xiv.] UNITS AND STANDARDS OF MEASUREMENT. 327 



intensity of magnetisation has the same dimensions. In the 

 science of electricity physicists have to deal with numerous 

 kinds of quantity, and their dimensions are different too in 

 the electro-static and the electro-magnetic systems. Thus 

 electro - motive force has the dimensions M^L^T \ in 

 the former, and M^T~ 2 in the latter system. Capa- 

 city simply depends upon length in electro- statics, but 

 upon L~ 1 T 2 in electro-magnetics. It is worthy of par- 

 ticular notice that electrical quantities have simple dimen- 

 sions when expressed in terms of density instead of mass. 

 The instances now given are sufficient to show the diffi- 

 culty of conceiving and following out the relations of the 

 quantities treated in physical science without a systematic 

 method of calculating and exhibiting their dimensions. It 

 is only in quite recent years that clear ideas about these 

 quantities have been attained. Half a century ago pro- 

 bably no one but Fourier could have explained what he 

 meant by temperature or capacity for heat. The notion 

 of measuring electricity had hardly been entertained. 



Besides affording us a clear view of the complex relations 

 of physical quantities, this theory is specially useful in 

 two ways. Firstly, it affords a test of the correctness of 

 mathematical reasoning. According to the Principle of 

 Homogeneity, all the quantities added together, and equated 

 in any equation, must have the same dimensions. Hence 

 if, on estimating the dimensions of the terms in any equa- 

 tion, they be not homogeneous, some blunder must have 

 been committed. It is impossible to add a force to a velo- 

 city, or a mass to a momentum. Even if the numerical 

 values of the two members of'a non-homogeneous equation 

 were equal, this would be accidental, and any alteration in 

 the physical units would produce inequality and disclose 

 the falsity of the law expressed in the equation. 



Secondly, the theory of units enables us readily and 

 infallibly to deduce the change in the numerical expression 

 of any physical quantity, produced by a change in the 

 fundamental units. It is of course obvious that in order 

 to represent the same absolute quantity, a number must 

 vary inversely as the magnitude of the units which are 

 numbered. The yard expressed in feet is 3 ; taking the 

 inch as the unit instead of the foot it becomes 36. Every 

 quantity into which the dimension length enters positively 



