3 
metrical contradiction — we may postulate the 
description of an equilateral and equiangular 
polygon of any number of sides whatever. 
The ring being thus formed, let a second circu- 
lar cylinder be placed vertically on the table so 
as to envelope the ring of balls externally, and, 
further, suppose that the second cylinder is 
tightly pressed, by its own elasticity or other- 
wise, round and against the ring of balls. 
Next, remove the inner cylinder. Equilibrium 
will still subsist and the ring of balls will re- 
main undisturbed, for the resulting pressures 
of the tightened envelope will act in the 
direciion of a point in the axis of the cylinder 
at a distance from the table equal to the radius 
of the balls, and will be counteracted by the 
mutual pressures of the balls, Vvhicli, as a 
moment’s reflection will show, tend outwardly, 
as those of the envelope tend inwardly, to the 
ring. Or, to use an argument like Ostrograd- 
sky’s, the pressure of the envelope tends to 
force the balls towards the centre of the ring, 
as we may call the point just spoken of: but, 
the balls being equal and similar and similarly 
situated, there is no sufficient reason why any 
one rather than another should approach the 
centre, and, inasmuch as the balls, supposed to 
be hard and incompressible, cannot all ap- 
proach it together, no motion of any ball 
towards the centre can ensue. And 
if we substitute for the tightened envelope its 
physical equivalent, an attractive force tend- 
ing towards the centre of the r-ing } we may 
admit Ostrogradsky’s result in the case of 
a system of spherical particles, ranged in a cir- 
cularring round a centre of attraction ; for the 
same argument will show that no ball can, so 
to say, be squeezed out of the ring, and forced 
to move away from the centre. 
6. But, by an argument which I shall endea- 
vor to illustrate, he calls upon us, implicitly at 
least, to go further. Conceive a smooth homo- 
geneous perfect sphere, of the size of the earth, 
endued with attraction like that of the ea th, 
but destitute of any motion either of transla- 
tion or rotation, uninfluenced by the attrac- 
tion of any of the heavenly bodies and covered 
by a film, layer or ocean of a homogeneous in- 
compressible liquid, sjy water, supposed, for 
the present purpose, to be absolutely incom- 
pressible and ot the same density at all tem- 
peratures. Suppose, further, that there is no 
cohesion between the liquid and the sphere, 
and that by a decrease of temperature or by 
any other means the spherical earth contracts — 
according to Ostrogradsky t lie fluid, film, layer, 
ot* ocean will remain motionless, a liquid vault 
interposed between the infinite space external 
to the fluid, and the void and finite expanse 
occasioned by the contraction of the earth. 
The suppositions just made are, in substance, 
the same as the conditions of Ostrogradsky ; 
for we know that, according to the law of 
gravitation, the resultant attraction of a system 
of concentric homogeneous spherical layers or 
shells, is the same as if the matter of all the 
shells were to be concentrated at their common 
centre. His argument is substantially the 
same as that which I have employed above ; 
the particles of the liquid being supposed to be 
equal, similar, and similarly situated, the 
symmetry of the arrangement round the 
centre shows that, for want of sufficient 
reason in that behalf, no one particle 
can approach the centre unless all the par- 
ticles situate on the same spherical surface do 
so simultaneously. But this they cannot do, 
since the liquid is supposed to be incompres- 
sible. Nor, by the like principle of sufficient 
reason, can any one of the particles on the same 
spherical surface be squeezed or pressed out- 
wards, that is to say, in a direction away from 
the centre. Hence, the inference that the 
liquid vault will remain in equilibrium. 
7. This inference seems to me to be erro- 
neous, unless W3 impress an arbitrary constitu- 
tion on the fluid, and have recourse to the un- 
necessary hypothesis that a fluid is absolutely 
continuous. Conceive the contraction to be con- 
tinued until all the nutter of the supposed 
spherical earth is concentrated at its centre, 
and we formally, as well as substantially, h>ve 
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 con- 
tact of all the particles situated on that surface, 
and all the paints of contact of any one pariicle 
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 sym- 
metry of the supposed arrangement shows that 
the closed figure so formed will be a regular 
(.quila'eral and equiangular) polygon. And 
that symmetry further indicates that each par- 
ticle will afford the construction of a similar 
polygon, that all the polygons so formed are 
equal, and that each side of each polygon is 
common to two adjacent particles, and forms 
the edge of a regular polyhedron. But, we 
know that there are only five regular solids ot 
polyhedra — namely, the regular pyramid (or 
tetrahedron), bounded by four equal and 
equilateral triangles, the cube (or hexahedron) 
by six squares ; the octahedron by eight equal 
and equilateral triangles ; the dodecahedron 
by twelve equal and equilateral pentagons j 
and the icoshedron by twelve equal and 
equilateral triangles. Consequently, how- 
ever we adjust the magnitude of 
the spherical balls or particles in refer- 
ence to that of the geometrical sphere, if 
we require a system of balb such that each ball 
shall be capable of being placed in contact 
with the adjacent balls while each shall be 
equidistant from the centre of the geo* 
