The contributions of Carl Friedrich Gauss (1777-1855) are found in two brief unpublished letters and notes. He begins with a different definition of parallel.
Definition: Given a line 
and a point P not on , let AP be the perpendicular from P to . A line 
is parallel to through P if for any line 
with S in the interior of 
and 
, it follows that intersects .
So it would seem that as we consider the set of lines through the point P sweeping up from 
, the parallel is the first line not meeting . Unless we assume Euclid's Postulate V, we do not know that this line is unique. Thus it may depend on the side of we choose. Therefore, we speak of parallel lines in a direction. These lines will be called nonintersecting lines to distinguish them from parallel lines.
Theorem 11.8: In a given direction, being parallel is an equivalence relation.
Proof: We need to show three things:
is parallel to and S is any point between P and Q, then 
is parallel to .
is parallel to in a gven direction then is parallel to in the same direction..
is parallel to in a given direction and is parallel to 
in the same direction, then is parallel to in that direction.
and consider any line through S, 
, such that C is interior to and 
. Construct 
. By our assumption intersects . Thus, has entered a triangle, so by Pasch's Theorem it must intersect .
. Let S be a point in the interiors of 
and 
, and construct the line 
.