February i8, 1909J 



NA TURE 



475 



computation prohibitive. We are dealing here with more 

 or less discontinuous effects that cannot be imitated by 

 continuous functions without leaving behind a train of 

 residuals, precisely as though we were to try to fit to the 

 actual configuration of the earth some standard pattern 

 of our own. Let me ask what phenomenon have we, in 

 fact, which will admit of the determination of forty-eight, 

 or even of twenty-four, physical constants? 



It had been my intention to say a few words on the 

 value and limitation of that much-used as well as abused 

 mathematical instrument of research, the method of least 

 squares. Properly employed, it is a most useful adjunct 

 to investigation ; but, as intimated, the true significance 

 of formulae established by this method is at times pushed 

 way beyond the limitations. What the tenor of my re- 

 marks might be will be sufliciently evident to you if I 

 submit this query for your consideration. What actual laws 

 of nature have been discovered by the method of least 

 squares ? 



It is an extremely interesting and suggestive fact that 

 the greatest experimental discoveries to-day arc not made 

 in the older, well-recognised sciences, but on their border- 

 lands — in the " twilight zone " of more or less related 

 sciences. I have but to mention the words " physical 

 chemistry," "physical geology," "astrophysics," "bio- 

 chemistry," &c., and you will readily grant the assertion 

 made. In the overlapping regions there seem to be the 

 greatest opportunities afforded for solid, thorough, and at 

 the same time remarkably rapid, experimental achieve- 

 ments ; and so we are having produced almost daily new 

 specialities or new subspecialities. 



What is the effect on the general broad-mindedness of 

 man of this extreme specialisation, so necessary for the 

 production of the best and most far-reaching results? Is 

 the modern specialist more narrow-minded than the 

 i;eneralist of a centurv or two ago? In view of the fact 

 that the prime instrument of research is, after all, the 

 mind, the question is not an irrelevant one. We find state- 

 ments occasionally made which would imply an aftirmative 

 answer to our question; but I, for one, would most 

 emphatically protest against such an inference. I should 

 maintain that the specialist, other things being equal, is 

 likely to be a broader man than he who has no speciality, 

 but simply a general knowledge of some particular science. 

 The reason for my positive statement would be found in 

 the fact mentioned, that the greatest part of the research 

 work to-day is being done on the border-lands of the 

 general sciences, for he who wishes to take part in this 

 very active competition must needs be far better equipped 

 than the mere generalist. The physical chemist, to be 

 most successful, must have a very intimate knowledge of 

 both physics and chemistry, and the more mathematical 

 skill he possesses the better. The astrophysicist must be 

 a physicist, a chemist, a mathematician, besides being an 

 astronomer. And so with regard to the geophysicist. 



Only a few names need be cited — like those, for example, 

 of Faraday, Maxwell, Kelvin, von Helmholtz, Mascart — 

 to support the contention that the broadest physicists are, 

 as a rule, those who have regarded their laboratory ex- 

 periments and deductions therefrom merely as a means to 

 an end, not an end in themselves, and who have accord- 

 ingly sought to apply the knowledge gained to the solution 

 of some of the great problems affecting the general welfare 

 of man. There is the greatest need in America of well- 

 trained and well-equipped physicists in the solution of the 

 many perplexing problems of the earth's physics with re- 

 gard to the phenomena of seismology, vulcanology, meteor- 

 ology, atmospheric electricity, terrestrial magnetism, &c. 

 When the investigator makes the attempt to apply some 

 of his laboratory facts to geophysical and cosmical pheno- 

 mena, he has opened to himself a world of which he never 

 ■ dreamed ; he finds zest in familiarising himself with the 

 fundamental facts of other sciences in which until now he 

 could take no interest. 



It is always interesting to know what was the precise 

 course followed in the discovery of a great law. How- 

 ever, no two investigators have ever pursued, or at least 

 but rarely, precisely the same paths, and we must there- 

 fore be content with the statement of the general prin- 

 ciples of research such as has already been given. 



A prevalent fault is observed in scientific publications 



\c). 2051, \'oi.. 79] 



whenever the investigator has had good training only on 

 the observational side, and but very little experience in 

 scientific computing, f-le is very apt to violate one of the 

 first and fundamental principles of good observing, viz. 

 to employ such a method or scheme of observing as will 

 yield but one definite result, and that with the highest 

 possible accuracy and with the least amount of computation. 

 Oftener than may be thought, schemes of observation are 

 used which leave an arbitrary element to the computer, 

 and in consequence a different result is forthcoming, accord- 

 ing to who makes the computation. Had we time, apt 

 illustrations could readily be given from published works. 

 The point made, that the observer must also bear in mind 

 the computation side, and work up his results as soon as 

 possible, is of fundamental importance in research work. 



It may be worth while to consider briefly the insatiable 

 desire of the analyst to ring in a series of sines and cosines 

 to resemble the course of some natural phenomenon of 

 which he does not know the exact law. Is this the old 

 story over again, though in somewhat altered garb, of the 

 epicycles and deferents of ancient astronomical mechanics, 

 which received its highest development in the Ptolemaic 

 system of the universe? You will recall that Ptolemy, 

 building on the suggestions of .^pollonius and of Hippar- 

 chus, supposed a planet to describe an epicycle by a 

 uniform revolution in a circle the centre of which was 

 carried uniformly in an eccentric round the earth. By 

 suitable assumptions as to his variable factors he was thus 

 able to represent with considerable accuracy the apparent 

 motions of the planets and to reproduce quite satisfactorily 

 other astronomical facts. This was the artifice employed 

 by the astronomer of the period before the modern and 

 niore subtle art of simulating nature, by the sine-cosine 

 method, had become known. 



What seemed so intricate and complex in Ptolemy's time 

 could be expressed in very simple language indeed, when a 

 Kepler discovered the true functions as embodied in his 

 three fundamental laws. The present method of hiding 

 our ignorance of the real law seems at times to exert such 

 a mesmerising influence as to make us inistake the 

 fictitious for the real. 



Of course I do not mean to discard this useful and, in 

 fact, indispensable tool of research, but simply wish to 

 direct 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 wTath. 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, somewhat outside of the 

 range for which the expression was established. Eagerly 

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

 formula 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. 



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 pheno- 

 menon is resolved into component parts with the aid of a 

 Fourier scries. The formula as generally adopted includes 

 the four terms having, respectively, periodicities of 24, 12, 

 S, and 6 hours. For ordinary 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 semi-diurnal " as " the 

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

 the semi-diurnal term unfolds itself twice in twenty-four 

 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, e.g., that in addition 

 to the diurnal term we also have a semi-diurnal one? 

 Even with four terms the series does not represent each 

 hourlv observation of the twenty-four with the same degree 

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

 are nearly of the same order of inagnitude as the observed 

 quantities. If the physical existence of the 12-hour term 



