OF THE BRITISH ASSOCIATION. 219 



I do not here refer to the fact that all quantities, as such, are subject 

 to the rules of arithmetic and algebra, and are therefore capable of being sub- 

 mitted to those dry calculations which represent, to so many minds, their only 

 idea of mathematics. 



The human mind is seldom satisfied, and is certainly never exercising its 

 highest functions, when it is doing the work of a calculating machine. What 

 the man of science, whether he is a mathematician or a physical inquirer, 

 aims at is, to acquire and develope clear ideas of the things he deals with. 

 For this purpose he is willing to enter on long calculations, and, to be for a 

 season a calculating machine, if he can only at last make his ideas clearer. 



But if he finds that clear ideas are not to be obtained by means of pro- 

 cesses the steps of which he is sure to forget before he has reached the 

 conclusion, it is much better that he should turn to another method, and try 

 to understand the subject by means of well-chosen illustrations derived from 

 subjects with which he is more familiar. 



We all know how much more popular the illustrative method of exposition 

 is found, than that in which bare processes of reasoning and calculation form 

 the principal subject of discourse. 



Now a truly scientific illustration is a method to enable the mind to grasp 

 some conception or law in one branch of science, by placing before it a con- 

 ception or a law in a different branch of science, and directing the mind to 

 lay hold of that mathematical form which is common to the corresponding ideas 

 in the two sciences, leaving out of account for the present the difference 

 between the physical nature of the real phenomena. 



The correctness of such an illustration depends on whether the two systems 

 of ideas which are compared together are really analogous in form, or whether, 

 in other words, the corresponding physical quantities really belong to the same 

 mathematical class. When this condition is fulfilled, the illustration is not only 

 convenient for teaching science in a pleasant and easy manner, but the recog- 

 nition of the formal analogy between the two systems of ideas leads to a 

 knowledge of both, more profound than could be obtained by studying each 

 system separately. 



There are men who, when any relation or law, however complex, is put 

 before them in a symbolical form, can grasp its full meaning as a relation 

 among abstract quantities. Such men sometimes treat with indifference the 

 further statement that quantities actually exist in nature which fulfil this 



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