16i 



SCIENCE. 



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



function, wliieli consequently will be a differential 

 coefficient. Integrals must always be so divided 

 into intervals that the elements under each in- 

 tegral sign always change gradually and remain 

 unbroken. 



In more detail Lobachevski treated this 

 question in his work, ' On the Convergence 

 of Trigonometric Series/ in which are also 

 contained very interesting general consid- 

 erations on functions. 



It seems, he writes, that we can not doubt the 

 truth that everything in the world can be repre- 

 sented by numbers, nor the truth that every 

 change and relation in it can be expressed by 

 analytic functions. At the same time a broad 

 view of the theory admits the existence of a de- 

 pendence only in the sense that we consider the 

 numbers united with one another as if given to- 

 gether. 



Now biology deals largely with aggre- 

 gates of individuals, and then, like the pure 

 theory of numbers, its variables are dis- 

 crete, and must change by jumps of at least 

 one individual. 



A mathematics proper to such investiga- 

 tions has not been accessible to the biolo- 

 gist, for not only has his calculus been 

 founded solely on continuity, but also his 

 geometry has been developed for him on 

 continuity assumptions from the very be- 

 ginning. 



The very first proposition of Euclid is to 

 describe an equilateral triangle on a given 

 sect (a given finite straight line). It be- 

 gins: 'Let AB be the given sect. From 

 the center A with radius AB describe the 

 circle BCD. From center B with radius 

 BA, describe the circle ACE. From the 

 point C, at which the circles cut one an- 

 other, etc' But the whole demonstration 

 is the assumption of this point C. Why 

 must the circles intersect? Not one word 

 is given in proof of this, which is the whole 

 problem. 



You may say the circle is a continuous 

 aggregate of points. If so, then the circle 



can not represent a biologic aggregate of 

 individuals. 



Geometry can be treated without any 

 continuity assumption, without continuous 

 circles, in fact without compasses. 



Such a geometry, a geometry for biolo- 

 gists, is my own 'Rational Geometry,' the 

 very first text-book of geometry in the 

 world without any continuity assumption. 



How biology has been misled in its 

 mathematics you will realize when you re- 

 call that geometry and calculus have been 

 the basis of mechanics, mechanics the basis 

 for astronomy and physics, physics the 

 basis for physical chemistry, while even the 

 theory of probability had no discontinuous 

 mathematics specially its own. 



Therefore, biologists had clapped over 

 their eyes spectacles of green continuity, 

 and these spectacles colored biologic the- 

 ories with the following characteristics as 

 enumerated by the Russian Bugaiev: (1) 

 the continuity of phenomena; (2) the per- 

 manence and unchangeableness of their 

 laws; (3) the possibility of characterizing 

 a phenomenon by its elementary manifesta- 

 tions; (4) the possibility of unifying ele- 

 mentary phenomena into one whole; (5) 

 the possibility of sketching precisely and 

 definitely a phenomenon for a past or fu- 

 ture moment of time. 



These ideas make the very essence, the 

 framework, the skeleton of modern biologic 

 theories. They have forced their way in 

 and imbedded themselves as being neces- 

 sary to make possible the application of 

 the methods of continuity-mathematics to 

 the investigation of nature. They follow 

 out the fundamental characteristics of con- 

 tinuous analytic functions. Therefore, we 

 may designate our modern biology as a 

 continuity-biology. 



Thus, as the Russian Alexeieff has point- 

 ed out, after the continuity world-scheme 

 had captured the fundamental natural sci- 

 ences, geometry, mechanics, astronomy, 



