VIII] OF LISTING'S TOPOLOGY 609 



rarest of all looks Hke supporting the conjecture. Moreover, in all 

 the commonest types we have a more or less equable division; but 

 on the other hand, the number of contacts in "type /is just the same 

 as in b, but the latter occurred thrice as often, and k, which is as 

 equable as any, was one of the least frequent of all. Coincidences 

 are weighed down by discrepancies, and we are left pretty much in 

 the dark as to why some types are much commoner than others. 



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 apphed them; 

 for in this study of a segmenting egg we are on the verge of a subject 

 adumbrated by Leibniz, studied more deeply by Euler, and greatly 

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

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

 Spatial Complexes or Topology of Johann Benedict Listing*. It 

 begins with regions, boundaries and neighbourhoods, but leads 

 to abstruse developments in modern mathematics. Leibniz 

 had pointed outf that there was room for an analysis of mere 

 position, apart from magnitude: "je croy qu'il nous faut encor une 

 autre analyse, qui nous exprime directement situm, comme I'Algebre 

 exprime magnitudinem.'' There were many things to which the 

 new Geometria Situs could be applied. Leibniz used it to explain 

 the game of sohtaire, Euler to explain the knight's move on the 

 chess-board, or the routes over the bridges of a town. Vandermonde 

 created a geometric de tissage%, which Leibniz himself had foreseen, 

 to describe the intricate complexity of interwoven threads in a satin 

 or a brocade§. Listing, in a famous paper ||, admired by Maxwell, 

 Cayley and Tait, gave a new name to this new "algorithm," and 

 shewed its apphcation to the curvature of a twining stem or tendril, 



* Cf. Clerk Maxwell, On reciprocal figures. Trans. R.S.E. xxvi, p. 9, 1870. 



t In a letter to Huygens, Sept. 8, 1679; see Hugenii Exercitationes math, et 

 philos., etc., ed. Uylenbroeck, p. 9, 1833. 



X Remarques sur les problemes de situation, Mem. Acad. Sci. Paris (1771), 

 1774, p. 566. 



§ A problem developed by many eminent mathematicians, and which Edouard 

 Lucas shewed to be intimately related to the construction of Magic Squares: 

 Recreations mathem. i, p. xxii, 1891. 



II Vorstudien iiber Topologie, Gottinger Studien, i, pp. 811-75, 1847; Der Census 

 raumlicher Complexe, ibid, x, pp. 97 seq., 1861. 



