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BIOLOGICAL RELATIONSHIPS OF THE 1, 2, 4, SERIES 



In a 1, 2, 4 series, let PN be the final product and N its series number, 

 then P N = 2 N -'; thus, 16 = 2 5 " 1 . 



Similarly in a simple organism, at any particular instant in its growth, 

 let PN have a corresponding value, that is to say, be the number of cells 

 that either actually exist or would have arisen by the uniform and continuous 

 dichotomous division of the single primal cell. Such simple and easily 

 understood organisms occur among the lower forms, and in the early embry- 

 onic stages of the higher forms, but are rare among the adult stages of the 

 higher forms, because in these latter some cells lag or cease in their dichotomy, 

 and because of losses of cells from various causes. Hence, the number of 

 cells actually present in an organism at any particular instant is likely to be 

 PN minus a certain number of cells, (X), due to delay or failure in some part 

 or parts of the dichotomy, or to loss. In this discussion account is taken of 

 all the cells that have been produced during the growth, whether present 

 in the organism at the proposed instant or not. This is in order to allow for 

 worn out or wasted cells; these, possibly vanished, cells are included in PAT 



The general 1, 2, 4 equation of an organism thus becomes P N = 2 N-1 X, 



1st. Jtaoe 



Ectoderm +, 



Fig. 3. Diagram of 8 generations of cells produced by dichotomous divisions; as, for 

 instance, in a segmenting egg. Three general characters of tissue are shown: (1) Sexual, 

 (2) intestinal and related tissues, (3) ectoderm and related tissues. The sexual and in- 

 testinal tissues are shown to have lagged behind those of the ectoderm, so that P N in this 

 instance equals 71. 



in which X is a whole number and a function of one or more "p's" of a lower 

 order, i.e., of the 1, 2, 4 character, or p = 2 n-1 character, in which, of course 

 p is smaller than P and n is smaller than N. These smaller (ascertainable) 

 groups are 1 , 2, 4 groups of cells due to the lag or failure of "earlier" generations 

 than N. (See the loop (X) in Fig. 3.) 



PN = 2 N-1 X is a general equation, which, when X = 0, represents a 

 strictly uniform and continuous mathematical dichotomy, found only in the 

 lower organisms or in the early embryonic stages of the higher ones. 



The various "p's" from PN down to P = 1, (the primal number) become, 

 therefore, historical insignia, indicating particular generations of cells, and 

 may be made the basis of a definite and fundamental mathematico-biological 

 nomenclature applicable to the generations of cells in an organism, and hence 

 to the organism itself. Applications of the equation are endless. 



