Physical Significance of the Least Common Multiple. 709 



our fractions and therefore fix also the numerator and the 

 L.CM. Then (1) leads immediately to the following rule, 

 which is that actually employed by Prof. Baly : — Express 

 the values with the same number of decimal places ; remove 

 the decimal point ; take the product of the resulting- integers 

 and replace the decimal place in the product. Thus the 

 L.C.M. of 7*7 and 5*6 is 431'2. It is to be observed that 

 we must take the product and not the L.C.M. of the 

 integers. For the number of decimal places expressed are 

 those which represent amounts greater than the experi- 

 mental error ; we do not know what the remaining places 

 are. If we took the L.C.M. of 77 and 56 (viz. 616) in 

 place of the product 4312, we should be assuming that the 

 unexpressed places were all zeros, and for such an assump- 

 tion there is not, of course, the slightest justification. 



This rule, which is imposed on us by decimal notation, 

 implies the choice of one out of the infinite number of 

 alternatives for the ratio p/q which is to represent the value. 

 Moreover, the choice is determined whollv by the radix of 

 the notation and the physical unit. But the L.C.M. is 

 determined by the choice • and the L.C.M. arrived at is thus a 

 quite arbitrary selection from an infinite number of pos- 

 sible alternatives. 



5. In place of a general discussion it will perhaps be 

 better to take a single example. Suppose that the real value 

 of the two mao-nitudes is 7'69438 . . . and 5*62936 . . . 



o 



inches, and that the possible experimental error is not less 

 than 1 per cent. Then we shall choose the numbers already 

 given, 7-7 and 5*6, and find 431-2 as the L.C.M. But if we 

 had been measuring in centimetres and had never heard of 

 inches (obviously a permissible supposition) the real values 

 would have been 19*5438 ... and 14*2986 . . . ; remem- 

 bering our 1 per cent, error, we shall choose 19*5 and 14*3, 

 of which the L.C.M. is 2788*5 cm. or 1976*6 inches— a 

 result quite different from the value 431*2 at which we arrived 

 before. (It is, of course, 10/2'54 times as great.) Accor- 

 dingly a change of unit leads, by the rule to which we are 

 forced, to the identification of a perfectly different length 

 to represent the L.C.M. of the original lengths. 



Now let us change our radix to 3, and express number in 

 the scale of 3 in italics. Then 7*69438 . . . =2120020202 . . . 

 and 5*62936 . . . = 12'12122221 .... With an accuracy 

 of 1 in 243 we shall choose 21-200 and 12*121. Oi' these 

 the L.C.M. is 1120212*200 = 1157'1 — again quite a different 

 result from our original 431*2. 



Indeed it is not difficult to see that by an appropriate 



