180 THE PRINCIPLES OF SCIENCE. 



which assert the equivalence of groups of atoms. Ordinary 

 fermentation is represented by the formula 



C H 12 6 = 2C 2 H 6 O + 2C0 2 . 



Three kinds of units, the atoms respectively of Carbon, 

 Hydrogen, and Oxygen, are here intermingled, but there is 

 really a separate equation in regard to each kind. Mathe- 

 maticians also employ compound equations of the same 

 kind ; for in a + b^/i = c + d *J i, it is impossible by 

 ordinary addition to add a to b J i . Hence we really 

 have the separate equations a = c, and b = d c . Similarly 

 an equation between two quaternions is equivalent to 

 four equations between ordinary quantities, whence in- 

 deed the origin of the name quaternion. 



Analogy of Logical and Numerical Terms. 



If my assertion is correct that number arises out of 

 logical conditions, we ought to find number obeying all 

 the laws and conditions of logic. It is almost super- 

 fluous to point out that this is the case with the funda- 

 mental laws of identity and difference, and it only remains 

 for me to show that mathematical symbols do really obey 

 the special conditions of logical symbols which were 

 formerly pointed out (p. 39). Thus the Law of Com- 

 mutativeness, is equally true of quality and quantity. As 

 in logic we have 



AB = BA, 



so in mathematics it is familiarly known that 

 2x3 = 3x2, or~xxy = yxx. 



The properties of space, in short, are as indifferent in 

 pure multiplication as we found them in pure logical 

 thought. 



c De Morgan's 'Trigonometry and Double Algebra,' p. 126. 



