THE NORMAL CURVE 95 
being included which lie between 550 : 449 and 449 : 550. 
The points thus obtained are so close together that the 
eye can scarcely distinguish whether they are joined by 
straight or curved lines. We have, in fact, arrived at 
a close approximation to the normal curve. 
The curve thus approximately indicated may be 
seen to be closely similar to the one shown in Fig. 5 ; 
in fact, the two curves are of such a kind that by 
altering the vertical and horizontal scales in one of 
the figures in a suitable ratio their form could be made 
practically identical. 
The figure arrived at in this way approximates to a 
mathematical curve which is intelligible to the mathe- 
matician from the formula y=e-**. The theoretical 
curve is really arrived at by supposing m in the ex- 
pression (I +1)* to become indefinitely great. Prac- 
tically, by making » very large we can get as near an 
approximation as we may wish to the normal curve of 
theory. Even in the case of relatively small values of 
» the approximation to the normal curve is fairly close, 
as may be seen by comparing together Figs. 6 and 7. 
The example of tossing up coins was only taken 
as a means of illustrating the more general assumption 
of an event or a magnitude depending upon a number 
of causes of equal strength, which in the long-run act 
with equal frequency in two opposite directions. We 
can understand that human stature may afford a 
comparable case, when we consider the large number 
of bones and cartilages the lengths of which must be 
added together in order to make up the total stature 
of any individual, and that the separate length of each 
