Abstract

The Book I of Euclid’s Elements begins with three propositiones that ask for the solution of three problems. Unlike other propositiones, these do not assert properties or relationships between geometric objects. In them, some of these objects are assumed as given, and actions are demanded in order to obtain other objects. The solution to this type of problem is a construction, and its foundations can be found in the definitions and postulates of the Geometry of Euclid. Furthermore, each construction is normally accompanied by the resolution of a second problem. This consists in demonstrating that the previously offered solution is correct, i.e., that the presented construction indeed produces what was demanded in the formulation of the initial problem. The first theorem of the Elements is exactly the fourth proposition. We will consider, in a propaedeutic manner, the implications of adopting an approach to geometry where all propositions are interpreted as problems and the solution to any specific construction problem consists in offering a procedure such that, globally, the execution of this procedure on a workspace by a geometer will be simulated by an Euclid machine—which is the central object of analysis of the present article. From this perspective, the Geometry of Euclid would admit an interpretation in which it would be, in addition to a foundational milestone in the history of mathematics, also a foundational milestone in the development of the notion of effective procedure that would only fully come to light in the twentieth century.