Correlation 105 



parts, are more precise but are even further from a satisfactory biological 

 explanation. They are inherited, but the genetic mechanisms involved 

 have hardly begun to be explored. At present we can simply describe and 

 classify these correlations. 



The various structures in a growing organic system tend to increase 

 together and thus to be correlated in size. In a given organ its dimensions 

 are likewise correlated. Since growth usually is not uniform, as develop- 

 ment proceeds, the relations between the parts of the system or between 

 the dimensions of the organ may change progressively and thus produce 

 differences in form. Growth is usually exponential in character, and there- 

 fore the relationship between the sizes of two structures growing at dif- 

 ferent rates may best be found by plotting the logarithms of their sizes 

 against each other. If the rates are different but the relation between the 

 two is constant, these values will fall along a straight line the slope of 

 which measures the growth of one structure relative to that of the other. 

 It is noteworthy that in most cases where two parts of the same growing 

 system, or two dimensions of a growing organ, are compared, their 

 relative rates are found to be constant, however different their absolute 

 rates may be. 



This relationship can be described simply by an equation. If y is the 

 size of one variable, .t that of the other, b the value of y when x is of some 

 arbitrary size, and k the ratio of the growth rate of y to that of x, then 



y = bx k 

 or 



log y — log b + k log x 



This phenomenon of constant relative growth (heterauxesis) has been 

 observed by many biologists but was first widely emphasized by Julian 

 Huxley (1932). He termed this type of growth heterogony, a term now 

 replaced in much of the literature by allometry. The constant b measures 

 differences in level, or at the beginning of growth, between two variables. 

 The constant k provides a measure of relative growth rate and may some- 

 times offer a clue to the mechanisms involved. It may be used to express 

 differences when these are based on genetic, environmental, embryologi- 

 cal, biochemical, or even evolutionary factors. This method of analysis 

 has proved useful in the study of many kinds of growth correlations. 



Correlations of Part and Whole. Among the familiar growth correla- 

 tions are those between an organ and the rest of the body or between 

 members of a series of multiple parts and the structure that they con- 

 stitute. In animals, large individuals typically have their organs cor- 

 respondingly larger than those of small ones. In plants, however, with 

 their lower level of organization, their often indeterminate growth, and 

 their multiple organs, this relationship is not so simple. In beans, for 



