Every time I run across of (complex to me at least) formulas like those to find inductance for an air-core coil or the noise in a resistor, where so many factors are involved, I wonder what was the process followed by the originator in constructing such an expression.

When there are linear relationship into play I could understand that someone with good insight and experience in the matter could express that, but how when there are roots and powers and fractions with (incredible) values...

And then they came the constants...

a) Thinking of those that came out from experimental work, is it a well outlined process to implement such complex expressions?

b) And an even and more insidious one: how in heaven the proved they were right?

 

In many cases, the process probably resembled the process by which sausage is made -- not a pretty sight to behold.

Some things are arrived at through purely theoretical derivation from first principles. Then ton's of data are taken to see if the theoretical relationships match reality.

Other things are arrived at through purely empirical processes where lots of data are taken and then people start trying different mathematical relationships to see which match the best. Sometimes a particularly good match is found and that equation then fosters new theoretical insights which might eventually result in someone figuring out how to derive that equation, now that they know the form they are trying to end up with, from first principles.

In most cases, however, the way that the formulas are taught to later generations seldom bears any resemblence to the erratic and often contentious process that originally brought the formula forth.