Irrotational ray systems

My question is regarding a famous publication by Sommerfeld and Runge, see the translation from the original German here 1. I am quoting in full the last paragraph on page 4 (corrected for an obvious typo $mathfrak S$ instead of $S$). In the quote below, $mathfrak S$ denotes the $textit{unit tangent vector field}$ of the rays. (For the special case of a rectilinear ray system discussed on page 4 the meidum is isotropic homogeneous, $mathit n =const$ and $rm{rot} mathfrak S =0$; for propagation in a general medium $rm{rot}(mathcal{n}mathfrak S) =0$.)

One can make the rotation character of the general ray bundle more intuitive thus: Once one has distinguished a “principal ray,” one considers the rays of the system that are infinitely close to it to be an “infinitely-thin bundle,” and marks the points at which a plane $E$ that is perpendicular to the principal ray, as well as a parallel plane $E’$ that is at an infinitely-small distance $delta$ from it, are met by the rays of the bundle. The associated points of $E$ and $E’$ are related by a general affine transformation. From the fundamental theorem of the kinematics of plane continua, it can always be decomposed into a deformation along two mutually-perpendicular directions (a transformation of an even
character in the coefficients) and a rotation (a transformation of an odd character). If one draws an infinitely-small circle in the plane $E$ around its intersection point with the principal ray then it will be converted into an ellipse by the deformation; this ellipse will be rotated by the rotation. The angular velocity – i.e., the infinitely-small rotation, divided by $delta$ – will now be equal to $frac{1}{2} rm{rot} mathfrak S$, which is similar to the vorticial velocity in hydrodynamics; the rotation must then be calculated at the midpoint of the plane $E$ (or $E’$ ) and represents the component of that vector along the principal ray […]

My question is about the sentence: "If one draws an infinitely-small circle in the plane $E$ around its intersection point with the principal ray then it will be converted into an ellipse by the deformation; this ellipse will be rotated by the rotation. The angular velocity – i.e., the infinitely-small rotation, divided by $delta$ – will now be equal to $frac{1}{2} rm{rot} mathfrak S$, […]"

Why is the angular rotation of the ellipse divided by $delta$ equals $frac{1}{2} rm{rot} mathfrak S$?