LOGARITHMIC SPIRALS 153 



Ponse, 1930) ; these have shown that if the regeneration-bud 

 from the tail of a Urodele be removed while still in the 

 indifferent stage and grafted on to the stump of a freshly- 

 amputated limb, it will grow not into tail but into limb ; 

 whereas if it has been left a couple of days longer on the tail 

 before being transplanted, it would have been irrevocably 

 determined as tail, and would have become tail even in its 

 new situation. They can only be interpreted as meaning 

 that what has conveniently been called a ' morphogenetic 

 field ' permeates the whole body even of the adult Amphibian. 

 It normally is without effect — in a sense a by-product, we 

 may say, of the construction of the animal ; but so soon as 

 indifferent material is placed under its influence, it reveals 

 its presence by the effect which it exerts on that material's 

 differentiation. 



Whether growth-gradients and morphogenetic effects on 

 differentiation are both results of one and the same organismal 

 field, or whether two essentially different, separate field- 

 mechanisms are at work, is very difficult to say. Further 

 discussion of this and related points will be deferred to 

 Chapter VI. 



Thus the horns of rhinoceroses, when considered from the 

 point of view of relative growth, even without further experi- 

 mental analysis, reveal interesting and unexpected properties 

 of the animal body. The same point of view, applied to other 

 structures of the same nature, is equally illuminating. The 

 horns of rhinoceroses are median ; we should therefore not 

 expect to find a difference of growth between their lateral 

 margins. But as soon as we deal with non-median structures, 

 we should expect, if growth-fields permeate the animal body, 

 to find a difference in growth-intensity not only between 

 anterior and posterior, but also between median and lateral 

 margins. If this is so, the resultant growth will be in what 

 is popularly called spiral form, i.e. not merely curving in a true 

 logarithmic spiral in one plane, but corkscrewing up at right- 

 angles to the first. 



This is what actually occurs in almost all horns of sheep, 

 goats and antelopes. Sometimes the lateral growth-difference 

 is very slight, and the ' shear ' at right-angles to the primary 

 plane of the horns' spiral is scarcely perceptible, as in the 

 Sable and other antelopes. At other times it is considerable, 

 and we get the horns of certain sheep corkscrewing out at 

 right angles to the side of the head. It appears that there 



