December 7, 1900.] 



SCIENCE. 



865 



Gotha 8ur la t.h6orie des probabilit6s appli- 

 ques aux sciences morales et politiques,' 

 published in Brussels, 1846, and translated 

 into English by O. G. Downes, 1849, Que- 

 telet applied the mean and probable error 

 to the designation of the peculiarities of 

 dififerent races of men ; and he even com- 

 pared an observed distribution of frequen- 

 cies of human statures with a theoretical 

 one calculated from the mean and the prob- 

 able error by the use of the formula, 



2 f _, 



y = —7^ I e ' dt. 



The possibilities of an extension of this 

 method of quantitative analysis to bio- 

 logical variation in general were, however, 

 during twenty years unrecognized, for the 

 time was not yet ripe. 



Meanwhile, on the biological side the 

 clouds of a revolution were rising. During 

 the fifties the variation of animals in nature 

 was much discussed. In 1856 Wollaston 

 wrote a book ' On the Variation of Species,' 

 and at a meeting of the British Association, 

 in the same year, Jenyns called for exact 

 data on the variation of organisms. Then 

 came the ' Origin of Species,' 1858, 1859, 

 which used the facts of variation to enforce 

 the conclusion of the mutability of species, 

 At that time the study of the variation 

 of species received a great impetus, but for 

 forty years the observations have been for 

 the most part qualitative or only roughly 

 quantitative. Among the most important 

 synoptic works on variation have been Dar- 

 win's 'Variation of Animals and Plants 

 under Domestication,' 1868, and Bateson's 

 ' Materials for the Study of Variation,' 

 1894. 



To the rule that until a decade ago 

 studies in variation were qualitative, the 

 anthropological studies have formed a strik- 

 ing exception. Anthropologists have been 

 forced to measure by the necessity of mak- 

 ing fine discriminations, and have sought 



by statistical methods to get the most out 

 of the resulting data. The first to advance 

 beyond Quetelet was Francis Gal ton, cousin 

 of Darwin, and already well known on ac- 

 count of his studies on heredity. In 1870, 

 in his book on ' Hereditary Genius,' he 

 used Quetelet's method of applying the law 

 of error to organisms. He used it espe- 

 cially to get a quantitative definition of his 

 grades of ability, A to G. In 1879, Galton 

 made a further and important step. He 

 pointed out that " an assumption which 

 lies at the basis of the well-known law of 

 ' Frequency of Error ' (commonly expressed 

 by the formula, y = k . e""" ) is incorrecb in 

 many groups of vital and social phe- 

 nomena." The assumption which Galton 

 combats is ' •' that errors in excess or in de- 

 ficiency of the truth are equally probable or 

 conversely, that if two fallible measure- 

 ments have been made of the same object 

 their mathematical mean is more likely to 

 be the true measurement than any other 

 quantity that can be named." Galton 

 goes on to show that this assumption 

 cannot be justified in vital phenomena ; 

 for example, in guesses at a color con- 

 taining 8 parts of white, we are equally 

 apt to err by selecting one with 16 parts 

 and one with 4 parts, yet the error in 

 one case is twice the error in the second 

 case. Conversely, in two guesses at a mid 

 tint, the most reasonable conclusion is not 

 the arithmetic mean of the two, but the geo- 

 metric mean. If the guesses are 4 and 16, 



the most probable value is not — - — = 10 



but v/4xl6 = 8; for 4:8::8:16. Gal- 

 ton then extends this case to biological 

 measures in general and calls for a law of 

 error based on the geometric mean. At 

 his suggestion Mr. Donald McAllister 

 worked out a more general form of the 

 probability curve, applicable to a distribu- 

 tion of frequencies based on geometric 

 error, and obtained the result, 



