Below follow an extract from Fourier's paper "THEORY OF HEAT" in which he says:
Consider the variable state of a solid whose heat is dispersed into air, maintained at the fixed temperature 0. Let $ω$ be an infinitely small part of the external surface, and $μ$ a point of $ω$, through which a normal to the surface is drawn ; different points of this line have at the same instant different temperatures. Let $v$ be the actual temperature of the point $μ$, taken at a definite instant, and $w$ the corresponding temperature of a point $ν$ of the solid taken on the normal, and distant from $μ$, by an infinitely small quantity $α$. Denote by $x, y, z$ the co-ordinates of the point $μ$, and those of the point $ν$ by $x + δx, y + δy, z + δz$ ; let $f(x, y, z) =0$ be the known equation to the surface of the solid, and $v = Φ(x,y,z,t)$; the general equation which ought to give the value of $v$ as a function of the four variables $x,y,z,t$.
Differentiating the equation $f(x, y, z) = 0$, we shall have:
$$mdx+ndy+pdz=0$$
$m,n,p$ being functions of $x,y,z$.
(...)
Now, it follows from the principles of geometry, that the co-ordinates $δx,δy,δz$ which fix the position of the point $ν$ of the normal relative to the point $μ$ satisfy the following conditions:
$$pδx=mδz$$ and $$pδy=nδz$$
My question is about the geometrical principle he make use to derive the last expressions?
You can find the paper here: page 115-116
$$m\,dx+n\,dy+p\,dz=0$$ is the equation of the tangent plane, thus $(m,n,p)$ is the direction of the surface normal, which is orthogonal to all directions in the tangent plane. As $(δx,δy,δz)$ has the same direction is has to be a multiple of the normal direction, which can also be formulated as the condition that all the $2\times2$ minors of $$\pmatrix{m&n&p\\δx&δy&δz}$$ have to be zero, which gives the cited equations.