MORPHOLOGY AND MATHEMATICS. 86 L 



attempted to compare organisms separated far apart in Nature and in zoological 

 classification. We are limited, not by the nature of our method, but by the whole 

 nature of the case, to the comparison of organisms such as are manifestly related to 

 one another and belong to the same zoological class. 



Our inquiry lies, in short, just within the limits which Aristotle himself laid 

 down when, in defining a " genus/' he showed that (apart from those superficial 

 characters, such as colour, which he called "accidents") the essential differences 

 between one " species " and another are merely differences of proportion, of relative 

 magnitude, or (as he phrased it) of " excess and defect." " Save only for a difference 

 in the way of excess or defect, the parts are identical in the case of such animals as 

 are of one and the same genus ; and by ' genus ' I mean, for instance, Bird or Fish." 

 And again : " Within the limits of the same genus, as a general rule, most of the 

 parts exhibit differences ... in the way of multitude or fewness, magnitude or 

 parvitude, in short, in the way of excess or defect. For ' the more ' and ' the less ' 

 may be represented as ' 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 transformed 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 structure 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.' ' 



* Historia Animalium, i, 1, 



