206 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1952 



though the language appropriate to their more complete setting in the 

 fundamental frame might be unintelligible to them. We see this 

 kind of thing happening in almost every branch of physics that finds a 

 utilitarian service. 



Of course, the reason for this state of affairs lies in the fact that the 

 fundamental laws frequently express their consequences in forms 

 which are not conveniently adaptable to problems. However, it is 

 possible to dress them up in different kinds of clothes suitable to the 

 various realms in which they have to function, and sometimes the 

 clothes are so different that it is hard to recognize in them the same 

 individual. However, from the practical standpoint, this existence 

 of different forms of raiment is not to be decried, however much it 

 may irritate the master of fundamentals to see his beloved concepts 

 so variously represented. Occasionally the whole success of utiliza- 

 tion of the concepts depends upon the raiment. An example is to be 

 found in a problem which, for a considerable time, taxed the genius of 

 the great Newton. This problem concerned the proof of the fact 

 that a spherically symmetrical distribution of matter obeying the law 

 of inverse squares acts, at external points, as though all the matter 

 were concentrated at the center. By the utilization of Gauss's theorem, 

 it is possible to prove this result, which baffled the great Newton for so 

 long, and to write the proof on the area of a postage stamp ; yet Gauss's 

 theorem is no more than the law of inverse squares garbed in such a 

 form as to make it particularly at home in the realm of spherical 

 symmetry. 



And so, in meditating upon our ancient and medieval architects, 

 I have to believe that, though unconscious of the general fundamental 

 laws, they nevertheless sensed many things which were true and were 

 able to mold them into a frame of procedure which was sufficiently 

 concrete and self-consistent to serve as a guide in their operations. 

 We should feel very insecure in this frame, fearing that something 

 had been forgotten, unless we could see the elements of that frame as 

 consistent parts of the more complete whole. However, there is an- 

 other respect in which the rule-of-thumb procedure has something to 

 be said for it. Many problems which are controlled by fundamentals 

 perfectly well known to us are incapable of solution by known mathe- 

 matical procedures and we are driven to illustrate the matter by appeal 

 to simplified cases. If I tap a table with my finger, I say that I 

 understand in a general way how the vibrations which ensue come 

 about. I think of those interactions of elasticity and inertia which 

 are fundamental to the solution of the problem. I can even write 

 down the differential equations which control the solutions, and yet 

 no mathematician on earth is clever enough to work out the solution 

 for this particular case. 



