Euclid's theory of magnitudes

Question

Euclid based much of his geometry on a theory of magnitudes that looks roughly like this:

1. A general theory of whole and part and how they are related in size (eg: the whole is greater than the part).
2. A general theory of the properties of magnitudes (eg: equals added to equals are equal).
3. A basic rule that allows one to determine in specific cases that one magnitude is equal to another (eg: all radii of a circle are equal; all right angles are equal).

Euclid applied this theory to lines, angles, and figures (meaning the area). From these simple foundations, he is able to prove the equivalence of all sorts of things that are not related by the basic rule alone.

I've been looking for a modern development of this theory, and though there is some interest in it (Robering), modern work on mereology and mereotopology that I've been able to find online all does things that seem to violate the spirit of Euclid's work; in particular:

1. They treat points as parts of lines and lines as parts of the plane (Robering comments on this). Euclid did not do that. In Euclid the parts of a line are lines and the parts of a plane figure are plane figures.
2. They assume a single universal whole of which everything is a part. This is related to the first issue. Obviously there isn't a single universal line of which all lines are a part. Similarly for angles.
3. Also related to the first issue: the works on mereotopology make use of the concept of an interior part, which basically means, for example, the part of a line segment not including the endpoints or the plane figure excluding the boundary. This doesn't makes sense when points aren't parts of lines and lines aren't parts of plane figures.
4. The works on mereology assume that any two objects form a whole. Euclid only considered connected wholes.
5. They assume completed infinities. Euclid avoided those.
6. Robering treats an angle as the infinite plane section enclosed by the rays of the sides of the angle. I assume this is to turn angles into parts of the plane so they can be part of the universal whole.
7. They assume that there is no maximum-sized whole, but this doesn't work for angles if you treat them as Euclid does as their own kind of magnitude.

Can anyone suggest a body of work that I can find online that deals with some of these issues?