SCIENCE PROGRESS 



RECENT ADVANCES IN SCIENCE 



APPLIED MATHEMATICS. By S. Brodetsky, M.A., Ph.D., F.Inst.P.. 

 etc., University, Leeds. 



It is not an easy matter to decide what is, and what is not, to 

 be included under the category of AppUed Mathematics. If 

 an attempt is made to interpret the term generously there is 

 serious danger of overlapping with other branches of study 

 like astronomy, physics, and engineering. On the other hand, 

 a limited view of the subject is liable to lose sight of researches 

 that affect the very fundamentals on which mechanics is based. 

 This applies with particular force to the theory of relativity. 

 Based on optical experiments, using the methods of pure 

 mathematics, and seeking verification in the results of astro- 

 nomical calculation and observation, the theory of relativity 

 is at bottom an attempt to put the theoretical side of mechanics 

 on consistent and logical foundations, and the applied mathe- 

 matician is at least as concerned in the fortunes of the theory as 

 the physicist or astronomer. No excuse need, therefore, be 

 offered for beginning the present article with an account of the 

 recent work on relativity. 



Far more has been written about relativity during the past 

 year than can be conveniently discussed in a single article. 

 Mathematicians and physicists have been busy examining the 

 theory in all its bearings. Perhaps the most useful papers to 

 examine first are those that attempt a critique, favourable or 

 unfavourable. Of such papers the most important are those 

 of the French mathematicians. P. Painleve devotes consider- 

 able space to a comparison of Einstein's treatment of gravitation 

 with that of Newton, Comptes Rendus, 173, 1921, 873-87. After 

 a brief statement of the fundamental postulates of the classical 

 mechanics, Painleve considers the Principle of Invariance, 

 which plays so leading a part in Einstein's work. He declares 

 his assent to this principle, but in a different form : " It is 

 possible from the laws of nature to derive consequences which 

 are invariant for all changes of the space-time frame of reference, 

 and which define these laws for any such change." Painleve 

 then proceeds to examine the way in which Einstein uses this 

 principle, emphasising the fact that Einstein always tries to 

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