August 31, 1917] 



SCIENCE 



201 



of the action of salt solutions on permea- 

 bility, growth, etc., involve even greater 

 complications produced by the interrela- 

 tion of conditioning factors. 



In order to get an accurate statement of 

 the Law of Minimum, it is necessary to 

 get away from the custom of discussing 

 causes, however difficult this may be.^^ 

 The idea of causation invariably indicates 

 incomplete analysis. A biological phenom- 

 enon is dependent not on a single variable, 

 but on a complex or constellation of factors, 

 as we have seen in the case of carbon as- 

 similation. It should be discussed there- 

 fore in terms of all the conditioning fac- 

 tors, not in terms of that one which tempo- 

 rarily happens to be a limiting factor. The 

 term "function" is valuable in this con- 

 nection. The amount of carbon assimila- 

 tion is a function of the temperature ; it is 

 another function of the illumination, etc. 

 With this idea of function in mind, the Law 

 of the Minimum may be stated in the fol- 

 lowing form. When a quantity is depend- 

 ent on a number of variable factors and 

 jnust be a function of one of them, the 

 quantity is that function which gives the 

 minimum value. Expressed in plain Eng- 

 lish this means that a chain is no stronger 

 than its weakest link. The Law of the 

 Minimum is only too obvious. Its applica- 

 tion is often so self-evident that it is made 

 as a matter of course. 



But the most interesting thing about the 

 law is not how it works, but when it does 

 not work. There is a fundamental discrep- 

 ancy between the Law of the Minimum and 

 Galton's Law of averages. In the current 

 text-books on genetics and plant physiol- 

 ogy^^ the following ingenious explanation 

 of Galton's Law is given. Assume that the 



11 Cf. B. E. Livingston, loc. cit. 



12 E. Baur, "Einfiihrung in die experimentelle 

 Vererbungslehre, " 2'"' Auflage, 1914. L. Jost, 

 " Vorlesungen iiber Pflanzenphysiologie, " 3'° Au- 

 flage, 1913. 



size of a bean is determined by only five 

 variables, each of which must occur in one 

 of two categories ; in one case the size of the 

 bean will be increased by one unit of size, 

 in the other it will be decreased by the same 

 amount. Considering all the possible per- 

 mutations of these five variables, we get the 

 following arrangement: 



The beans will be of six sizes, -\-5, +3, 

 + 1, — 1, — 3, — 5, and out of a very large 

 number (to), n/S2 will be -f 5, 5n/d2 wiU 

 be 4- 3, IOto/32 will be + 1, 10?i/32 will be 

 — 1, 5?i/32 will be —3, and to/32 will be 

 ■ — ■ 5. The six sizes are in the ratio 

 1 : 5 : 10 : 10 : 5 : 1. If we plot the sizes of 

 the various classes of beans against the fre- 

 quency of their occurrence, we get an ap- 

 proximation to the familiar curve of nor- 

 mal error. For the sake of simplicity, the 

 number of variable factors was made five 

 and the number of categories in which each 

 might occur was limited to two. If the 

 variables and the categories are made sufiS- 

 ciently numerous, the curve of normal error 

 can be approximated within any desired 

 degree of exactitude. It is unnecessary to 

 point out the empirical fact that when the 

 sizes, weights, etc., of organisms or their 

 parts are divided into classes and the 



