Analysis of Heterogeneity in Some Simplest Chaeactees 201 



savannah ; and the second lot was collected later in the season at a time when it 

 was raining daily, the only dry place was the gravel-bed, and here the individuals 

 were found that were discontinuous from the general population. These statis- 

 tical discontinuities can be multiplied without number, and are sharp changes, 

 often only produced in a few individuals by chance conditions of location, but 

 the majority of the population was not so influenced. It was not observed that 

 the influence had produced any change in the shape of the spots or in the different 

 directions in which this spot varied ; it was simply in one case an accentuation 

 and in the other a diminution in the amount of the pigments formed as measured 

 by the area darkened by them. Both were simple climatic aberrations and on 

 testing failed to appear in the next generation so that statistical discontinuity or 

 aberration means little or nothing. 



Is there any discontinuity in the presence and absence of these spots ? Care- 

 ful examination shows the gradual diminution of any or all of these spots until 

 the traces are no longer visible to the unaided eye, but proper preparation shows 

 on microscopic examination a long scale of decreasing color values. In their 

 lowest terms these simplest characters may be invisible to the eye, only micro- 

 scopic traces persisting, often a faint light yellow-brown color in the lower por- 

 tion of the cuticula which gradually vanishes completely. In these characters 

 there is no visible discontinuity in the present and absent conditions ; neverthe- 

 less the two extreme conditions are capable of fixation in " lines," and when 

 crossed these show clear dominance of the presence over the absence and segre- 

 gation in Fj. 



There is divergence between the different directions of variation of these spots. 

 Spot b varies in an antero-posterior and in a medianward direction, rarely in 

 others. Between these directions of variation no intermediate directions have 

 been found. Occasionally a direction not hitherto observed arises suddenly in 

 some individual or individuals. The new direction of variation starts from the 

 center and goes in its own direction with remarkable precision. There is diver- 

 gence in the directions of change in these characters but whether there is discon- 

 tinuity depends upon definition. 



In this section the answer to the question turns in a way not unlike that of 

 the first section ; there is diversity, divergence, with discontinuity of end-results 

 of diverse sorts, but the proposition of the unity of direction in " continuous 

 variations " seems entirely unfounded. There are statistical discontinuities, but 

 from the method of examination these can exist only in plus and minus aberra- 

 tions ; and further, this method of statement " lumps " so many different condi- 

 tions as one homogeneous class that it is extremely doubtful if the results of sta- 

 tistical study are in reality biological at all. There remains the undoubted fact 

 that in these simplest characters there are differences that are continuous, in that 

 there are plenty of intergrades, and those that are divergent, giving discontinu- 

 ous end-results between which present knowledge has found no intermediates. 

 Actually, the " variations " are diverse relations in constitution, amount, space 

 relations, and interactions with the rest of the mechanism, which result in differ- 

 ing arrangements in the visible product deposited as the outcome of these pro- 

 cesses. That intergrades are present in some aspects of the complex, lacking in 

 others, is not a basis for the creation of sharply marked categories. That inter- 

 grades are present or absent is no criterion of the nature of the processes, and it 

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