Deriving special relativity using alternative axioms Einstein’s paper on SR takes as axioms that (1) the laws of physics are the same in all inertial frames, and (2) the vacuum speed of light is the same for all observers, independent of the relative speed of the source.
I saw a reference once to an alternative derivation, dating to around 1907 (after Einstein’s) that instead used (2') any two observers would see each other moving at the same speed.
This derivation essentially replaces $\gamma = (\sqrt{1-\frac{v^2}{c^2}}) ^{-1}$ with $\gamma = (\sqrt{1-\alpha v^2})^{-1}$ and changes the velocity sum formula to $\frac{v+u}{1+\alpha vu}$. If $\alpha = 0$, you get Gallilean invariance and Newton's mechanics. If $\alpha > 0$, you get Lorentz invariance and SR. 
Unfortunately I failed to record the details of the citation, so I have not been able to find the paper or modern treatments of this approach.
Can anybody explain this derivation, or point me to a source? 
 A: I think you are looking for
"Einige allgemeine Bemerkungen über das Relativitätsprinzip,"
 Physikalische Zeitschrift. 11. (1910), pp. 972–976
by Vladimir Ignatowski
https://de.wikisource.org/wiki/Einige_allgemeine_Bemerkungen_%C3%BCber_das_Relativit%C3%A4tsprinzip
translated as
"Some general remarks on the relativity principle" 
by Vladimir Ignatowski 
https://en.wikisource.org/wiki/Translation:Some_General_Remarks_on_the_Relativity_Principle
It begins

When Einstein introduced the relativity principle some time ago, he simultaneously assumed that the speed of light ${\displaystyle c}$ shall be a universal constant, i.e. it maintains the same value in all coordinate systems. Also Minkowski started from the invariant ${\displaystyle r^{2}-c^{2}t^{2}}$ in his investigations, although it is to be concluded from his lecture "Space and Time"[1], that he attributed to ${\displaystyle c}$ the meaning of a universal space-time constant rather than that of the speed of light.
  
  Now I've asked myself the question, at which relations or transformation equations one arrives when only the relativity principle is placed at the top of the investigation, and whether the Lorentzian transformation equations are the only ones at all, that satisfy the relativity principle.

The passage of interest follows Eq. (24)

$${\displaystyle p={\frac {1}{\sqrt {1-q^{2}n}}}}\qquad\mbox(24)$$

  From (24) it follows, that ${\displaystyle n}$ (which we can denote as a universal space-time constant) is the reciprocal square of a velocity, thus an absolute-positive quantity.
  

  We see that we obtained transformation equations similar to those of Lorentz, except that ${\displaystyle n}$ is used instead of ${\displaystyle {\tfrac {1}{c^{2}}}}$. However, the sign is still undetermined, because we could have set the positive sign under the square root in (24) as well.
  

  Now, in order to determine the numerical value and the sign of ${\displaystyle n}$, we have to look at the experiment...
  
  ...
  ... This gives
  $$
{\displaystyle n={\frac {1}{c^{2}}}}\qquad\mbox{(25)}$$
  And only from that it follows, that ${\displaystyle c}$ is constant for all coordinate systems. At the same time we see that the universal space-time constant ${\displaystyle n}$ is determined by the numerical value of ${\displaystyle c}$.
  
  Now it is clear that optics lost its special position with respect to the relativity principle by the previous derivation of the transformation equations. By that, the relativity principle itself gains more general importance, because it doesn't depend on a special physical phenomenon any more, but on the universal constant ${\displaystyle n}$.
  
  Nevertheless we can grant optics or the electrodynamic equations a special position, though not in respect to the relativity principle, but in respect to the other branches of physics, namely in so far as it is possible to determine the constant ${\displaystyle n}$ from these equations.

For more early papers on Relativity, 
this is a useful starting point:
https://en.wikisource.org/wiki/Portal:Relativity

FOOTNOTE:

Here is another reference by Ignatowski. This reference is a more thorough introduction to the idea.
"Das Relativitätsprinzip"
Archiv der Mathematik und Physik 17: 1-24, 18: 17-40 (1910)
https://de.wikisource.org/wiki/Das_Relativit%C3%A4tsprinzip_(Ignatowski)
I had to run it through Google Translate.
