On Maxwell's discovery of electromagnetic waves and the gauge condition

The original 'theoretical discovery' of electromagnetic waves by Maxwell is analysed and presented in modern notation. In light of Maxwell's well-known prophetic dedication to the concept of the vector potential, it is interesting to reveal his derivation of the wave equation for this potential without the application of any gauge condition. This is to contrast with typical approaches students learn from standard textbooks for the derivation of the wave equation in various forms. It is in our opinion that intuition and insight, rather than logical deduction, must have played a more significant role in Maxwell's original discovery, as is not uncommon with discoveries made by the pioneers in science.