simply, 

 modified mathematical expressions of the fundamental mathemati co-biologi- 

 cal phenomena inevitably arising from the fact that cells (as well as some of 

 their components) divide in accordance with the 1, 2, 4 series? 



Variations of the 1, 2, 4 series, as expressed in cell multiplication, say in a 

 segmenting egg, can be readily diagrammed. (See Fig. 3.) If in such a cell- 

 division diagram any particular multiplying cell or cells be pictured as halted, 

 while the others continue to divide, the next step will bring about a variation 

 from the geometrical series. If the reader will draw a few simple diagrams, 

 he will find it easy, by such variations, graphically to represent, as existing 

 at successive early stages in the imagined ontogeny, numbers of cells, say, 

 from 1 to 10 inclusive, and will see that conceivably this could go on indefi- 

 nitely, and that therefore any number whatever is a possible biological variation 

 of the 1, 2, 4 series. But this broadening of the possibilities must not be 

 allowed to obscure the basic fact that the numbers are neverthe- 

 less definite mathematical variations of the 1,2,4 series due to the 

 binary division of cells and their components; -which in turn seems 

 compulsory owing to the nature of matter itself. Our problem seems 

 to be: Which of these numerous variations are the more sig- 

 nificant, and what are their mathematical and biological relationships? 1< 



A new triplonch, Tylenchus cancellatus n. sp. (Figs. 1 and 2), 

 infesting the roots of peonies, will serve, in a very limited way, to 

 illustrate the foregoing remarks. The figures (Fig. 1) show the 

 existence, near the head, of sixteen external longitudinal grooves. 

 Near the middle of the neck this number changes to eighteen by 

 the splitting, on each side of the nema, of one of the lateral, or 

 sublateral, elements of the series, so that most of the body presents 

 18 grooves. Posteriorly this number reduces to 14, 10, then 8. 

 (Fig. 1.) 



This emphasizes the value of pondering the variants of the 

 1, 2, 4 series. If the numbers of the various elements were con- 

 fined to the 1, 2, 4 series, they would be less significant, hence less 

 useful; -e.g., in the interpretation of relationships. But variations 

 abound, and are, as yet, for the most part unexplained; probably 

 often highly complex. It is certain, however, that if these variations Peo ^' 2 r ' oot 

 can be envisaged and understood, they will serve as basic data. ( ^naiis) a nat~ 



There seems at present no way of stating exactly the upper f^^d 11 !* 

 limit of the numbers representing these variations of the 1, 2, 4 S^fcaSfc 

 series as exemplified in an organism. It may in some organisms latus - 

 reach twenty figures, and therefore the discovery and interpretation of some 

 of the highest members of this modified geometrical series, as exemplified in 

 organisms, may be beyond our present compass. Nevertheless, does it not 

 seem likely that relationships traced in this manner may at .least be set upon 

 a firmer basis than is the case when data of other sorts are used, or even 

 upon an entirely new basis? 



