Before we knew what light was, James Clerk Maxwell was studying electricity and magnetism. He developed a series of equations describing how the two forces related, and realized that these equations could describe an oscillation in space that could travel on its own without a nearby source. It only works if you quickly change the electric field, though, and a constant field won’t do this.
He did the math, and discovered that all such waves would travel at a specific speed. Around this time, the speed of light was measured by someone else, and they turned out to be the same the speed.
Further experimentation revealed that these waves behaves differently at different frequencies of oscillation. Higher frequencies became more like visible light, and higher still like x-rays and gamma rays.
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