Perhaps the most marvelous thing about Einstein's Special
Relativity derivation is the math he used to get from his
tau function in t and x' to his tau=f(t,x) transform.

[We let his a=phi(v)=1, as he concludes later.]

[1]  tau = (t-vx'/(cc-vv)).

[2]  tau = (t-vx/cc)/sqrt(1-(v/c)^2).

First of all, to get to [2], we certainly have
to rid [1] of x'.   x'=x-vt.

[3]  tau = (t-v(x-vt)/(cc-vv))

  = (tcc-tvv-vx-vvt)/(cc-vv)

  = (tcc - vx)/(cc-vv)

Now, divide numerator and denominator on the right
by cc:

[4]  tau = (t-vx/cc)/(1-vv/cc).

There's only one way to get [2] from [4]. Let
tau<>tau, a logical absurdity in this situation;
Einstein has proceeded far beyond tau the unknown
function. The only unknown is a, which he later
says is phi(v)=1.

And if it is legal to get [2] by multiplying only one side
by sqrt(1-vv/cc), then it is also correct to multiply
only one side by (1-vv/cc), and get the galilean transform.

Or to multiply one side by pi and get "t and -vx/cc
are really circle diameters" transforms. [You know,
the circumference of a circle is Pi*diameter?]

But in all cases - both the absurd Einsteinian and Pi
transforms - it is not legal to treat only one side of
an equation in a non-identity fashion. The left side of
the tau function would not be tau, but gamma*tau or Pi*tau.

The appearance of gamma is just as magically marvelous
in the X' transform (we used X' for the moving system
x value coordinate, remember?):

   X' = ccx'/(cc-vv).

      = (ccx-ccvt)/(cc-vv)

      = (x-vt)/(1-vv/cc).

Not X' = (x-vt)/sqrt(1-vv/cc).