2o8 PROBLEMS OF RELATIVE GROWTH 



effects of differences in absolute size, and that when an island 

 race is smaller, its small size is probably due in the main to 

 environmental stunting. This is supported by the findings of 

 Antonius (1920), who shows that in the dwarf race of deer on 

 Sardinia, the length of the face relative to the cranium is 

 reduced. Since the face is positively heterogenic, this again 

 is to be expected if the animal is stunted. The stunting has 

 produced a strain which, because dwarfed, has permanently 

 juvenile proportions. See also Klatt (1913) on dog skulls. 1 



I have spent some time on this case, as its analysis is fairly 

 complete. We may presume that similar differences in rela- 

 tive antler-size found in geographical races of Roe-deer are 

 secondary, though here the differences in absolute size 

 (which on the whole grades upwards from west to east across 

 Europe) are themselves more likely to be genetic, since the 

 type of habitat of the Roe-deer is approximately the same 

 over its whole range. 



It is equally clear that many other percentage measurements 

 — e.g. of chela-size or abdomen-size in crabs — will in them- 

 selves be of no systematic value. What are taxonomically 

 distinctive in such organs are their growth-constants — the 

 values of the growth-ratio and of the partition-constant in 

 the heterogony formula, and the absolute body-sizes at which 

 heterogony (if it is not continuous or uniform) begins, ends 

 or alters. These numerical values, if properly established, 

 would be true taxonomic characters ; since, however, they 

 are far less convenient to ascertain (though probably of much 

 greater biological importance) than the usual type of specific 

 difference, which is selected for its obviousness and ease of 

 recognition, they are not likely to be much employed. 



A peculiarly interesting example, constituting in a sense a 

 test-case, is provided by the Lucanid beetle Cyclommatus 

 tarandus, with markedly heterogonic male mandibles. This 

 has been studied by Dudich (1923), whose measurements and 

 conclusions have been further analysed by me (1927, 1931c). 

 Coleopterists have distinguished a number of forms of the male 

 of this stag-beetle — five main types, or if we include sub-types, 

 no fewer than seven, all based on structural characteristics of 

 the mandible. (See also Griffini, 1919, Natura, 10, 13.) 



They are as follows (I omit the sub-types) : 



1 A slightly different case of retention of juvenile proportions is 

 given by Drennan (1932) who discusses the skull of the pre-Bushman 

 race of South Africa. This appears more akin to paedomorphosis (p. 239) . 



