BioLoGcy AND MATHEMATICS 24 
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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 continuity, 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 equt- 
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 necessary to make 
