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I'm trying to learn and understand how Coulomb derived his formula. This is from Wikipedia:

He used a torsion balance to study the repulsion and attraction forces of charged particles, and determined that the magnitude of the electric force between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

Now the question and I didn't find anything about it. How did he charge and measure the charge on the balls? It was before batteries were invented (and before voltages were known) and even if he created charges on the ball, how did he know its exact number.

In other words: How did he use the torsion balance along with the little information known at that time to come up with his equation?

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I am not sure what Coulomb did historically, but he didn't need to know the charges of the balls. If he doubles one of the charges, he can notice that the force is doubled. And so he can know that the force is proportional to the charge. The same discussion could be applied to the distance.

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    $\begingroup$ He halved the charge by touching the charged ball on to an identical uncharged ball. Clearly he could halve it again and so on. $\endgroup$
    – Philip Wood
    Commented May 26, 2021 at 17:58
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    $\begingroup$ @vino rue. Thanks, but the question how he exact double the charge? $\endgroup$
    – Aminadav Glickshtein
    Commented May 26, 2021 at 19:39
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    $\begingroup$ @AminadavGlickshtein As Philip said, he could half the charges by touching identical uncharged balls. So, before touching, the ball has double the charge it has after touching. $\endgroup$
    – Vino Rue
    Commented May 27, 2021 at 18:35
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Conveniently you he didn't need to know the exact charges. Observing the change in force over distance with the torsion balance just requires that the charges are different and constant. The ration of distance to force can be seen regardless.

To get spheres of relative charge one could charge a sphere, insert it in the balance and measure the force. Then touch it to an uncharged sphere to half the charge and test again. I'm no expert in the science of the time but you can't let our modern methods of more exact charging and measurement color how you would do the experiment. :)

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