Ruler and Compass Construction
You might have noticed that Euclid’s five axioms don’t contain anything about measuring distances or angles. Up to now, this has been a key part of geometry, for example to calculate areas and volumes.
However, at the times of Thales or Euclid, there wasn’t a universal framework of units like we have today. Distances were often measured using body parts, for example finger widths, or arm lengths. These are not very accurate and they vary for different people.
To measure longer distances, architects or surveyors used knotted cords: long pieces of string that contained many knots at equal intervals. But these were also not perfectly accurate, and different string had the knots placed at slightly different distances.
Greek mathematicians didn’t want to deal with these approximations. They were much more interested in the underlying laws of geometry, than in their practical applications.
That’s why they came up with a much more idealised version of our universe: one in which points can have no size and lines can have no width. Of course, it is
Euclid’ axioms basically tell us what’s possible in his version of geometry. It turns out that we just need two very simple tools to be able to sketch this on paper:
A straight-edge is like a ruler but without any markings. You can use it to connect two points (as in Axiom 1), or to extend a line segment (as in Axiom 2).
A compass allows you to draw a circle of a given size around a point (as in Axiom 3).
Axioms 4 and 5 are about comparing properties of shapes, rather than drawing anything. Therefore they don’t need specific tools.
You can imagine that Greek mathematicians were thinking about Geometry on the beach, and drawing different shapes in the sand: using long planks as straight-edge and pieces of string as compass.
Even though these tools look very primitive, you can draw a great number of shapes with them. This became almost like a puzzle game for mathematicians: trying to find ways to “construct” different geometric shapes using just a straight-edge and compass.
The Greek Mathematician