INSTRUMENTS AND METHODS OF RESEARCH 195 



pensable tool of research, but simply wish to call attention to its 

 limitations and to the importance of not overlooking the fertile by- 

 products, the residuals which, because of our neglect of them, may 

 some day rise and smite us in their wrath. Each one of us at one 

 time or another has doubtless established, by least squares, an empirical 

 formula of some kind which so beautifully fits the observations as to 

 make us bold and venturesome. Now comes a new observation, some- 

 what outside of the range for which the expression was established. 

 Eagerly the test is applied, and we find, to our chagrin, that the for- 

 mula on which so much work had been spent will not fit the new 

 result, and that we have a " counterfeit " and not the real law. 



A graphical process, like the crucial and decisive experiment, may 

 at times reveal an essential fact that the mind of even the greatest 

 of mathematicians has failed to extract from his formulas. 



Let us suppose, for illustration, we are dealing with a phenomenon 

 which almost entirely unfolds itself during the time between sunrise 

 and sunset — the well-known diurnal variation of the earth's magnetism 

 is a striking case of the kind. Following the usual method, the phe- 

 nomenon is resolved into component parts with the aid of a Fourier 

 series. The formula as generally adopted includes the four terms 

 having, respectively, periodicities of 24, 12, 8 and 6 hours. For ordi- 

 nary magnetic latitudes the striking result is obtained that the second 

 term — the 12-hour one — is as important as the first, or 24-hour, one; 

 so we might equally as well say " the semidiurnal " as " the diurnal 

 variation of the earth's magnetism." In fact, as the semidiurnal term 

 unfolds itself twice in 24 hours, it is in reality more important than 

 the purely diurnal one. 



Does the resolution into Fourier terms of a phenomenon of the 

 kind given really prove their existence in nature? Can we conclude 

 without question that in addition to the diurnal term we also have 

 a semidiurnal one? Even with four terms, the series does not repre- 

 sent each hourly observation of the 24 with the same degree of 

 precision. In fact, the residuals for the night hours are nearly of the 

 same order of magnitude as the observed quantities. If the physic- 

 al existence of the 12-hour term is not proved, then there is no need 

 of racking our brains as to its physical origin. 



The difficulty disclosed by this example is of the same kind as 

 the one treated in spherical harmonics, viz., that we are attempting 

 to represent a more or less non-continuous function having a duration 

 commensurate with that of the daylight hours by functions running 

 smoothly through their individual courses for 24 hours. 



Babbage, the inventor of the calculating machine named after him, 

 said he once had the following question put to him : " Pray, Mr. 

 Babbage, if you put into the machine wrong figures, will the right 



