TRANSACTIONS OF THE SECTIONS. 3 



the work of the Section we shall have abimdant examples of the successful appli- 

 cation of this method to the most recent conquests of science ; but I wish at present 

 to direct your attention to some of the reciprocal effects of the progress of science 

 on those elementary conceptions which are sometimes thoxight to bo beyond the 

 reach of change. 



If the skill of the mathematician has enabled the experimentalist to see that the 

 quantities which he has measured are connected by necessary relations, the disco- 

 veries of physics have revealed to the mathematician new forms of quantities 

 which he could never have imagined for himself. 



Of the methods by which the mathematician may make his labours most useful 

 to the student of nature, that which I think is at present most important is the 

 systematic classification of quantities. 



The quantities which we study in mathematics and physics may be classified in 

 two different ways. 



The student who wishes to master any particular science must make himself 

 familiar with the various kinds of quantities which belong to that science. When 

 he understands all the relations between these quantities, he regards them as form- 

 ing a connected system, and he classes the whole system of quantities together as 

 belonging to that particular science. This classification is the most natui-al from 

 a physical point of^view, .and it is generally the first in order of time. 



But when the student has become acquainted with several diflerent sciences, he 

 finds that the mathematical processes and trains of reasoning in one science resemble 

 those in another so much that his knowledge of the one science may be made a 

 most useful help in the study of the other. 



When he examines into the reason of this, he finds that in the two sciences he 

 has been dealing with systems of quantities, in which the mathematical forms of 

 the relations of the quantities are the same in both systems, though the physical 

 nature of the quantities may be utterly different. 



He is thus led to recognize a classification of quantities on a new principle, 

 according to which the physical nature of the quantity is subordinated to its 

 mathematical fonn. This is the point of view which is characteristic of the 

 mathematician; but it stands second to the physical aspect in order of time, 

 because the human mind, in order to conceive of different kinds of quantities, must 

 have them presented to it by nature. 



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 submitted 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 processes 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 conception 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 cori'esponding 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 



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