60 DARWINISM TO-DAY. 



random and examined for variation in any character, say total 

 length of body, not only would there be found a larger number of 

 individuals of medium length than of any other length between the 

 two extremes, represented by the longest and shortest individuals, 

 but that the various lengths between the mean and the longest and 

 between the mean and the shortest, would be represented by groups 

 of individuals regularly decreasing in number as the length in- 

 creased or decreased on either side of the mean, but of equal number 

 if compared at equal amounts of difference away from the mean. 



The curve expressing graphically the law of probabilities or, 

 better, the frequency of error, is determined by the formula for this 

 frequency deduced originally by Gauss at the beginning of the last 

 century. It would lead us too far afield to reproduce here the 

 mathematical proof of the formula or method of its determination, 

 but Vernon's excellent concrete illustration of how such a formula 

 could be deduced directly from a study of biologic variation 

 may be quoted. "Supposing," says Vernon in "Variation in Animals 

 and Plants,'' pp. n and 12, 1903, "a group of developing organisms 

 be taken, of which the growth can be affected in a favourable or an 

 unfavourable manner by their surroundings. Let us suppose that 

 there are twenty different agencies, each of which would produce 

 an equal, favourable effect on growth, and twenty which would pro- 

 duce just as great an effect in the opposite direction. Suppose, also,, 

 that each organism is subjected to only half of these forty different 

 agencies ; then it would follow, according to the laws of chance, 

 that a larger number of the organisms would be acted upon by 10- 

 favonrable and 10 unfavourable agencies, than by any other com- 

 bination ; i.e., they would, on our hypothesis, remain absolutely 

 unaffected in their growth. A somewhat smaller number would be 

 acted upon by 11 favourable and 9 unfavourable agencies, or on the 

 whole, would have their growth slightly increased. A still smaller 

 proportion would be acted on by 12 favourable and 8 unfavourable 

 agencies, or would have their growth rather more increased. Finally, 

 the number of organisms acted on by 20 favourable and o unfa- 

 vourable agencies would be extraordinarily small, but in this case the 

 effect on growth would be extremely large. Similar relationships, 

 only in the reverse direction, would of course be found in those 

 cases in which the number of unfavourable agencies exceeded the 

 number of favourable. If desired, the proportional numbers of organ- 

 isms acted on by all the different combinations of agencies may be 

 readily determined by expanding the binomial ( l / 2 + /^) 20 - It 

 is found, for instance, that for each single time the organisms are 

 acted on by the whole 20 favourable agencies, they are acted on 190 

 times by 18 favourable and 2 unfavourable; 15,504 times by 15 



