The second and third propositions work on a construction transferring the line segment BC onto the ray AD. This means that the goal is to construct a point H that lies on AD such that the length of AH is equal to the length of BC.  In Euclidean geometry, the existence of these propositions might seem pointless, because if you were to try and transfer a line segment onto a ray using a ruler and compass, you could just line the compass onto the original line segment, and move the compass. However, in hyperbolic geometry, it's not that easy to transfer a line segment onto a ray, since you can’t just “move the compass.”

 

Instead, Euclid first uses proposition 1 to construct an equilateral triangle off AB, calling the resulting point E. The line segments EA and EB are then extended into rays. The next step is to construct a circle with center B and radius BC, and identifying the intersection this makes with EB as a new point G. Another circle is then constructed, with center E and radius EG. The intersection between this circle and ray EA is labeled L. This is where Euclid decides to end the second postulate and begins the third. What Euclid has done is that he has transferred BC over to A, resulting in the line segment AL, but this doesn't necessarily lie on AD. It's not too complicated to verify that the length of BC is indeed equal to the length of AL. As a final step, the circle with center A and radius AL is constructed, and it's intersection with ray AD is called H.