From Euclid's definitions, postulates, and common notions, can you prove that a straight line is the shortest distance between two points, or is that basically an assumption of the way lines are measured?
Here is an online copy of much of the text of Euclid's Elements.
Proposition 20 is:
In any triangle the sum of any two sides is greater than the remaining one.
This does prove the theorem for the case where one straight line is shorter than two straight lines at an angle, and it's obvious how to prove from that that any chain of straight lines is longer than a single straight line, but I don't see anything that rules out that another sort of curve might be shorter. Maybe you could prove it from proposition 20 using the method of exhaustion?
What about modern formulations of Euclidean geometry? Do any of them make it a theorem rather than an axiom that the shortest distance is a straight line?
Best Answer
There are certain things that you have to consider. First of all, the shortest distance among what paths? You have to note that planar paths defined by arbitrary functions don't necessary exist in classical Euclidean geometry. For example, the path $ x(t) = t , y(t) = \exp(t) $ doesn't exist in classical Euclidean geometry, but our objects are lines, line segments, circles, angles, and things like that. If you consider paths consisting of $n$ line segments, you can prove your claim by proposition 20 and induction on $n$.
Secondly, even if you allow arbitrary paths defined by functions from $\mathbb{R} \to \mathbb{R}^2$ in your system, before proving your claim, you have to come up with a plausible definition of the arc length. What does it really mean when we say that some curve is longer than another? One agreed upon definition of the arc length is as follows:
$$f: \mathbb{R} \to \mathbb{R}^2,\ f(t) = ( f_1(t), f_2(t)),\ a \leq t \leq b.$$
Consider $\Sigma = \{P\mid \text{$P$ is a partition of $[a,b]$}\}$, suppose $P_0 \in \Sigma$ and that $P_0 = \{x_0, x_1, x_2,\dotsc, x_n \}$, where $x_0=a$ and $x_n=b$. Define $\Gamma(P_0) = \sum_{k=1}^{n}|f(x_k)-f(x_{k-1})|$. Now, the arc length can be defined to be $\Gamma = \sup \{ \Gamma(p) \mid P \in \Sigma\}$, provided that it exists.
So, we intuitively define the arc length to be the supremum of all possible finite line segments following each other from $f(a)$ to $f(b)$. By this definition and by the proposition 20 and proving the case for $n$ line segments, it easily follows that the shortest path is the straight line segment from $f(a)$ to $f(b)$.