1094 THE THEORY OF TRANSFORMATIONS [ch. 



Ah algebraic curve has its fundamental formula, which defines 

 the family to which it belongs; and its parameters, whose quantita- 

 tive variation admits of infinite variety within the limits which 

 the formula prescribes. With some extension of the meaning of 

 parameters, we may say the same of the families, or genera, or other 

 classificatory groups of plants and animals. We cross a boundary 

 every time we pass from family to family, or group to group. The 

 passage is easy at first, and we are led, along definite lines, to more 

 and more subtle and elegant comparisons. But we come in time 

 to forms which, though both may still be simple, yet stand so far 

 apart that direct comparison is no longer legitimate. We never 

 think of "transforming" a helicoid into an ellipsoid, or a circle into 

 a frequency-curve. So it is with the forms of animals. We cannot 

 transform an invertebrate into a vertebrate, nor a coelenterate into 

 a worm, by any simple and legitimate deformation, nor by anything 

 short of reduction to elementary principles. 



A "principle of discontinuity," then, is inherent in all our classifi- 

 cations, whether mathematical, physical or biological; and the 

 infinitude of possible forms, always limited, may be further reduced 

 and discontinuity further revealed by imposing conditions — as, for 

 example, that our parameters must be whole numbers, or proceed 

 by quanta, as the physicists say. The lines of the spectrum, the six 

 families of crystals, Dalton's atomic law, the chemical elements 

 themselves, all illustrate this principle of discontinuity. In short, 

 nature proceeds from one type to another among organic as well as 

 inorganic forms; and these types vary according to their own 

 parameters, and are defined by physico-mathematical conditions of 

 possibility. In natural history Cuvier's "types" may not be per- 

 fectly chosen nor numerous enough, but types they are; and to seek 

 for stepping-stones across the gaps between is to seek in vain, for 

 ever. 



This is no argument against the theory of evolutionary descent. 

 It merely states that formal resemblance, which we depend on as 

 our trusty guide to the affinities of animals within certain bounds or 

 grades of kinship and propinquity, ceases in certain other cases to 

 serve us, because under certain circumstances it ceases to exist. Our 

 geometrical analogies weigh heavily against Darwin's conception of 

 endless small continuous variations; they help to show that dis- 



