What Arnold Sommerfeld is doing here?

Aug 2019
I have started Sommerfeld's Lecture on Theoretical Physics Volume II (Mechanics of Deformable Bodies), in the very first page I couldn't understand this (I have attached the picture of that page also)

" Let P be a point of the volume element under consideration and \(\displaystyle x, ~,y, ~z\) its coordinate in a rectangular system the origin of which, O, lies within the element. In a general motion of the body the points P and O will both experience change of position, which we denote by \(\displaystyle \xi,~\eta,~\zeta\) and \(\displaystyle \xi_0,~\eta_0,~\zeta_0 \) respectively, referring to the chosen space-fixed coordinate system. Taylor's formula then gives for the displacement of P $$ \xi = \xi_0 + \frac{\partial \xi}{\partial x} x + \frac{\partial \xi}{\partial y} y + \frac{\partial \xi}{\partial z}z+ .... \\
\eta = \eta_0 + \frac{\partial \eta}{\partial x}x + \frac{\partial \eta}{\partial y}y+ \frac{\partial \eta}{\partial z}z +... \\
\zeta = \zeta_0 + \frac{\partial \zeta }{\partial x}x + \frac{\partial \zeta }{\partial y}y + \frac{\partial \zeta}{\partial z}z + ...
\label{1}$$ "

Now, I must say that I'm shocked! First of all the notation are very very unusual, is \(\displaystyle \xi\) the same thing as \(\displaystyle \Delta x\) and similarly \(\displaystyle \eta\) is \(\displaystyle \Delta y\) and \(\displaystyle \zeta \) is \(\displaystyle \Delta z\) ? And second question how he got those equations and what do they mean? I know something like this $$ \Delta f \left(x,y,z\right) = \frac{\partial f}{\partial x} \Delta x + \frac{\partial f}{\partial y}\Delta y + \frac{\partial f}{\partial z}\Delta z$$ which is quite understandable as \(\displaystyle \frac{\partial f}{\partial x} \Delta x = \Delta f~\textrm{in x direction}\) and similarly for other terms and hence adding all of them give us net change in \(\displaystyle f\). But what does those equations mean? What is the idea behind them?