306 
minutest diameter conceivable as well as for 
those of the most colossal size we can imagine, 
and for all intermediate sizes. 
Now regard the spheres as equal masses of 
homogeneous matter endowed with the prop- 
erty of mutual attraction or gravity. They 
will tend to collect together in a group if free 
to move relatively, and to remain so. The 
cubical arrangement would be entirely con- 
sistent with complete equilibrium of the at- 
tracting forces, but this can not be permanent 
since it is not a formation of maximum den- 
sity or concentration. It does not fully sat- 
isfy the collecting tendency under the forces 
of mutual attraction, and the equilibrium of 
the cubical formation must be unstable. If 
arranged on the cubical system the group will 
collapse on the slightest disturbance and the 
members will seek another arrangement per- 
mitting greater concentration. The rhombic 
dodecahedral grouping affords maximum con- 
-centration but it too fails to give complete 
stability, for it is not perfectly symmetrical 
and the forces of attraction can not be per- 
manently balanced or in complete equilibrium 
throughout. Any one group of thirteen of 
the balls or spheres would be “ satisfied ” as to 
concentration and balance or equilibrium by 
the regular dodecahedral arrangement, but as 
above set forth this could not possibly obtain 
as a fixed condition throughout a group of 
more than thirteen of the spherical bodies. 
For a group of an indefinite number of the 
equal spheres greater than thirteen, there 1s no 
stable and permanent arangement possible. 
We can now in imagination expand the 
diameter of the balls to any extent limited 
only by infinity—which means without limit— 
and likewise their size may be reduced to any 
dimensions short of zero, while their number 
may be multiplied also without restriction. 
The above relations are true for the smallest 
units of matter that can exist as well as for 
the most gigantic bodies. Furthermore the 
truth of these principles is not dependent on 
the complete occupation by matter of each of 
the individual spherical spaces or volumes con- 
sidered. These spherical spaces may be only 
the respective “fields” or space loci of one 
SCIENCE 
[N. S. Vou. XLVIITI. No. 1241 
or more separate portions of matter in a state 
of motion respecting neighboring portions in 
other similar spaces or fields—all haying mo- 
tions of revolution, of vibration, of oscillation, 
or of pulsation, with limitless combinations 
and variations as to size-and number of the 
individual portions, their velocity, direction 
and amplitude of, movement, etc. 
Every assemblage or group of matter tends 
to assume the form with an external bound- 
ing surface of spherical shape under the mu- 
tual attractions of its parts, but however large 
or small such an assemblage or whatever may 
be the number of its individual members, its 
internal structure is governed by the prin- 
ciples above outlined. This indicates an ex- 
planation of the paradox involved in the first 
assumptions or impressions above referred to. 
The conceit that a perfectly symmetrical 
grouping of equal spheres with maximum con- 
centration can be made, at first seemingly en- 
tirely simple and even axiomatic, turns out to 
be inconsistent with elementary facts of geom- 
etry and therefore impossible. 
A direct corollary of this proposition:is that 
a plenum of matter in any form, or any mate- 
rial “continuous medium,” is impossible and 
non-existent. All material substances affected 
by gravity—which is equivalent to saying all 
real matter whatever—must be atomic or 
“oranular” in its structure and in its be- 
havior, and this does not depend upon an as- 
sumption of “ intermolecular repulsion ” or of 
“kinematical energy,” nor indeed even upon 
the theory of energy as a separate entity, nor 
on any other extraordinary force or attribute. 
Plain gravitational attraction with the result- 
ant unrelenting stress and struggle for a status 
which is geometrically unattainable is all- 
sufficient. This may even be made to ac- 
count for apparent repulsion. 
The reason for the conviction and belief 
that these principles have an intimate and 
fundamental relation to the universal and 
eternal unrest of matter and to all physical 
phenomena of whatever nature will now be 
apparent and we have at least an interesting 
and suggestive side light on Boltzmann’s dem- 
onstration of “the indispensability of atom- 
