VIII] THE PARTITIONING OF SPACE 385 



The rules and principles which we have arrived at from the 

 point of view of surface tension have a much wider bearing than is 

 at once suggested by the problems to which we have applied them ; 

 for in this elementary study of the cell-boundaries in a segmenting 

 egg or tissue we are on the verge of a difficult and important 

 subject in pure mathematics. It is a subject adumbrated by 

 Leibniz, studied somewhat more deeply by Euler, and greatly 

 developed of recent years. It is the Geometria Situs of Gauss, the 

 Analysis Situs of Riemann. the Theory of Partitions of Cayley, 

 and of Spatial Complexes of Listing*. The crucial point for the 

 biologist to comprehend is, that in a closed surface divided into 

 a number of faces, the arrangement of all the faces, lines and 

 points in the system is capable of analysis, and that, when the 

 number of faces or areas is small, the number of possible arrange- 

 ments is small also. This is the simple reason why we meet in 

 such a case as we have been discussing (viz. the arrangement of 

 a group or system of eight cells) with the same few types recurring 

 again and again in all sorts of organisms, plants as well as animals, 

 and with no relation to the lines of biological classification : and 

 w^hy, further, we find similar configurations occurring to mark 

 the symmetry, not of cells merely, but of the parts and organs of 

 entire animals. The phenomena are not " functions," or specific 

 characters, of this or that tissue or organism, but involve general 

 principles which lie within the province of the mathematician. 



The theory of space-partitioning, to which the segmentation 

 of the egg gives us an easy practical introduction, is illustrated in 

 much more complex ways in other fields of natural history. A 

 very beautiful but immensely comphcated case is furnished by 

 the "venation" of the wings of insects. Here we have sometimes 

 (as in the dragon-flies), a general reticulum of small, more or less 

 hexagonal "cells" : but in most other cases, in flies, bees, butter- 

 flies, etc., we have a moderate number of cells, whose partitions 

 always impinge upon one another three by three, and whose 

 arrangement, therefore, includes of necessity a number of small 

 intermediate partitions, analogous to our polar furrows. I think 



* Cf. (e.g.) Clerk Maxwell, On Reciprocal Figures, etc., Trans. R. 8. E. xxvr, 

 p. 9, 1870. 



T. G. 25 



