Is Coulomb's law the earliest mathematical formula describing electricity?

Question

Is Coulomb's law the earliest formal equation of electricity?

Before Coulomb, many scientists and engineers conducted experiments of electricity.

Answer 1

Sort of, if we interpret "electricity" and "formal equation" narrowly. Then the only other contender is Cavendish, who in 1775 did experiments with various capacitors, established that charge on them was proportional to voltage, and introduced the notion of capacitance as the proportionality constant, so $Q=CV$. Of course, he did not call voltage "voltage", but "degree of electrification", and he is the one who introduced that too. Unfortunately, Cavendish did not publish his results, they were only published by Lord Kelvin in 1879, so the "priority" goes to Coulomb in 1785.

But Coulomb's achievement was more experimental than theoretical. That electric force falls as inverse square was already conjectured by Priestley in 1766 based on Franklin's observation that a charge is not attracted to the walls of electrified can when inside it. Priestley recalled that Newton proved in Principia that a mass inside a spherical shell is not attracted to its walls either, and that the force has to fall as inverse square of the distance for this to happen. Cavendish came to the same conclusion in 1771, but "indifferent to fame he neglected to communicate this and other work of importance", as we already know. His anticipations of the notions of current, resistance and the Ohm's law (albeit only qualitatively, he "measured" current by electrocuting himself) shared the same fate.

Laws of magnetic force were also established earlier. Newton wrote in Principia that the "power of magnetism... in receding from the magnet, decreases not in the duplicate but almost in the triplicate proportion of the distance, as nearly as I could judge from some rude observations", the result confirmed by the publishers of the 1742 edition Le Seur and Jacquier. And in 1750 Michell clarified that in a magnet "each pole attracts or repels exactly equally" and "attraction and repulsion of magnets increases, as the squares of the distances from the respective poles increase". See Whittaker's History of the Theories of Aether and Electricity and Timeline Of History Of Electricity

Although that light is an electromagnetic wave was not established until 19th century by Maxwell, the law of reflection was already known to Euclid c. 300 BC if not earlier, one of the first quantitative laws of physics to be discovered. Ptolemy knew the approximate law of refraction (for small incidence angles) c. 150 AD, and measured some refraction coefficients, and the exact law (with sines) was established by Snell and Descartes in 17th century. And in 1619 Kepler gave the famous geometric argument for the inverse square law applied to the intensity of light from point sources:"there is as much light in the narrower spherical surface, as in the wider... therefore the form will be made weaker in unequal spheres in the ratio of the square of its distance". Bullialdus extended it to gravity in 1645, and eventually it reached Newton (possibly through Hooke), who confirmed it for gravity based on Kepler's planetary laws, and influenced Michell and Priestley. See Who was first to explain intuitively the inverse square law of gravity?