188 THE PRINCIPLES OF SCIENCE. [chap. 



is 2^—1—8 or 247. If an organ has eleven stops we 

 find in the twelfth line the numbers of ways in which we 

 can draw them, i, 2, 3, or more at a time. Thus there are 

 462 ways of drawing five stops at once, and as many of 

 drawing six stops. The total number of ways of varying 

 the sound is 2048, including the single case in which no 

 stop at all is drawn. 



One of the most important scientific uses of the arith- 

 metical triangle consists in the information which it gives 

 concerning the comparative frequency of divergencies 

 from an average. Suppose, for the sake of argument, that 

 all persons were naturally of the equal stature of five feet, 

 but enjoyed during youth seven independent chances of 

 growing one inch in addition. Of these seven chances, 

 one, two, three, or more, may happen favourably to any 

 individual ; but, as it does not matter what the chances 

 are, so that the inch is gained, the question really turns 

 upon the number of combinations of o, i, 2, 3, &c., things 

 out of seven. Hence the eighth line of the triangle gives 

 us a complete answer to the question, as follows : — 



Out of every 128 people — 



One person would have the stature of 



7 persons „ „ 



21 persons „ „ 



35 persons „ „ 



. 35 persons „ „ 



21 persons „ „ 



7 persons „ „ 



I person „ „ 



By taking a proper line of the triangle, an answer may 



be had under any more natural supposition. This theory 



of comparative frequency of divergence from an average, 



was first adequately noticed by Quetelet, and has lately 



been employed in a very interesting and bold manner 



by Mr. Francis Galton,' in his remarkable work on 



" Hereditary Genius." We shall afterwards find that the 



theory of error, to which is made the ultimate appeal in 



cases of quantitative investigation, is founded upon the 



^ See also Galton's Lecture at the Royal Institution, 27th February, 

 1874 ; Catalogue of the Special Loan Collection of Scientitic Instru- 

 ments, South Kensington, Nos. 48, 49 ; and Galton, Philosophical 

 Magazine, January 1875. 



