Paul Gordan's theses were published in <A HREF="https://books.google.nl/books?id=9u09AAAAcAAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false">De linea geodetica</A> and digitised by Google, from which I reproduce the relevant page:

<IMG SRC="https://ilorentz.org/beenakker/MO/Gordan.png"/>

Translation:

 - I. The method of <A HREF="https://en.wikipedia.org/wiki/Cournot_competition">functional division,</A> proposed by the respectable <A HREF="https://en.wikipedia.org/wiki/Antoine_Augustin_Cournot">Cournot</A> as an analytical and empirical method, appears unsuitable, since one does not have fully empirical functions.
 - II. The method of the infinitely-small is, I claim, no less
   precise than the method of limits. 
 - III. It is of greater interest to
   investigate the implicit properties of a function defined by a
   differential equation than to investigate in terms of which known
   functions it can be expressed. 
 - IV. The principles of Democritus
   remain to the present day as the foundation of the theory of atoms.

One can also learn from this publication that the opponents of Gordan at the March 1, 1862 *Disputatio* were Dr. Kretschmer, Mr. Geiser, and Mr. Rathke, presumably <A HREF="https://www.genealogy.math.ndsu.nodak.edu/id.php?id=141906">E.E. Kretschmer</A> and <A HREF="https://www.genealogy.math.ndsu.nodak.edu/id.php?id=103791">C.F. Geiser</A>. Gordan acknowledges as his teachers in mathematics the professors <A HREF="https://en.wikipedia.org/wiki/Ernst_Kummer">Kummer,</A> <A HREF="https://en.wikipedia.org/wiki/Ferdinand_Joachimsthal">Joachimstal,</A> <a href="https://en.wikipedia.org/wiki/Friedrich_Julius_Richelot">Richelot,</a> <A HREF="https://en.wikipedia.org/wiki/Johann_Georg_Rosenhain">Rosenhain,</A> <A HREF="https://en.wikipedia.org/wiki/Heinrich_Schröter">Schroeter,</A> and <A HREF="https://en.wikipedia.org/wiki/Johann_Gottfried_Galle">Galle</A> --- but it is unclear whether any of these were present at the defense ceremony.

Concerning the question "what was actually said at the defense"; at least in my University (Leiden) the practice through the centuries has been that the examination itself is not recorded, the secretary records only the outcome (pass/fail/cum laude etc.).