August 11, 1905.] 



SCIENCE. 



11 



teaches the production of man from lower 

 biologic forms by wholly natural causes. 



If this be so, then skipping the funda- 

 mental puzzle as to how a living thing gets 

 any conscious knowledge, any subjective 

 representation of that independent world, 

 it remains of the very essence of the doc- 

 trine of evolution that man's knoAvledge of 

 this independent world, having come by 

 gradual betterment, trial, experiment, 

 adaptation, and through imperfect instru- 

 ments, for example, the eye, can not be 

 metrically exact. 



In the easiest measurements it is said Ave 

 can not even with the best microscopes go 

 beyond one one millionth of a meter; that 

 is, we are limited to seven significant figures 

 at most. "What is the meaning then of the 

 mathematics which, as in the case of the 

 evaluation of tt, has gone to seven hundred 

 places of significant figures'? 



If then we are to hold to evolution, sci- 

 ence must be a construction of the animal 

 and human mind; for example, geometry 

 is a system of theorems deduced in pure 

 logical way from certain unprovable as- 

 sumptions precreated by auto-active animal 

 and human minds. 



So also is biology. But here the assump- 

 tions are more fluctuating, and many of 

 them are still on trial. 



Since every science strives to characterize 

 as to size, number, and, where possible, 

 spatial relations, the phenomena of its do- 

 main, each has need of the ideas and 

 methods of mathematics. One of the 

 fundamental ideas of mathematics is the 

 idea of variation, the variable, qualitative 

 and quantitative variability. 



AA^hen related quantities vary, one may 

 vary arbitrarily. This is called the inde- 

 pendent variable. Others may vary in de- 

 pendence upon the first. Such are called 

 dependent variables or functions of the 

 independent variable. The change of the 

 variables may be continuous or discontin- 



uous. The blind prejudice for the assump- 

 tion of continuity is so profound as to be 

 unconscious. 



But if biologists did but know it, the 

 characteristics, peculiarities and methods of 

 investigation for continuous functions dif- 

 fer essentially from those for discontinuous 

 functions. 



Our calculus assumed continuity in all 

 its functions, and also that differentiability 

 was a necessary consequence of this con- 

 tinuity. 



Lobachevski, the creator of the non- 

 Euclidean geometry, emphasized the dis- 

 tinction between continuity and differentia- 

 bility, therein also being half a century in 

 advance of his contemporaries. 



The mathematicians of the eighteenth 

 century did not touch the question of the 

 relation between continuity and differentia- 

 bility, presuming silently that every con- 

 tinuous function is eo ipso a function hav- 

 ing a derivative. 



Ampere tried to prove this position, but 

 his proof lacked cogency. The question 

 about the relation between continuity and 

 differentiability awoke general attention be- 

 tween 1870 and 1880, when "Weierstrass 

 gave an example of a function continuous 

 within a certain interval and at the same 

 time having no definite derivative within 

 this interval (non-differentiable). 



Meanwhile, Lobachevski, already in the 

 thirties, showed the necessity of distin- 

 guishing the 'changing gradually' (in our 

 terminology, continuity) of a function and 

 its ' unbrokenness ' (now, differentiability). 



AVith especial precision did he formulate 

 this difference in his Russian memoir of 

 1835 : 'A Method for Ascertaining the Con- 

 vergence,' etc. 



A function changes gradually wlien its increment 

 diminishes to zero together with the increment 

 of the independent variable. A function is un- 

 broken if the ratio of these two increments, as 

 they diminish, goes over insensibly into a new 



