OF PHYSICAL QUANTITIES. 259 



I need not say that this is not true, and that mathematicians, in solving 

 physical problems, are very much aided by a knowledge of the science in which 

 the problems occur. 



At the same time, I think that the progress of science, both in the way of 

 discovery, and in the way of diffusion, would be greatly aided if more attention 

 were paid in a direct way to the classification of quantities. 



A most important distinction was drawn by Hamilton when he divided the 

 quantities with which he had to do into Scalar quantities, which are completely 

 represented by one numerical quantity, and Vectors, which require three numerical 

 quantities to define them. 



The invention of the calculus of Quaternions is a step towards the know- 

 ledge of quantities related to space which can only be compared for its impor- 

 tance, with the invention of triple co-ordinates by Descartes. The ideas of this 

 calculus, as distinguished from its operations and symbols, are fitted to be of 

 the greatest use in all parts of science. 



We may imagine another step in the advancement of science to be the 

 invention of a method, equally appropriate, of conceiving dynamical quantities. 

 As our conceptions of physical science are rendered more vivid by substituting 

 for the mere numerical ideas of Cartesian mathematics the geometrical ideas of 

 Hamiltonian mathematics, so in the higher sciences the ideas might receive a 

 still higher development if they could be expressed in language as appropriate 

 to dynamics as Hamilton's is to geometry. 



Another advantage of such a classification is, that it guides us in the use 

 of the four rules of arithmetic. We know that we must not apply the rules 

 of addition or subtraction unless the quantities are of the same kind. In certain 

 cases we may multiply or divide one quantity by another, but in other cases 

 the result of the process is of no intellectual value. 



It has been pointed out by Professor Rankine that the physical quantity 

 called Energy or Work can be conceived as the product of two factors in many 

 different ways. 



The dimensions of this quantity are - , where L, M, and T represent 



the concrete units of length, time, and mass. If we divide the energy into two 

 factors, one of which contains L", both factors will be scalars. If, on the other 

 hand, both factors contain L, they will be both vectors. The energy itself is 

 always a scalar quantity. 



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