The third and fourth definitions in Euclid's Elements say:
- The ends of a line are points.
- A straight line is a line which lies evenly with the points on itself.
The fourth definition is usually regarded by modern mathematicians as meaningless or puzzling at best. A good definition we are told is that a straight line is the shortest distance between two points. However, Euclid's third, sixth and seventh definitions can provide some guidance on how to read the fourth definition.
- The edges of a surface are lines.
- A plane surface is a surface which lies evenly with the straight lines on itself.
I would argue the proper way to appreciate seventh definition is through the eye of a builder or surveyor. The straight line in this context is a builder's ideal straight edge which is used to gauge the flatness of a surface. As an ideal straight edge slides over and around the surface one looks from the edge of the surface for gaps between the edge and the surface. If no gaps are seen then the "plane surface lies evenly with straight lines on itself".
By analogy with the surface, the straight line in the fourth definition should be viewed from its end points. From this perspective if no part of the line lies away from the end points, then the line "lies evenly with the points on itself" which makes it a straight line.
Source for definitions: