Can Euclid prove that a straight line is the shortest distance between two points?

Question

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?

Answer 1

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 $f(b)$.

Answer 2

This question fundamentally comes down to "How do you measure the length of a curve?"

Classical Greek mathematics doesn't really have the ability to either satisfactorily ask or answer this question. Archimedes could describe the lengths of circular arcs but, according to the linked article, the fact that a straight segment is the shortest path between two points is taken as axiomatic; there are probably other classes of curves which the Greeks could compute the lengths of, but the general problem was well beyond their mathematics. Indeed,

• even Archimedes' argument about the value of $\pi$ relies on an infinitesimal argument, which is more akin to calculus than the rest of Greek mathematics, and
• the Greeks didn't really even have a notion of "general curve"—they were comfortable with straight lines and conic sections, but didn't do a great deal beyond these somewhat elementary curves.

Basically, the question doesn't make sense in the context of Greek mathematics; and even if it did, the tools to answer the question in a rigorous way simply don't exist.

In modern mathematics, we have a more precise definition of distance: if $\gamma : [a,b] \to \mathbb{R}^n$ is some curve which is "nice enough" (injective and continuously differentiable, perhaps—this is basically what is required for a "rectifiable" curve; more general notions exist, but I don't think that these really conform to the usual notion of what a "path" is), then the length of this curve is given by something like $$ \operatorname{Length}(\gamma) = \lim_{\max_j \Delta t_j \to 0} \sum_j \left|\frac{\gamma(t_j) - \gamma(t_{j-1})}{\Delta t_j}\right| \Delta t_j = \int_{a}^{b} |\gamma'(t)|\,\mathrm{d}t, $$ where $\{a=t_0<t_1<\dotsb<t_n=b\}$ is some partition of $[a,b]$, and $\Delta t_j = t_j - t_{j-1}$. See, for example, arc length on MathWorld. This formula comes from approximating $\gamma$ with very short line segments, and then taking a limit as the "mesh" of those segments goes to zero (basically, the length of the longest segment is taken to zero).

The argument that a straight line segment is the shortest path between two points then comes down to basically two observations:

1. The triangle inequality: if $A$, $B$, and $C$ are three points, then $$ d(A,C) \le d(A,B) + d(B, C). $$ From the triangle inequality, it can be deduced that any piecewise linear path from $A$ to $C$ is as least as long as the straight line segment.

2. Inequalities behave "nicely" with respect to limits: if $a_n \le b_n$ for all $n$, then $\lim_{n\to\infty} a_n \le \lim_{n\to\infty} b_n$. Thus \begin{align} &d(\gamma(a),\gamma(b)) \le \sum_{j} \left| d(\gamma(t_j), \gamma(t_{j-1}) \right| \\ &\qquad\implies d(\gamma(a), \gamma(b)) \le \int_{a}^{b} |\gamma'(t)|\,\mathrm{d}t = \operatorname{Length}(\gamma). \end{align}

Answer 3

I see that neither of the other answers mentioned the Pythagorean theorem, which seems to me to be the main point here. If one has a polygonal curve connecting a pair of points $A$ and $B$, each side of the polygon can be included as the hypotenuse of a right-angle triangle one of whose sides is parallel to $AB$. Since the hypotenuse is longer than each of the sides by the Pythagorean theorem, it follows that the curve is longer than the segment $AB$. Exhaustion-type arguments when a curve is approximated by a polygon were certainly familiar to the ancient Greeks. If one doesn't use the Pythagorean theorem, one won't be able to prove that it is the shortest, rather than merely a shortest. For example, in $\ell^\infty$ there are many minimizing paths joining a pair of points.