My recent proof of non-existence of Green's function for magnetic field was sounded as "the Biot-Savart law is wrong" casts a shadow onto their names that is wrong itself. In my recent report I specially made an excursion to history to explain that J.-B. Biot and F. Savart have not made any errors and what we are indebted to them for. I did not do it in my article on the subject [Z. Turakulov. On applicability of the method of Green's function in Magnetostatics {Preprints Theor-Phys.org theor-phys.org/prepr/Particles and Fields} tp-4663-01] because the article was destined only for exposing the proof itself strictly as a mathematical fact. Now I would like to do it for everyone who reads my materials.
I have learned from the work of J.D. Jackson and L.B. Okun' [J.D. Jackson, L.B. Okun, Historical roots of gauge invariance. http://arxiv.org/vc/hep-ph/papers/0012/0012061v3.pdf] that the well-known integral representation of the vector potential obtained vrom the vector of current by the method of Green's function which majority of authors call "Biot-Savart law" belongs to F. Neuman who derived it about two decades after Biot and Savart have published their ides. This fact rises a question, what exactly they said. I find impossible for myself to get and read their original work so I have to guess about everything myself. Below I outline my views on their role in develpment of our views on current-current interactions.
F. Klein in his lectures on development of mathematics in XIX century [Лекции о развитии математики в XIX столетии. М.: Наука, 1989] wrote that they have discovered exact law of the force acting on a magnetic pole from electric circuit, but he gave no details. Whatever exact law they would obtain for this particular interaction, it was much more important that they have introduced the notion of interaction of current and, necessarily, the notion of vector potential. I believe that their ideas go as follows.
Point-like charge is a scalar, so it produces scalar potential. Unlike it, current is a vector, so it produces vector potential. Hardly the notion of vector potential appeared before them, so, they had to explain what does this notion mean. Most probably, they answered this question as follows. A vector of current has potential energy equal to scalar product of the current and the vector potential. In classical mechanics of a charged particle this idea leads to the notion of velocity depending potential and correct description of its motion. If scalar multiplication was not known at the time, we are indebted to them for this operation too. Thus, most probably, their main achievements were the great ideas of vector potential, vector interaction and, probably, scalar product of two vectors. That is what in my opinion, we are indebted to J.B. Biot and F. Savart for.
