[From the Proceedings of ike London Mathematical Society, Vol. in. No. 34.] 



XL VI. On the Mathematical Classification of Physical Quantities. 



THE first part of the growth of a physical science consists in the discovery 

 of a system of quantities on which its phenomena may be conceived to depend. 

 The next stage is the discovery of the mathematical form of the relations 

 between these quantities. After this, the science may be treated as a mathe- 

 matical science, and the verification of the laws is effected by a theoretical 

 investigation of the conditions under which certain quantities can be most 

 accurately measured, followed by an experimental realisation of these conditions, 

 and actual measurement of the quantities. 



It is only through the progress of science in recent times that we have 

 become acquainted with so large a number of physical quantities that a classifi- 

 cation of them is desirable. 



One very obvious classification of quantities is founded on that of the 

 sciences in which they occur. Thus temperature, pressure, density, specific heat, 

 latent heat, &c., are quantities occurring in the theory of the action of heat on 

 bodies. 



But the classification which I now refer to is founded on the mathematical 

 or formal analogy of the different quantities, and not on the matter to which 

 they belong. Thus a finite straight line, a force, a velocity of rotation, &c., 

 are quantities, differing in their physical nature, but agreeing in their mathe- 

 matical form. We may distinguish the two methods of classification by calling 

 the first a physical, and the second a mathematical classification of quantities. 



A knowledge of the mathematical classification of quantities is of great use 

 both to the original investigator and to the ordinary student of the science. 

 The most obvious case is that in which we learn that a certain system of 

 quantities in a new science stand to one another in the same mathematical 



VOL. ii. 33 



