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The problem of Pappus of Alexandria from the 4th century AD was a fundamental impetus for the birth of analytic geometry in the 17th century. This article aims to demonstrate the historical relevance of the problem, describe the Cartesian method for solving quadratic equations with one unknown, and illustrate its application in a modern problem involving four lines. This article adopts an applied deductive methodological approach. It begins with a historical overview of Pappus’s problem and its relationship to the discovery of analytic geometry. It then explains how Descartes solved a type of quadratic equation with one unknown using his innovative method, which was groundbreaking at the time. Subsequently, a modern problem involving four lines is solved, representing a contemporary interpretation of Pappus’s problem. This article is significant because it establishes the relationship between Pappus’s problem, the discovery of analytic geometry, and conic sections. Another important result is that the solution to the modern four-line problem is the set of points that form a conic section specifically, a hyperbola and an ellipse. In conclusion, Descartes’ solution to Pappus’s problem ensured both the procedure and the result of the modern four-line problem presented in this article. Pappus’s problem is considered fundamental to modern scientific development, as the method of combining algebra with geometry that solves it led to the emergence and advancement of analytic geometry, an essential branch for the development of science.
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