apotelesmata

Apotelesmata is a term from ancient Greek mathematics used to designate construction problems—tasks that require creating or producing a geometric object to meet specified conditions, rather than proving a relation or theorem. The singular form is apotelesma; the plural is apotelesmata. The word derives from Greek roots meaning "that which is to be accomplished" or "a result to be achieved.

In the classical tradition, apotelesmata are the construction problems embedded within Euclid's Elements and other Greek mathematical writings. They stand in contrast to theorems (theoremata), which assert relationships that follow from axioms and previously proved results. An apotelesma typically requests the creation of a geometric configuration satisfying given constraints—such as drawing a line through a point parallel to a given line, or constructing a triangle with prescribed sides and angles—using a straightedge and compass. These problems were central to ancient geometry and served as a primary means of demonstrating geometric methods and principles.

The term was used by later Greek commentators and continued in medieval and Renaissance mathematical literature to distinguish geometry problems from theorems. In modern mathematics, apotelesmata are mainly of historical interest, but they remain a useful label for describing a particular category of ancient geometric tasks.

Etymology: from Greek apotelesma, "that which is accomplished" or "result." See Euclid's Elements and related commentaries in the tradition of ancient geometry.