The field concept in Ampère’s magnetostatics

See footnote 14 in Ref. 6, which quotes the Blunn translation of Ampère’s memoir of Ref. 1 which, in turn, appears in Ref. 2. The item in question is an extensive footnote in Ampère’s paper referring to the Biot-Savart experiment and to an important logical error that they committed. Blunn tones down Ampère’s indictment of Biot and Savart considerably. In his original footnote in Ref. 1, Ampère makes it plain that Biot and Savart are guessing and complains about their failure to acknowledge a presentation by Felix Savary correcting the error.

A function is a special kind of relation. Thus, we can express $y⃗=F(x⃗1,x⃗2,…,x⃗m)$ in the form $y⃗−F(x⃗1,x⃗2,…,x⃗m)=0$⁠, thus giving rise to a relation in $m+1$ variables. If $F(Qv⃗,ϕx⃗1,Qv⃗,ϕx⃗2,…,Qv⃗,ϕx⃗m)=Qv⃗,ϕy⃗$ for any choice of $v⃗$ and $ϕ$⁠, we say $F$ is an isotropy or is an isotropic mapping. If $F$ is a scalar function, then no rotation operator multiplies $y$ and $F$ is a scalar invariant. We will use the definition of isotropy in this form applied to the element pair force function $F⃗mn$⁠.