Given only a compass and straightedge, Greeks believed that many constructions were impossible to complete.

The Greeks, using only a compass and straightedge, believed that many constructions were impossible to complete. This belief was based on their understanding of geometry and the limitations of these tools.

Step 1: Understanding the Tools A compass is a tool for drawing circles or arcs. A straightedge is a tool for drawing straight lines. These are the only tools allowed in classical Greek geometry.

Step 2: Understanding the Limitations The Greeks discovered that there were certain geometric constructions that seemed impossible with these tools. For example, they could not trisect an arbitrary angle or construct a square with the same area as a given circle (a problem known as squaring the circle).

Step 3: Proving the Impossibility It was not until many centuries later, with the development of algebra and the concept of irrational numbers, that mathematicians were able to prove definitively that these constructions are indeed impossible with only a compass and straightedge.

So, the Greeks' belief was correct, but they lacked the mathematical tools and concepts to prove it.