## Line Geometry and Euclid proposition 4 and 8 in book I

Geometric formulae, word problems, theorems and proofs, etc.
Spacium Rex
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### Line Geometry and Euclid proposition 4 and 8 in book I

I was reading Euclids book and it is really good.

He starts with plane geometry and then he does solid geometry and after that I want to read some other peoples books about hyperspace geometry but plane geometry is 2 dimensional so that means he skipped 1 dimensional geometry. Is there a book which has axioms for geometry on a line?

Also I do not get proposition 4 and 8 because they use superposition but should that not be an axiom? Because superposition is just placing the triangles on each other and then seeing that they are the same but then you would have to measure them and both measuring and the triangles in real space would be imperfect and leading to errors because Euclidian space is more precise than anything that is earthy and mechanical.

nona.m.nona
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### Re: Line Geometry and Euclid proposition 4 and 8 in book I

Is there a book which has axioms for geometry on a line?
Possibly ("probably"?) not. However, some information is available, such as (this) online article. Most other resources appear to be quite advanced, usually treating one-dimensional "cases" of other topics, such as "projective foliations", etc.

Also I do not get proposition 4 and 8 because they use superposition but should that not be an axiom?[/quote]
You are not alone in making this argument. One scholarly article (link, starting on page 26) and an online "book" (link) concur. One might say that this is a philosophical question which, in a sense (since Euclid is long dead), must remain unanswered. We, in the present day and within modern mathematics, are left to develop our own, more rigourous, presentations.