18 Growth 



should be the same for all races, but actually in the small-fruited races 

 this period is much less than in the large-fruited ones. 



The mathematical analysis of growth involves many complexities and 

 has been developed in much greater detail than is possible in the present 

 brief discussion. For a fuller treatment of this subject the reader is re- 

 ferred to the work of Pearl ( 1939 ) , D'Arcy Thompson ( 1942 ) , Erickson 

 (1956), and others. 



It is evident that no single mathematical statement will express all types 

 of growth nor perhaps any of them with complete exactness. Growth is 

 a very complex process involving many variables, and it is not to be ex- 

 pected that it can be compressed into a single equation. Even if it could 

 be, this would not tell us a great deal for, as D'Arcy Thompson well says, 

 a formula "which gives a mere coincidence of numbers may be of little 

 use or none, unless it go some way to depict and explain the modus 

 operandi of growth." That growth under some conditions proceeds as at 

 compound interest and at others like an autocatalytic process is of some 

 importance in providing a clue to the mechanism of growth, but so far 

 mathematical analysis has added comparatively little to our understand- 

 ing of the fundamental character of growth itself. For this we must look 

 to a more concrete study of the growth process in terms of genetics, bio- 

 chemistry, and physiology. 



Variation in Growth. Many structures do not show the simple sigmoid 

 growth discussed in the preceding section. Just as the smooth course of 

 growth in mammals is interrupted by birth and by puberty, it is modified 

 in various ways in plants. In fruits of peach and cherry, for example, 

 Tukey and Young (1939) and others have shown that after these struc- 

 tures are partly grown there is then for some time no increase in volume. 

 This is the period in which the endocarp, or "stone," is being formed. 

 Later the fruit begins to enlarge again, so that a curve like that in Fig. 

 2-5 results. Duncan and Curtis ( 1942 ) have shown a somewhat similar 

 growth curve in the fruit of certain orchids where one epoch of growth is 

 associated with meiosis and a later one with seed maturity. In vegetative 

 structures, aside from annual periods, there are also sometimes discon- 

 tinuous cycles, as in the pear shoots studied by Reed ( 1927 ) , where there 

 may be three such in one season. The dandelion scape shows a some- 

 what similar growth pattern (Chao, 1947). It is rapid during flower de- 

 velopment, much slower after the flower opens, and then accelerates 

 greatly as the fruit becomes mature. 



Borriss ( 1934a ) reports that etiolated stipes of Coprinus show a marked 

 periodicity of growth with maxima 3.5 to 4.5 hours apart. This and similar 

 cases may be manifestations of endogenous rhythms such as have fre- 

 quently been reported in other processes ( p. 322 ) . 



