a collapsing compass
Imagine a pair of compasses so floppy that the moment you lift the pencil tip off the paper, the legs snap shut. You can draw a circle as long as both points stay on the page, but you cannot carry a fixed gap from one place to another the way you would with a stiff modern compass. This is the tool Euclid actually assumed, and it sounds much weaker than what sits in a school pencil case.
Formally, a collapsing compass can do exactly one thing: given two points A and B already marked, draw the circle centred at A passing through B (radius |AB|). The instant it leaves the paper the radius is forgotten, so you may NOT set it to |AB| at A and then go draw a circle of that same radius somewhere far away around an unrelated point C. The straightedge, meanwhile, is unmarked: it only draws the line through two given points, never measures.
The surprise is that this crippled compass is no weaker at all. Euclid's very first propositions show that with a collapsing compass you can still copy any length to any new spot (that is exactly what copying-a-segment achieves). So mathematicians speak of the 'compass equivalence theorem': everything a rigid compass can construct, a collapsing one can too. Because of this, all later constructions are described as if the compass were rigid, knowing the collapsing version could reproduce them.
Mark two dots A and B. With the collapsing compass you draw the circle centred at A through B and the circle centred at B through A. Lift the compass, and the radius is gone — yet by Euclid's Proposition I.2 you can still transfer the length |AB| to a third point.
The compass forgets its radius the moment it leaves the paper, yet loses no constructive power.
A common misconception is that the collapsing compass is genuinely weaker than a modern one — it is not. The compass equivalence theorem proves they construct exactly the same set of points.