150 PROBLEMS OF RELATIVE GROWTH 



or foraminifera ; but the horns of antelopes, sheep, oxen, 

 rhinoceroses, etc., as well as the teeth of Vertebrates, also 

 fall into this category. At first sight the forms engendered 

 by this type of growth appear so different from those we 

 have hitherto been considering that we do not even expect 

 to find that the underlying growth-mechanisms have any- 

 thing in common. The differences, however, depend almost 

 entirely upon the basic difference between any multiplicative 

 and any additive type of growth. It is to my mind one of 

 the most interesting results of these growth-studies that 

 we are able to demonstrate the same fundamental fact of 

 growth-gradients operating to produce these two apparently 

 unrelated types of organic form. 



Let me first take the horn of rhinoceroses as example. It 

 has been admirably handled by D'Arcy Thompson in his 

 Growth and Form. I here base myself on his lucid analysis, 

 which, like so many other important ways of thinking that 

 enable us to see familiar facts in a new light, seems self-evident 

 once grasped ; but I add one or two detailed points, and 

 link it up with the ideas which emerged from the study of 

 multiplicative growth. 



The horn of a rhinoceros, then, is produced by intensive 

 production of keratin in special form and abundance over a 

 limited area of the head epidermis. The restriction of horn- 

 producing potency to a limited area is doubtless of the same 

 nature as the other restrictions of potency which occur during 

 early development and sooner or later convert the germ from 

 a plastic construction capable of marked regeneration to a 

 determined construction which we can designate as a chemical 

 mosaic (see Huxley, 1924c). The potency of producing eye, 

 ear, brain or limb becomes similarly restricted and localized 

 in the amphibian and other embryo, and the restriction of 

 horn-potency to a localized area is only another result of this 

 mosaic-producing chemo-differentiation. 



On this horn-area, keratin is being produced so as to 

 accumulate at right angles to the surface. In addition, the 

 horn-area itself is enlarging over the surface as the animal 

 grows. Whether, as seems likely, it is enlarging somewhat 

 more rapidly than the surface of the head as a whole cannot 

 be stated with certainty until detailed measurements have 

 been made ; but this is immaterial to our present purpose. 



