Although, it seems very easy to find the magnetic field produced by a solenoid or a toroid, all we got to do is to make a suitable an Amperian Loop and take the $\mathbf B$ out of the integral and so on.
But the main step is to find the direction of magnetic field. For example see these derivations of Griffiths:
I want to know who actually drived those equations, who actually worked out for the directions of magnetic field (Griffiths has done his)?
I searched for magnetic field by solenoid and toroid but found nothing historical. I aimed at finding something from Wikipedia’s “Refernces” or “External Links”. I read somewhere that Wilhelm Eduard Weber did work out the field for some geometrical shapes.
So, I request for some refernces where I can find the actual derivations of magnetic field produced by solenoid and toroid.