TRANSACTIONS OP THE SECTIONS. d 



twofold caution so often given by Lord Bacon against over-generalization on the 

 one hand, and against over-specialization on the other, is still as deserving as ever 

 of the attention of mankind. But in our time, v^^hen vague generalities and empty 

 metaphysics have been beaten once, and we may hope for ever, out of the domain of 

 exact science, there can be but little doubt on which side the danger of the natural 

 philosopher at present lies. And perhaps in our Section, as at present constituted, 

 there is a freer and fresher air ; we are, perhaps, a less inadequate representation 

 of '• that greater and common world " of which Lord Bacon speaks, than if we 

 were subdivided into as many parts as we include, I will not say sciences, but 

 groups of sciences. Perhaps there is something in the very diversity and multi- 

 plicity of the subjects which come before us which may serve to remind us of the 

 complexity of the problems of science, of the diversity and multiphcity of nature. 



On the other hand, it is not, as it seems to me, difficult to assign the nature of 

 the unity which underlies the diversity of om- subjects, and which justities to a 

 very great extent the juxtaposition of them in our Section. That unity consists 

 not so much in the nature of the subjects themselves as in the nature of the 

 methods by which they are treated. A mathematician at least (and it is as a 

 mathematician I have the privilege of addressing you) may be excused for con- 

 tending that the bond of union among the physical sciences is the mathematical 

 spirit and the mathematical method which pervades them. As has been said with 

 profound truth by one of my predecessors in this chair, our knowledge of nature, 

 as it advances, continuously resolves difi'erences of quality into diflerences of 

 quantity. All exact reasoning (indeed all reasoning) about quantity is mathe- 

 matical reasoning ; and thus, as our knowledge increases, that portion of it which 

 becomes mathematical increases at a still more rapid rate. Of all the great subjects 

 which belong to the province of this Section, take that which at first sight is the 

 least within the domain of mathematics ; I mean meteorology Yet the part which 

 mathematics bears in meteorology increases every year, and seems destined to 

 increase. Not only is the theory of the simplest instruments of meteorology essen- 

 tially mathematical, but the discussion of the observations — upon which, be it 

 remembered, depends the hopes which are already entertained with increasing 

 confidence of reducing the most variable and complex of all known phenomena to 

 exact laws — is a problem which not only belongs wholly to mathematics, but 

 which taxes to the utmost the resources of the mathematics which we now possess. 

 So intimate is the union between mathematics and physics that probably by far 

 the larger part of the accessions to our mathematical knowledge have been 

 obtained by the efforts of mathematicians to solve the problems set to them by 

 experiment, and to create " for each successive class of phenomena a new calculus 

 or a new geometry, as the case might be, which might prove not wholly inadequate 

 to the subtlety of nature." Sometimes, indeed, the mathematician has been 

 before the physicist ; and it has happened that when some great and new qu.estion 

 has occurred to the experimentalist or the observer, he has found in the armoury 

 of the mathematician the weapons which he has needed ready made to his hand. 

 But much oftener the questions proposed by the physicist have transcended the 

 utmost powers of the mathematics of the time, and a fiesh mathematical creation 

 has been needed to supply the logical instrument reqidsite to interpret the new 

 enigma. Perhaps I may be allowed to mention an example of each of these two 

 ways in which mathematical and physical discovery have acted and reacted on 

 each othei". I purposely choose examples which are well known, and belong, the 

 one to the oldest, the other to the latest times of scientific history. 



The early Greek geometers, considerably before the time of Euclid, applied 

 themselves to the study of the various curve lines in which a conical figure may 

 be cut by a plane — curve lines to which they gave the name, never since forgotten, 

 of conic sections. It is difiicult to imagine that any problem ever had more 

 completely the character of a "problem of mere curiosity " than this problem of 

 the conic sections must have had in those earlier times. Not a single natural 

 phenomenon which in the state of science at that time could have been intelligently 

 observed was likely to require for its explanation a knowledge of the nature of 

 these curves. Still less can anj' application to the arts have seemed possible ; a 

 nation which did not even use the arch were not likelv to use the ellipse in anv 



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