Maxwell's equations

Maxwell's four equations say, in order: electric field lines begin and end on charge; magnetic field lines never begin or end at all, because isolated magnetic poles do not exist; a changing magnetic field creates a circulating electric field; and a magnetic field circulates around both electric current and a changing electric field. The first three summarised what was already known. The last clause of the fourth was Maxwell's own addition, and it changed physics.

The reason it mattered is that it closed a loop. A changing electric field makes a magnetic field, which as it changes makes an electric field, which makes a magnetic field. Once started, the pair can sustain each other and travel with no charges or currents present at all. Maxwell computed the speed of this self-sustaining disturbance from two constants measured in laboratory experiments on capacitors and coils, and got the speed of light — which is how optics turned out to be a branch of electromagnetism.

That also settles what electromagnetic waves travel through: nothing. Each field is the source of the other, so the wave carries its own means of propagation and needs no medium. The nineteenth century spent decades searching for an ether anyway, and Michelson and Morley's failure to find it is one of the results special relativity was built on. Together the four equations cover radio, radiant heat, visible light, X-rays and gamma rays as one phenomenon at different frequencies.