158 Intelligence and Miscellaneous Articles. 



cessary hypothesis that a fluid is absolutely continuous. Conceive 

 the contraction to be continued until all the matter of the supposed 

 spherical earth is concentrated at its centre, and we formally as well 

 as substantially have the case discussed by Ostrogradsky. About 

 that centre describe geometrically a sphere passing through one of 

 the points of contact of the particles situate on the inner surface of 

 the liquid vault. Then, from the symmetry of the arrangement, 

 we know that the geometrical sphere will pass through all the points 

 of contact of all the particles situated on that surface, and all the 

 points of contact of any one particle will be in one plane. In a 

 plane, through the points of contact, draw geometrical tangents at 

 all the points of contact of any one particle with all the adjacent 

 particles. Then the symmetry of the supposed arrangement shows 

 that the closed figure so formed will be a regular (equilateral and 

 equiangular) polygon. And that symmetry further indicates that 

 each particle will afford the construction of a similar polygon, that 

 all the polygons so formed are equal, and that each side of each po- 

 lygon is common to two adjacent particles, and forms the edge of a 

 regular polyhedron. But we know that there are only five regular 

 solids or polyhedra, — namely, the regular pyramid (or tetrahedron), 

 bounded by four equal and equilateral triangles ; the cube (or hexahe- 

 dron), by six squares ; the octahedron, by eight equal and equilateral 

 triangles; the dodecahedron, by twelve equal and equilateral penta- 

 gons ; and the icosahedron, by twenty equal and equilateral triangles. 

 Consequently, however we adjust the magnitude of the spherical 

 balls or particles in reference to that of the geometrical sphere, if we 

 require a system of balls such that each ball shall be capable of being 

 placed in contact with the adjacent balls while each shall be equidis- 

 tant from the centre of the geometrical sphere, we are restricted to 

 systems of four, six, eight, twelve, and twenty balls, each touching 

 the others of the same system as follows : viz. three others in the 

 system of four, four others in the system of six, three others in that 

 of eight, five others in that of twelve, and three others in that of 

 twenty. A case of fluid equilibrium which can only occur where the 

 particles of the fluid do not exceed twenty in number, can scarcely 

 be held to affect the fundamental principle of hydrostatics. And 

 the fact that while the number of regular polygons is unlimited, 

 that of the regular polyhedra is limited, destroys (except in the par- 

 ticular instances just adverted to) the analogy between aline or circle 

 of particles in equilibrium and a sphere of like particles in equilibrium, 

 and prevents it from being urged in support of the new hydrostatical 

 paradox. I do not at present call to mind any investigations in which 

 a perfect continuity of the fluid is assumed, unless probably in some 

 of those of Professor Chains of Cambridge. But even if I am right 

 in thinking that he has assumed it, all the ends that he had in view 

 would probably be equally well served by changing the assumption 

 to that of particles or distances infinitesimally small in comparison 

 with the particles whose motion is discussed, or the mutual distances 

 of the latter particles. At all events an hypothesis assumed for a spe- 

 cial purpose ought not to influence the present discussion, unless it 

 explains phenomena to be explained in no other way. — Queensland 

 Daily Guardian, Wednesday, May 9, 1866. 



