How to add speeds; Galileo and Einstein won’t agree.

How to calculate the speed of anything, when their speed becomes closer to the speed-of-light. 
This article was originally a comment in the linked article;  Why Nothing Moves Faster Than Light.

— In order to correct the comment I have made earlier  ” unless something is completely mass-less in its rest-frame ” I also add the following. This is a fact which I have realized lately — or rather trapped myself to commit an inconsistent remark, by following the same comment in making other remarks elsewhere. 

But it’s better late than never to realize; when something is mass-less, it will never have a rest-frame, because by Einstein’s transformation rules, known as Theory of Relativity, to be consistent, a mass-less particle will always move at the speed of light c, no matter which frame we are looking at it from. This then leads to the velocity addition formula of Einstein.

Now we will discuss in a slightly more detail the two kind of velocity addition formula, one prior to Einstein and one that came from Einstein’s work. 

Prior to Einstein. 
According to Newton and Galileo ( Galileo Project ), known by a name Galilean Relativity, the following follows; if C moves at speed

Optical Path and Fermat’s Principle.

We can see ourselves in mirror and take our mirror reflected selfie as a consequence of Fermat’s Principle, the topic of discussion of the blog.

In the last few weeks I am trying to understand why light traverses straight lines and why it refracts. The other day I saw a little mug floating inside a bucket full of water. Inside water any object would look shortened, a phenomena known as refraction. That’s because light rays would “bend” inside water (towards a direction where they have to take a shorter path) as their speed must reduce, given to the fact that in the same time in a rarer medium light would have traveled a longer distance in the exact same time, which is no longer possible due to the crowd of molecules and subatomic ghetto that it meets along its way.

Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity by Steven Weinberg .. review

A highly sofisticated and an elegant text book on general Relativity and its application. I would say a super book. I have read this book and solved for now a few daunting problems in Physics by applications of the content in this book. My research is available on my website but for context I recommend this book for serious practising physicists.