I've started today the book of Landau "Field theory". He starts from the invariance of the speed of light, expresses it as the fact that $c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2=0$ is preserved when we change inertial frame, so he considers $ds^2=c^2dt^2-dx^2-dy^2-dz^2$, and says
"We have observed that if $ds=0$ in one frame then $ds'=0$ in another frame. But $ds$ and $ds'$ are infinitesimal of the same order. So it follows that $ds^2$ and $ds'^2$ have to be proportional that is $ds^2=ads'^2$..." and he goes on to prove that $a=1$.
How to translate this argument in a rigorous one? I'm really interested in this, both to understand this deduction and also to be able in future to make similar ones.
Thanks to everyone who will help
Bye!