Why when "fluxions" are zero can equation be considered linear? Reading an article form the 40's, there is a passage like this:

Why can the equations be considered linear? (I assume "fluxions" are the derivatives.)
P.S.: Article is "Effects of Controls on Stability, Ashby, 1945", p. 7 in Mechanisms of Intelligence, full PDF here.
 A: Fluxions are the old word for derivatives. The article is saying that a nonlinear set of equations can be linearized near an equilibrium--which is when derivatives are zero. In your case equilibria occur exactly at the simultaneous roots of the $f_i$.
A: $\newcommand{\Reals}{\mathbf{R}}$Let $F$ be a sufficiently smooth (real- or vector-valued) function of one or more real variables. The overarching principle is, the qualitative behavior of $F$ at a point $p$ is dominated by the value $F(p)$; if this value vanishes, the behavior is dominated by the first-degree approximation $DF(p)(x - p)$; if the first-degree approximation is zero, the behavior is dominated by the second-order terms $\frac{1}{2!} D^{2}F(p)(x - p)^{2}$ in the Taylor expansion, etc.
Familiar applications abound in calculus:


*

*A local extremum of a function can only occur at a critical point: If $F$ is a (continuously-differentiable) real-valued function of one variable and $F'(p) \neq 0$, then $F$ is locally monotone near $p$.
In case it helps, if $F$ is real-valued and twice continuously-differentiable, then for $x$ sufficiently close to $p$,
$$
  f(x) = f(p) + f'(p)(x - p) + \tfrac{1}{2!}f''(p)(x - p)^{2} + o(x - p)^{2}.
  $$
If $f(p) = 0$, this rearranges to
$$
  f(x) = (x - p)\bigl[f'(p) + \tfrac{1}{2!}f''(p)(x - p) + o(x - p)\bigr],
  $$
which is approximately linear, since the terms in square brackets are approximately $f'(p)$ for $x \approx p$.

*If $DF(p) = 0$ and $D^{2}F(p)$ is positive-definite, then $p$ is a local minimum of $F$. If instead $D^{2}F(p)$ is negative-definite, then $p$ is a local maximum of $F$.
Now for your actual question: Suppose $F$ is an $\Reals^{n}$-valued function on an open neighborhood of $p$ in $\Reals^{n}$, and you want to investigate the qualitative behavior of the first-order differential equation
$$
x'(t) = F(x(t))
\tag{*}
$$
near $p$.
If $F(p) \neq 0$, then the constant term dominates, and the solutions of (*) are approximately
$$
x(t) \approx x(0) + tF(p).
$$
If $F(p) = 0$ but the matrix $A = DF(p)$ is non-singular (invertible), then (in senses that can be quantified if necessary) the system (*) is approximately $x' = Ax$, and the solutions are approximately
$$
x(t) \approx \exp(tA) x(0).
$$
Particularly, if the eigenvalues and eigenspaces of $A = DF(p)$ are known (or generally, its Jordan canonical form is known), then the local qualitative behavior of the original system can be read off.
Presumably that's the paper's rationale for linear approximation.
