726 THE THEORY OF TRANSFORMATIONS [ch. 



'excess' and 'defect*.'" It is precisely this difference of relative 

 magnitudes, this Aristotelian "excess and defect" in the case of 

 form, which our co-ordinate method is especially adapted to 

 analyse, and to reveal and demonstrate as the main cause of what 

 (again in the Aristotelian sense) we term "specific ' differences. 



The applicability of our method to particular cases will depend 

 upon, or be further limited by, certain practical considerations 

 or qualifications. Of these the chief, and indeed the essential, 

 condition is, that the form of the entire structure under investi- 

 gation should be found to vary in a more or less uniform manner, 

 after the fashion of an approximately homogeneous and isotropic 

 body. But an imperfect isotropy, provided always that some 

 "principle of continuity" run through its variations, will not 

 seriously interfere with our method ; it will only cause our trans- 

 formed co-ordinates to be somewhat less regular and harmonious 

 than are those, for instance, by which the physicist depicts the 

 motions of a perfect fluid or a theoretic field of force in a uniform 

 medium. 



Again, it is essential that our struct^ire vary in its entirety, 

 or at least that "independent variants" should be relatively few. 

 That independent variations occur, that localised centres of 

 diminished or exaggerated growth will now and then be found, 

 is not only probable but manifest; and they may even be so 

 pronounced as to appear to constitute new formations altogether. 

 Such independent variants as these Aristotle himself clearly 

 recognised : " It happens further that some have parts that others 

 have not; for instance, some [birds] have spurs and others not, 

 some have crests, or combs, and others not; but, as a general 

 rule, most parts and those that go to make up the bulk of the body 

 are either identical with one another, or differ from one another 

 in the way of contrast and of excess and defect. For 'the more' 

 and 'the less' may be represented as 'excess' or 'defect.'" 



If, in the evolution of a fish, for instance, it be the case that 

 its several and constituent parts — head, body, and tail, or this 

 fin and that fin — represent so many independent variants, then 

 our co-ordinate system will at once become too complex to be 

 intelligible; we shall be making not one comparison but several 



* Historia Animalium i, 1. 



