176 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1912. 



one'should make use of the same for stating the properties of this 

 body in the most simple manner possible. So it is that the possibility 

 of seeing an isolated microbe under the microscope is not an indispen- 

 sable condition for the attenuation of the viruses and of the use of 

 vaccines. In the same way, in the reproduction of a masterpiece by 

 pliotogravure, it is not the individual knowledge of the points consti- 

 tuting the negative that interests us.^ 



From the abstract point of view, if we admit that all human theory 

 ought to be, in the last analysis, expressed by means of a finite and 

 relatively small number of data, it seems difficult to deny the possi- 

 bility of entu'ely constructing the theory without causing the inter- 

 vention of hypotheses which imply the existence of elements whose 

 number surpasses that which the investigation of man can conceive. 

 But the verification of this abstract possibility can not prevail against 

 the importance of services rendered by molecular theories in the 

 unification of apparently unrelated phenomena; and so it is per- 

 mitted to consider these reserves on the possibilities of the future, as 

 a simple matter of style. 



Is it possible to go still further, and suppress all reserve of this 

 kind? In order to answer this question it is necessary to examine 

 in detail all the phenomena that one explains by means of the molecu- 

 lar hypotheses and seek to ascertain whether an extremely large 

 number of parameters is really necessary to this explanation. Among 

 the discontinuous phenomena whose experimental laws are well 

 known, the most characteristic are those of spectra in series. One 

 knows that the positions of the spectral rays are determined with a 

 very great precision by formulae, of which the first and most simple, 

 due to Balmer, includes the difference of the reciprocals of the squares 

 of two integers. There, perhaps, is the most remarkable example of 

 the intervention of the integer in a natural law. If the laws of this 

 kind were more numerous and better known, we would perhaps be 

 led to cite arithmetic and the theory of numbers among the branches 

 of mathematics which one can connect with molecular })hysics. Can 

 one, by induction, admit that the formula of Balmer is exact, not 

 only for small integers concerning which the experimental verifica- 

 tion is rigorous, but for many other larger intergers concerning which 

 this verification is impossible ? And if it is thus, is not there one of 

 the discontinuous phenomena whose explanation requires a very 

 large number of parameters ? It does not seem so. On the one hand, 

 the formula with the variable integer, can contain with precision 



> This individual knowledge of points intervenes in the process of transmission of the negative to a 

 distance; but here these points, although numerous, are, however, finite, and accessible to our observation. 

 If we transmit by telephone an orchestral selection, we know that all the aesthetic beauties of the piece 

 result, in the last analysis, from certain vibrations which would require too much time to know individually; 

 but, in fact, these elementary vibrations present nothing visible in musical aesthetics; an excellent composer 

 of music can ignore their existence and an excellent physicist can be a wretched musician. 



