SECT. 3.] Averages. 439 



analogous question. And the answer, as just remarked, must 

 be in the negative. An annual increase of 1 p. c. continued 

 for 100 years will more than double the total; it will multiply 

 it by about 27. The true annual increment required is mea 

 sured by 1( y2 ; that is, the population may be said to have 

 increased on the average 07 p. c. annually. 



We are thus directed to the second kind of average dis 

 cussed in the ordinary text-books of algebra, viz. the geome 

 trical. When only two quantities are concerned, with a single 

 intermediate value between them, the geometrical mean con 

 stituting this last is best described as the mean proportional 

 between the two former. Thus, since 3 : J\5 :: ^15 : 5, 

 Jl5 is the geometrical mean between 3 and 5. When a 

 number of geometrical means have to be interposed between 

 two quantities, they are to be so chosen that every term in 

 the entire succession shall bear the same constant ratio to 

 its predecessor. Thus, in the example in the last paragraph, 

 99 intermediate steps were to be interposed between 1 and 2, 

 with the condition that the 100 ratios thus produced were to 

 be all equal. 



It would seem therefore that wherever accurate quantita 

 tive results are concerned, the selection of the appropriate 

 kind of average must depend upon the answer to the ques 

 tion, What particular intermediate value may be safely 

 substituted for the actual variety of values, so far as the 

 precise object in view is concerned ? This is an aspect of 

 the subject which will have to be more fully considered in 

 the next chapter. But it may safely be laid down that for 

 purposes of general comparison, where accurate numerical 

 relations are not required, almost any kind of intermediate 

 value will answer our purpose, provided we adhere to the 

 same throughout. Thus, if we want to compare the statures 

 of the inhabitants of different counties or districts in Eng- 



