76 
SIE B. C. BBODIE ON THE CALCULUS OF CHEMICAL OPERATIONS. 
assertion made in the symbol = be limited to identity of “weight,” x=x-\-l ; so that, 
as regards this identity, the assertion made in the “ normal ” equation 
l + 2 a |= 2 a +| 2 • ( 1 ) 
is the very same assertion as that made in the “ abnormal ” equation 
2a£=2a+£ 2 ; (2) 
it being perfectly immaterial, as before remarked, whether we say that the “ weight ” 
of a unit of space and the “ weight ” of two units of water is identical with the “ weight ” 
of two units of hydrogen and a unit of oxygen, or whether we simply say that the 
“ weight ” of two units of water is identical with the “ weight ” of two units of hydrogen 
and a unit of oxygen, just as it is immaterial whether we say that three and two are 
identical with five, or that nothing and three and two are identical with five, the 
selection among such expressions being dictated merely by algebraical convenience. Yet 
nevertheless a very real and appreciable distinction may be drawn between the two 
assertions, which from another aspect have by no means a strictly equivalent meaning. 
The former equation (1) necessarily includes every assertion which is made in the 
latter ; but the latter (2 ) does not include every assertion which is made in the former. 
For if we extend our notion of identity to identity of “ space ” as well as of “ matter,” 
it is open to us to assign an interpretation to the symbol of identity = in equation 
(1) which cannot be assigned to that symbol in equation (2). A unit of space and two 
units of water are identical, both as regards “ weight ” and as regards the “ space ” 
occupied by that “ weight,” with two units of hydrogen and a unit of oxygen. But it 
is not true that two units of water are identical, both as regards “ weight ” and as 
regards the “ space ” occupied by that “ weight,” with two units of hydrogen and a 
unit of oxygen ;■ for the space occupied by two units of water is 2000 cub. centims., 
but the space occupied by two units of hydrogen and a unit of oxygen is 3000 cub. 
centims. [I. Sec. I. (10, 11)]. Thence, in the reduction of a chemical equation to the 
“ normal ” form, we not only render the equation algebraically correct, but also extend 
and complete the idea expressed in the equation. 
Now this conception of a perfect and complete identity, both as regards “ space ” 
and matter, cannot arise so long as we confine our inquiries to the composition of 
the units of ponderable matter, as considered in the first part of this Calculus, but 
has its origin exclusively in those dynamical inquiries which relate to the changes 
of matter. For this there is an obvious reason. In investigating the laws of gaseous 
combinations, which are the special objects of this method, a most essential point to be 
considered, whether from the point of view of the chemist or of the physical philoso- 
pher, is the relation of “ matter ” to “ space ; ” and in the adequate consideration of a 
chemical change we are bound to take cognizance not only of changes in the compo- 
sition of the “ units of ponderable matter,” but also of changes in the “ bulk ” of matter, 
and to find the means of entering these changes in our equations. The contrivance 
