Euclidean Construction for Dummies

Here is a simple explanation of the famous Euclidean Constructions:

Why didn’t Euclid just measure things with a ruler and calculate lengths? For example, one of the basic constructions is bisecting a line (dividing it into two equal parts). Why not just measure it with a ruler and divide by two?
One theory is the the Greeks could not easily do arithmetic. They had only whole numbers, no zero, and no negative numbers. This meant they could not for example divide 5 by 2 and get 2.5, because 2.5 is not a whole number – the only kind they had. Also, their numbers did not use a positional system like ours, with units, tens , hundreds etc, but more like the Roman numerals. In short, it was quite difficult to do useful arithmetic.
So, faced with the problem of finding the midpoint of a line, they could not do the obvious – measure it and divide by two. They had to have other ways, and this lead to the constructions using compass and straightedge or ruler. It is also why the straightedge has no markings. It is definitely not a graduated ruler, but simply a pencil guide for making straight lines. Euclid and the Greeks solved problems graphically, by drawing shapes instead of using arithmetic.

This website has lots of nice animations of compass/ruler drawings. This is a popular pedagogical tool for Architectural Geometry education today as it reflects the need for referential systems derived from pure geometric (formal) relations, not arithmetic ones.