Does Einstein's derivation of the Lorentz transformation have a mathematical error? The article "Mathematical Inconsistencies in
Einstein's Derivation of the Lorentz Transformation" says that Einstein's derivation of the Lorentz transformation has some mathematical error, and his opinion seems to be true. Is Einstein really wrong?  (I'm not asking about whether the Lorentz transformation is wrong.)
 A: Using the equation number on the linked page:
A)  Einstein goes from 1) and 2) to 3) and 4).  This assumes linearity.
B)  Einstein goes from "3) holds for positive $x$" to "3 holds for all $x$".  This follows from linearity.
Einstein either has or has not previously established linearity.  If not, then there is a gap in the argument occurring before A).  If so, then there is no gap.
The author claims that there is a gap occurring between A) and B).  In neither case is this correct.
A: I do not know the precise way followed by Einstein, but interpreting what the weird site tries to quote, I think Einstein's idea is correct. Assume that the transformation of coordinates $$\mathbb R^2 \ni (x,t) \to (x',t')\in \mathbb R^2$$ is linear. If physics says that $$\mbox{$x\pm ct=0\quad $ if and only if $\quad x'\pm ct'=0$,}\tag{0}$$ then the form of the linear transformation between the two coordinate frames must necessarily satisfy $$x+ct = a(x'+ct')\tag{1}$$ $$x-ct = b(x'-ct')\tag{2}$$ for some constants $a,b$. 
PROOF. Pass to light coordinates $u = x+ct$, $v = x-ct$ and  $u' = x'+ct'$, $v' = x'-ct'$. The transformation of coordinates remains linear and the physical constraint (0) is now $$\mbox{$u=0\quad $ if and only if $\quad u'=0$}$$ $$\mbox{$v=0\quad $ if and only if $\quad v'=0$.}$$ The only possible linear relation between coordinates $(u,v)$ and coordinates $(u',v')$ is given by a diagonal matrix, which is exactly what (1) and (2) say.
