**Abstract:**

History records how Maxwell unified the work of Gauss, Faraday and other pioneers which led to the prediction of electromagnetic waves, because the d’Alembert wave equation is derivable from the Maxwell equations.

In contrast, I begin with three simultaneous algebraic-vector equations and show that these define the Maxwell equations and the properties of vacuum. Now instead of using the d’Alembert wave equation to define electromagnetic waves, we can use the three simultaneous algebraic-vector equations to define wave structures that can be three dimensional, e.g. ball lightning. Also, it shows that the electromagnetic phenomenon and the properties of the vacuum are dictated by mathematical requirements.

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