VIII] OF TOPOLOGICAL ANALYSIS 611 



their vertices with the symbol 3. There remains the quadrilateral 

 ABEH, containing two peripheral triangles ABE, EH A; mark B 

 and H each with the symbol 2. The residual points A, E are to be 

 marked 1 and 1. The polygon ABCDEFGHJ may now be read 

 off: 124313424; and this formula, resulting from the triangula- 

 tion, defines completely the system of chords and the topology of the 

 "island." This is one of the twenty-seven cases of a nine-celled 

 arrangement; and here are our twelve arrangements of eight cells, 

 recatalogued under the new method: 



a 11231323 | g 12341243 



b 1 1321323 



c 12312313 



d 12313132 



e 123 132 13 



/ 1234 1234 



h 12341342 



i 12 3 4 14 3 2 



j 12 4 3 12 4 3 



k 12431342 



I 1323 1323 



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 arrangements 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 fines of biological 

 classification: and why, 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, even "properties of space," 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 

 innumerable ways, some simple, some extremely compHcated, in 

 other fields of natural history ; and some serve the better to illustrate 

 the mathematical, and others the physical groundwork of the 

 phenomenon. 



