54 
SIE B. C. BEODIE ON THE CALCULUS OF CHEMICAL OPEEATIONS. 
Figure II. is constructed on the same principles, to indicate the combinations on 
hypothesis a 2 . The area within the outer line comprehends the region of possible com- 
binations, actual or conceivable, of the simple weights of the system* a , c , i, ., 
all of which are constituted of the matter of our actual elemental bodies, and are capable 
of being resolved into them. But this region is divided into a system of annular spaces? 
of which only each alternate ring is actually inhabited. The occupied regions are indi- 
cated by the shaded spaces, and are assumed to be tenanted by those substances which 
satisfy the law of even numbers ; while the unexplored regions, indicated by the open 
rings, are occupied by the imaginary substances which do not satisfy this law. It is 
therefore a mere mistake to consider hypothesis a to be in any sense more hypothetical 
than hypothesis a 2 . Both hypotheses indicate to us the possible existence of a system 
of unreal things ; but there is a wide difference in the relation subsisting between the 
realities and unrealities on the two views respectively. If we consider the operations of 
nature to be directed, as a flight of arrows at a target, to the construction out of the 
matter of our elemental bodies of a system of chemical substances made up of the matter 
of those elements, on hypothesis a the arrows go direct to the mark, this aim is pro- 
perly attained ; if no arrows appear in the outer rings of the target, it is that they lie 
wide of the mark ; whereas on hypothesis a 2 , while the object aimed at is the same, 
the arrows are placed exclusively in the alternate or coloured rings of the target, 
2 , 4 , 6 , . . ., while the white rings, 1, 3 , 5 , . . ., have no arrows placed in them at 
allf. The former hypothesis, therefore, leads to results perfectly consistent with 
the view that the operations of nature are directed to this end, while the results 
to which we are brought by the latter hypothesis are absolutely inconsistent with this 
view. 
Although the actual existence of those substances which lie external to our system is 
by no means a necessary consequence of our hypothesis (for we certainly cannot expect 
to be able to do all that is possible to be done), and the construction of these substances 
may be a problem altogether beyond the range of our experimental powers, yet it must 
be admitted by any who hold to the principles of this method that the appearance upon 
the scene of these “ ideal existences ” is at least a possible phenomenon ; and it is not 
without interest to consider our position in regard to this contingency. Two reasons 
* i.e. 5 , ® 2 , ... 
t In figure I. we should have within the central shaded space in the case of the combinations a n w 1 ' 2 . . . , 
those combinations in which the ratios . . . were severally not < 0 not >2, while in the open space the 
combinations would appear in which this condition was not satisfied. In figure II., considering the symbol 
a™ c m > i™*, . . ., we should have in the shaded spaces 2, 4, 6, . . ., the combinations of these letters taken 
2 and 2, 4 and 4, 6 and 6, . . . together, which combinations are represented in our actual system of things : 
while in the open spaces, 1, 3, 5, . . . we should have the combinations of these same simple weights taken 
singly, 3 and 3, 5 and 5, . . ., which are non-existent combinations. 
