In retrospective of history of mathematics, I am trying to reconstruct the answers to following fundamental questions:
- Who proved first in a certain geometry that "The shortest distance between two points is a straight line."? Please provide proof or give precise reference thereto.
- Is the statement "The shortest distance between two points is a straight line." within an Euclidean geometry an axiom or a theorem?
- Does Euclidean geometry (and/or any other geometry) define (per axiom), what a "point", what a "line", and what a "straight line" to be?
Remarks:
- The statement above could have been of course formulated over time in different ways, for instance instead of the notion "distance" I also recognized "path" and "curve".
- Precise references, explanation and proofs needed, non-trivial answers appreciated.
Thank you in advance for your support.
In his On the Sphere and Cylinder I, Archimedes gives some definitions, and then some assumptions. His first assumption reads (translated into English):
For "extremities," read "endpoints"; for "least," read "shortest." The notion that a straight line is the shortest distance between two points is often ascribed to Archimedes on the basis of this assumption. Given that, I suppose we might call this an axiom, rather than a theorem, and therefore it would not admit of a proof.
In Euclidean geometry, terms such as "point" and "line" are often called "undefined," meaning that they are defined by their behavior in the axioms, rather than explicitly in some kind of glossary.
The idea of a "straight line" (as opposed to a line that is not straight) on the other hand, appears to me to have been assumed as obvious by many of the ancient Greeks. For instance, Archimedes (in the same work) uses "straight line" in a few of the definitions, but nowhere defines it or states what conditions it must satisfy. Euclid says in Definition 4 of his Elements that
I think you'll agree that this is not exactly satisfying as a definition.