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**THE FIVE PROPOSITIONS
This web site uses the four fundamental theorems of the vector calculus,
in conjunction with an experiment conducted with **

*PROPOSITION I*

That**biology**is “the study of those systems that can replace their internal energy”; that it is described by the general equation*dU*=*Mdt =*δ*Q*- dH; M > 0 (see explanation of variables and terms); and that it is governed by the four laws of biology summarized upon the left.

*PROPOSITION II*

That**ecology**is “the study of the processes systems use to replace their internal energy”; that it is described by the equation*pdt + mdt = dh + du =*δ*q**; m*> 0 (see explanation of variables and terms); and that it is governed by the four maxims of ecology summarized upon the right.

*PROPOSITION III*

That all biological populations are subject to three constraints …:- the constraint of constant propagation, (where P is the population's total energy flux in watts);
- the constraint of constant size, (where M is the population's mass flux, and R is the average individual mass flux scaled per unit population size, i.e. per “biomole”);
- the constraint of constant equivalence, (where S is the population's “engeny” in watts per kgs per unit population size, i.e. per “biomole”).

*PROPOSITION IV*

That every species can be uniquely characterized by stating:- its generation length, T;
- its average individual mass flux,
*m̅*; and - its average individual energy flux,
*p̅*,

- with T being a boundary condition set by the Liouville theorem; and with
*m̅*and*p̅*being the decompositions of the Helmholtz theorem and so the scalar and vector potentials.

*PROPOSITION V*

That the Gibbs-Duhem equation

(A)

*m*̅μ =*m̅**dS*=*dU*+*dH*- Σ_{i}μ_{i}(*dv*_{i}-*dm*_{i})

governs all biological energy;

*(B)*That the Euler equation

μ =*dS*= (∂*S*/∂*U*)_{V,Ni}*dU*+ (∂*S*/∂*V*)_{U,Ni}*dV*+ Σ_{i}(∂*S*/∂*u*_{i})_{U,V,{Nj≠i}}*du*_{i}+ Σ_{i}(∂*S*/∂*v*_{i})_{U,V,{Nj≠i}}*dv*_{i}

governs all biological activity;

*(C)*That it is impossible for the summation terms in the above Gibbs-Duhem and Euler equations ever to be zero; and therefore that a population free from Darwinian competition and evolution is impossible.