I am reading Big Bang by Simon Singh. Here is the short story (as I understood it):
Einstein gets a contradiction by assuming that
1- The speed of light $c$ is constant with respect the ether.
2- Galilean relativity is true.
because if you were traveling at the speed of light, you wouldn't be able to see yourself in a mirror, since you would be moving with respect the ether at the same speed at the light leaving your face. So you would be able to tell that you are moving, contradicting galilean relativity.
From here seems logical to deduce (assuming galilean relativity) that you will see your face on the mirror, therefore the speed of light can't be constant with respect the ether. And here comes where I am stuck, to me the most obvious way of solving this problem is thinking that light will act as any other thing we know, if you are moving at the speed of light and you throw a ball in the same direction of the movement at a speed $v$, you will see the ball leaving you at $v$ and someone that doesn't move with respect to you will see it at $c+v$. But according to the book, Einstein's first thought was to say that light has a constant speed for every observer.
So finally here comes my question: Did Einstein (or anyone before him) consider the first argument I gave here and, if so, why did he reject it?
P.D. In case it is useful to know something about my background, I am a mathematician with very little knowledge of physics, just the things I read in these kind of divulgative books. That said, sorry about incongruences or inaccuracys, I would very much apreciate corrections.