Proof of conservation of spacetime interval

Question

In this paper as well as in Landau's textbook on classical field theory, there is a proof of the conservation of spacetime interval in which authors deduce, that the differentials of the interval in two different inertial reference frames are infinitesimals of the same order and therefore must be proportional. What does it mean and how can you prove it without refering to the properties of Lorentz transform, which is to be derived from conservation of spacetime interval.

Answer 1

$ds'$ is some function of $ds$ which we want to find:

$ds'=ds'(ds)$

We can taylor expand this function:

$ds'(ds)=ds'(0)+ads + \mathcal{O}(ds^2)$

for some $a$.

Since $ds'$ and $ds$ are both of first order, we neglect higher order terms.

From the propagation of light, we know that if $ds=0$ then also $ds'=0$ and therefore

$ds'(0)=0$

This leaves us with

$ds'=ads$

for some $a$.