No wonder I had a problem understanding what it was you were doing. It was as if you actually went out of your way to take a very simple thing and say it in a very complex way. You should have simply said the following: Let x be the distance traveled by a photon and a particle. Let v be the speed of a particle where v < c. Let t be the time it takes the particle travel the distance x and T the time it takes a photon to travel the same distance. Then x = vt = cT. Therefore T/t = v/c.
See how simple that was?
Yes, but he makes it even more complicated in the following section where he considers the moving frame. He uses the same T and t for the values seen by the observer in the rest frame when viewing the moving frame, but T and t are what the observer within the moving frame sees and are seen as reduced by the observer in the rest frame. He needs new symbols for these transformed values as he is making the mistake of equating unequal values.
I find the entire analysis over complicated and it is very unclear. To be honest I haven't even looked to see whether the maths is correct as there are too many false assumptions before you even start on the maths. No, no, I'll be really honest, it's because I'm not a mathematician and I don't really enjoy the maths bit