# Higher dimensional euclidean geometry books recommendation

I've recently finished reading (and doing exercises of) Kieselev's book on planimetry (2D Euclidean geometry) and started his second book on stereometry (3D analogue) so I was wondering if there exists similar works on higher Euclidean geometry (with 4 and more dimensions). I'd like to read a book which starts giving very elementary definitions (like points and line) and then proceeds step by step to prove fundamental theorems; a treatment of higher geometry in a sort of Greek fashion, with postulates and almost without coordinates, used at least to solve some particular problem and not to treat the whole argument.

I know that there is the obstacle in representing or even imaging objects in more than 3D but I'd prefer this kind of treatment instead of the simple "let's see what happens if we generalize this to many dimensions"

The problem is that it is difficult to make geometry rigorous if you use the 'greek style', instead of using $\mathbb{R}^n$. For example, already in his first proposition, Euclid assumes that if you have a line segment, draw two circles with the length of the line segment as radius and as centers the endpoints of the line segment respectively, then these two circles cross at at least one point. From a picture this is obvious. However, proving this requires some sort of intermediate value theorem. For 2D and 3D euclidean geometry this is not a big problem, since we could add things that are obvious from the drawing as an axiom or something like that, or even sloppier, use the drawing as 'proof'. But what kind of facts are 'obvious' for 4D or nD spheres and other geometric objects? Our intuition fails a little, this is why the coordinate approach is more rigorous. We need to define what space is before proving stuff about things in space.