what is length in euclidean geometry?

Question

When we say an equilateral triangle with side length 3, what do we mean by length?

Can we define length in euclidean-geometry?

Answer 1

This is actually a somewhat subtle question, depending on what you mean by "Euclidean geometry." If you mean geometry as Euclid did it, then as far as I know Euclid doesn't work directly with a notion of length, only with a notion of relative length.

Here's a clean way to set things up: two line segments $L_1, L_2$ have the same length if it's possible to rotate and translate one so that it coincides with the other. Two line segments $L_1, L_2$ are in relative proportion $p : q$, for positive integers $p$ and $q$, if it's possible to stack $q$ copies of $L_1$ together and $p$ copies of $L_2$ together such that they coincide with each other. If you fix a "measuring stick" line segment $L$, you can now use it to measure the length of any other line segment by stacking copies of them together and checking whether the corresponding lengths match. Formally this defines the length of any line segment whose length is a rational multiple of the length of $L$.

In particular when we say a line segment has length $3$ what we are really saying is that it's in relative proportion $3 : 1$ with some "measuring stick" line segment $L$ that we've fixed.

(You start to get a sense of what disturbed the Pythagoreans so much about irrational numbers: if two lines are in irrational proportion with respect to each other then it's impossible to establish their relative proportion by stacking copies of them and matching them up to each other. Scary!)

If you define Euclidean geometry by working explicitly in Cartesian coordinates, as is now standard, then the "measuring stick" line segment is, for example, the line segment from $(0, 0)$ to $(1, 0)$ in the Cartesian plane $\mathbb{R}^2$, but this gets disguised when we define the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ as $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$. This definition works (it must work by the Pythagorean theorem) and is the standard way to set things up formally but it sweeps some stuff under the rug. Other more axiomatic approaches are possible, e.g. Hilbert's axioms.