VIII 



GROWTH AND EVOLUTION 



239 



mm 



o 1 



3 4 5 6 



Fig. 40. Coordinate transformation of the outline of the fish Diodon into the moon fish, 

 Orthagoriscus. After Thompson, 1942. 



changes: the coordinate transformation according to D'Arcy Thompson (1942) 

 and the principle of allometry according to Huxley (1932). 



Thompson observed that, if the projection of an organ or a whole organism 

 is entered into a system of coordinates, it can frequently be transformed into 

 another related form by mere deformation of the coordinate system. For instance, 

 the shape of the fish, Diodon, drawn in rectangular coordinates, gives the bizarre 

 moonfish Orthagoriscus, if the coordinates are transformed into concentric circles 

 in the vertical axis, and hyperbolas in the horizontal (Fig. 40). 



Similarly, when the skull of Hyracotherium (Eohippus) is drawn in a Cartesian 

 coordinate system, it can be transformed into that of the modern horse, inter- 

 mediate stages appearing which correspond to evolutionary intermediates in the 

 equine series. Applied with care, Thompsonian transformation is a suitable 

 method for tracing evolutionary series and for excluding sidelines, as was shown 

 by Rummer's (1953) analysis of the evolution of the human skull {cf. p. 249). 



Brontotherium 



gigas 



Eotitanops 



Fig. 41. Evolution in Titanotheria. a Comparison of size of Eotitanops (eocene) and 

 Brontotherium (lower oligocence), as an example of Cope's rule, b Progressive change in skull 

 shape and orthogenetic increase of horns with increasing body size. After Osborn and Robb 



from Bertalanffy, 19513. 



Literature p. 253 



