SIE B. C. BEODIE ON THE CALCULUS OE CHEMICAL OPEEATIONS. 
69 
(1) Two units of hydrochloric acid are identical with a, unit of hydrogen and a 
unit of chlorine. 
(2) A unit of oxychloride of phosphorus and three units of water are identical 
with three units of hydrochloric acid and a unit of phosphoric acid. 
(3) A unit of alcohol and a unit of oxygen are identical with a unit of water and 
a unit of acetic acid. 
(4) Two units of ammonia and a unit of oxalic ether are identical with two units 
of alcohol and a unit of oxamide. 
(5) A unit of glycerine and three units of acetic acid are identical with three units 
of water and one unit of triacetine. 
(6) A unit of lactic acid and two units of hydriodic acid are identical with a unit 
of propionic acid, a unit of water, and a unit of iodine. 
(7) A unit of mannite and eleven units of hydriodic acid are identical with a unit 
of iodide of hexyl, six units of water, and five units of iodine. 
(8) Five units of hydrogen and a unit of pentoxide of iodine are identical with 
five units of water and a unit of iodine. 
Such examples might be greatly multiplied. Equations possessing this peculiar 
property are not due to accident, but to the simplicity of natural laws. In the chemical 
metamorphoses of which they express the results there is no change of gaseous volume. 
It would be by no means difficult thus to make a system of carefully selected equations 
to which, as satisfying the condition given above, the processes of algebra would be truly 
applicable. That such a system should be possible, that a system of “ normal ” equa- 
tions, subordinate to algebraical laws, should actually be found in the midst, so to say, of 
an abnormal system, is undoubtedly a most striking and suggestive fact. But neverthe- 
less, so far as any realization of the objects of a calculus are concerned, this circumstance, 
if taken alone, is totally inoperative. The result of constituting such a system would be 
to divide chemical equations into two classes — a class with which we could really deal alge- 
braically, and a class with which we could not so deal ; and corresponding to these classes 
we should have, as will presently be seen, two systems of phenomena — a system which 
we could realize, study, and comprehend, and a second system absolutely unintelligible to 
us. These anomalies, however, may be completely removed by a mathematical trans- 
formation of the equation, to which I must now request the attention of the reader. 
(3) It has been remarked (I. Sec. IV. 3), where the meaning and properties of the 
chemical symbol 1 are under discussion, that any number of numerical symbols 0, 1, 2, 3 
may be added to a chemical function without affecting the interpretation of that function 
as regards “ weight.” This may be inferred from the fundamental equation xy=x-\-y , 
which equation becomes if y= 1, x=x-\-l. It is also an immediate consequence of the 
interpretation assigned in this Calculus to that symbol, which regarded as a symbol of 
“ weight ” (being the symbol of an empty unit of space) is the symbol of “ no weight,” 
and regarded as the symbol of an operation is the symbol of (if we may so say) “ taking 
the unit of space as it is ” without performing upon it any operation resulting in “weight ” 
L 2 
