Abstract 

This is a paper about problematic analysis in ancient Greek geometry. A problem is a request to construct a geometrical object with given properties. Problematic analysis is the first-part of a two-part method in ancient Greek geometric, known as ‘analysis and synthesis’. This paper seeks to explain and demonstrate problematic analysis, and to reveal a previously unrecognized role. Efforts to understand analysis and synthesis focus on it as a method of discovery. There is more to problematic analysis than discovering a means by which to construct an object with certain properties. Problematic analysis also contributes to generalization in at least two inter-related ways: First, it resolves a problem associated with generalization by ensuring that a range of figures, each of which is a possible configuration from a set of instructions, works with the same proof. Second, this first result contributes to a geometer’s efforts to study the range of possible solutions. It identifies seemingly different solutions as being variations of the same solution. The first contribution pertains to generalizing a result from a proof that works with a particular case in Synthesis. The second contribution pertains to diorism, and it contributes to diorism’s role of identifying the limits of a solution, and the number and arrangement of possible solutions.