166 



SCIENCE. 



[N. S. Vol. XXII. No. 554. 



methods. Thence sprang a wholly new 

 branch of mathematics, enumerative geom- 

 etry. 



A third, an epoch-making universe of 

 discrete mathematics, is the wonderful in- 

 variant theory of the great Sylvester and 

 his brother-in-arms Cayley, two men whose 

 loss left the English-speaking world with- 

 out a single mathematician of first rank, of 

 the rank of Hilbert and Poincare. 



In chemistry this discrete mathematics 

 has shown itself of such use and power that 

 we may assuredly say chemistry owes its 

 present standpoint almost wholly to two 

 lines of advance, both discrete, the atomic 

 structure theory of Kekule and Mendel- 

 ieev's periodic system of the chemical ele- 

 ments. 



The brilliant and rapid advances in 

 chemistry have come not from suppressing' 

 but from stressing the individuality of the 

 elements. Its mathematics has been essen- 

 tially discrete. 



The arithmologic scheme of chemical re- 

 search, the atomic structure theory of Ke- 

 kule, coincides completely with the scheme 

 of the symbolic invariant theory, though 

 both were worked out independently. 



Now to biology and sociology, having to 

 do with single individuals differing from 

 one another, in biology cells, in sociology 

 human personalities, the continuity mathe- 

 matics with its universalism is so ill adapt- 

 ed by its nature that the discrete way of 

 thinking must here soon take the chief role, 

 giving as it does large and free play to the 

 individual peculiarities of the elements to 

 be studied. 



The continuity thought-way strives to re- 

 duce all phenomena of nature to a general 

 mechanism with fate-determined movement. 

 Just contrary to this then is the view that 

 living nature is a rationally-correlated 

 realm, in which everything is harmonic, 

 shows adaptation, strives toward perfec- 

 tion. 



Are not the mechanical form-phenomena 

 of the living organism only its most ele- 

 mentary properties, upon which are built 

 others higher, psychic? Now the psychic 

 properties of a living organism can not be 

 studied by observation and comparison of 

 the accompanying mechanical properties 

 unless they flow from these mechanical 

 properties. If these accompaniments be 

 unessential, the psychic properties can not 

 be concluded from them. Here is even yet 

 the battleground. 



Biologists are at present emphasizing the 

 statistical method, but upon this modern 

 mathematics has for them another message. 

 They rely upon the method of least squares 

 and mean value. But Chebyshev has dem- 

 onstrated that not the great number, but 

 the independence of the metric phenomena 

 plays the chief part in the application of 

 the theory of mean value. This independ- 

 ence is the essential requisite, and it is the 

 very thing whose unw^arranted assumption 

 vitiates much biologic research. 



An illustration may be drawn from fire 

 insurance. From the records of past con- 

 flagrations of single houses, if the burning 

 of each one is independent of that of every 

 other, the theory of mean value can get a 

 number which can be counted upon to recur 

 with slight variation from year to year, 

 and upon it can be based the charges for 

 insurance. 



To realize how completely this essential 

 requirement may be lacking, we have only 

 to remember the Chicago fire, or the Balti- 

 more fire. 



Biologists have treated their combina- 

 tions as if they were simple summations of 

 independent elements. 



More likely are the combinations com- 

 posed of interdependent factors whose 

 symbolization must be at the simplest a 

 product. 



A tremendous illustration of variation 

 under change of stimuli is given by Japan. 



