710 Dr. N. Campbell and Prof. E. C. C. Baly on the 



choice of unit or o£ radix, and employing always the rule 

 which Prof. Baly employs, it is possible to make out any 

 number whatsoever to be the L.C.M. of any other numbers 

 whatsoever. And this indefiniteness, it must be insisted, 

 does not result from Prof. Baly's use of the wrong rule ; so 

 long as there can be a right rule to determine something 

 that does not exist, it is the right rule. The plain fact of 

 the matter is that, if the numbers of which the L.C.M. 

 is to be taken are not known with perfect and complete 

 mathematical accuracy, there is no such thing as the 

 L.C.M. His rule merely adopts one out of an infinite 

 number of alternatives ; and the alternative which it adopts 

 depends wholly on the unit of measurement <*md the scale of 

 notation. 



6. Are we then to conclude that the numerical agreements 

 which he finds are pure coincidences ? The answer to that 

 question I leave to those familiar with spectroscopic work, 

 for they alone can determine whether the accuracy of the 

 measurements is such as to make the probability of the 

 necessary coincidence sufficiently great for that explanation 

 to be plausible. But it seems that if the answer is negative, 

 there is only one conclusion to be drawn. 



If the L.C.M. is to be significant the values must be 

 known with complete accurac} r . There is only one kind 

 of magnitude which has no experimental error and can be 

 known with complete accuracy ; that magnitude is the 

 number of something, which is necessarily an integer*. If 

 we were measuring electric charges and found values 

 9-6 xlO- 10 , 14-3 xlO" 10 , 286 xlO- 10 e.s.c.g.s. units, we 

 should doubtless conclude that we were measuring 2, 3, 6 

 electronic charges, and might justifiably attribute significance 

 to the fact that the third number is the L.C.M. of the first 

 two. And the conclusion would not be invalid because 

 there was experimental error ; the error arises in deter- 

 mining what is the charge of which the measured charges 

 are integral multiples ; we know without any error at all 

 that the charges are integral, and not fractional, multiples 

 of that charge, and since our measurements are accurate 

 •enough to distinguish between successive integers, we know 

 exactly what multiples they are of the unknown charge. 

 We know that the real charges are px, qx, rx where p, q, r 

 are integers; and we can draw the conclusion that qx is the 

 L.C.M. of px and rx without knowing x. The conclusion, 

 moreover, would be independent of any change of unit 

 (which would merely change x) or of change of radix. 

 * See ' Physics : The Elements,' Chap. xvi. 



