How did scientists/physicists originally discover that light has no mass? Relativity is founded upon the fact that light has no mass, and the equations prove this. However, what I am asking is how did scientists originally realize that light does not have mass? It was know that light had a speed, which is also a characteristic of classical objects forgetting relativity and that no object with mass could ever reach that speed. What led physicist to conclude that light has no mass.
 A: The basic fact is that a non-zero mass of the photon would change the inverse square law. There are a couple of ways to do that: one is to assume that you have an inverse $(2+\epsilon)$ law and set experimental limits on $\epsilon$; the other is to assume that the potential has the Yukawa form $e^{-\mu r}/r$ where $\mu = m_\gamma c/\hbar$ and set experimental limits on $\mu$.
There are both laboratory experiments and measurements of the Earth's magnetic field that give limits for $m_\gamma < 4\times 10^{-48} g$, which is 20 orders of magnitude smaller than the mass of the electron.
These measurements validate the inverse square law at distances of order $1 cm$ to $10^9 cm$ but with additional assumptions, it can be extended down to $10^{-15} cm$.
All of this comes from Jackson's "Classical Electrodynamics", 2nd (or 3rd) ed, section 1.2.
A: The simplest way to consider this is as follows: A photon of light has momentum and velocity. You might think that because it has momentum, it must have some mass, but if you extrapolate to zero velocity (where the momentum goes to zero) you discover that a photon of light has no rest mass left over- thus demonstrating that photons are massless. 
