Deobmber 7, 1900.] 



SCIENCE. 



867 



Lacerta muralis. Students of molluscs have 

 also been led in a few instances to quantita- 

 tive studies. One of the most important of 

 these was that of Bateson, who showed by a 

 series of measurements the gradual change 

 in form of the cardiums on the terraces of 

 an inland sea which is gradually becoming 

 denser. 



From 1890 on, the published works in the 

 field we are considering became more 

 numerous and appeared simultaneously in 

 several countries and stimulated from va- 

 rious sources. It was early in this decade, 

 too, that the remarkable series, by Pearson 

 and his pupils, of mathematical contribu- 

 tions to the theory of evolution, began to ap- 

 pear. Since these papers profoundly af- 

 fected biological work, it will be well to 

 consider them first. We may then consider 

 in turn the English biological school — es- 

 pecially the work of Weldon and his pupils 

 — the continental biological work, and 

 finally that of America. 



In 1894 Professor Karl Pearson published 

 the first of his valuable series of papers 

 on the mathematics of evolution, entitled 

 'On the Dissection of Frequency Curves.' 

 While the primary result of these investi- 

 gations has as yet proved of no great ser- 

 vice, the methods elaborated have formed 

 the basis of all Pearson's later analysis. 

 These methods consisted of the analytical 

 investigation of the frequency curve by re- 

 duction to the first to fifth movements about 

 an assumed vertical axis. The root of the 

 mean square error was employed as the best 

 measure of variability and was called the 

 standard deviation. 



Pearson's second paper : ' Skew Varia- 

 tion in Homogeneous Material,' 1895, is the 

 basis of the newer quantitative methods. 

 Starting with the recognition of the fact that 

 the vast majority of biological frequency 

 curves are not symmetrical but ' skew,' 

 Pearson undertakes an analysis of distribu- 

 tion curves in general, gets a general for- 



mula and concludes that there are five types 

 of curves altogether. These are the nor- 

 mal curve — -which is symmetrical and has 

 an infinite range ; the symmetrical curve 

 with limited range ; the skew curve with 

 range unlimited in both directions, with 

 range limited in both directions, and with 

 range unlimited in one direction and limited 

 in the other. The analysis of skew frequency 

 curves was not entirely new, for it had pre- 

 viously been made by an American, E. L. 

 De Forest, of Connecticut. But Pearson's 

 work, having especially a biological aim, 

 has become generally adopted by biolo- 

 gists. In 1897 there was published, posthu- 

 mously, an analysis of frequency curves by 

 Fechner ; but Fechner's treatment is much 

 clumsier than Pearson's. Within recent 

 years a fresh analysis of skew curves has 

 been made by the English mathematician, 

 Edgeworth in the Philosophical Magazine. 

 Sheppard, too, has contributed methods of 

 analysis of frequency curves. All these 

 will doubtless eventually be of service to 

 biologists. 



Pearson's third paper, 1896, was devoted 

 to the theory of correlation, which he placed 

 on an analytical basis. His method of 

 getting the index of correlation was a long 

 one, which Duncker has greatly simplified 

 in his paper published a year ago. 



Pearson's fourth paper, published jointly 

 with L. N. Q. Filon in 1898, dealt with the 

 probable errors of frequency constants and 

 is of great importance for measuring pre- 

 cisely the degree of unreliability of any re- 

 sult. Pearson's later papers, many of which 

 are published under the uniform title ' Data 

 for the Problem of Evolution in Man,' have 

 elaborated and extended the results of his 

 earlier works. 



The results of the last six years, then, have 

 placed the analysis of biological frequency 

 curves at once upon a satisfactory scientific 

 basis. But the limit of improvement in the 

 method of analysis is by no means reached. 



