Biology and Mathematics 243 



Now biology deals largely with aggregates of individuals, 

 and then, like the pure theory of numbers, its variables are 

 discrete, and must change by jumps of at least one individual. 



A mathematics proper to such investigations has not been 

 accessible to the biologist, for not only has his calculus been 

 founded solely on continuit}^ but also his geometry has been 

 developed for him on continuity assumptions from the very 

 beginning. 



The very first proposition of Euclid is to describe an equi- 

 lateral triangle on a given sect (a given finite straight line). It 

 begins: "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 another, 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 cannot represent a biologic aggregate of 

 individuals. 



Geometry can be treated without any continuity assump- 

 tion, without continuous circles, in fact without compasses. 



Such a geometry for biologists, is my own Rational Geom- 

 etry, 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 recall 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 theories 

 with the following characteristics as enumerated by the Russian 

 Bugaiev: 



(1) The continuity of phenomena; 



(2) The permanence and unchangeableness of their laws; 



(3) The possibility of characterizing a phenomenon by its 

 elementary manifestations; 



(4) The possibility of unifying elementary phenomena into 

 one whole; 



(5) The possibility of sketching precisely and definitely a 

 phenomenon for a past or future 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 necessar}^ to make 



