Os critérios de aceitabilidade geométrica e a representação de curvas em La Géométrie de René Descartes

Abstract

This work analyses Descartes different criteria for geometrical acceptability and representation of curves, as they are found in his book La Géométrie, taking into account the historical and scientific contexts of the first half of the seventeenth century, when the work of Descartes was written. The main purpose is to find out what Descartes regarded as a sufficient representation of a curve; which ways of representing curves he used; and which curves were geometrically admissible or inadmissible, according to his selection criteria. This dissertation first discusses Descartes Jesuitic education and the influence of scholastic thinking over his thought. Next, it describes some important steps in the historical development of algebra and geometry, and the improvement of the algebraic notation from the late fifteenth century up to the appearance of Descartes Géométrie. The analysis of the Regulae ad Directionem Ingenii helped to elucidate the meaning of the constructive procedure of Cartesian geometry. It was found that Descartes classification of curves was a direct outcome from the general principles of the Cartesian analytic method, as it appears in the Regulae. Descartes did not explicitly characterize geometrical curves as those admitting algebraic equations. He used two criteria for geometrical acceptability of curves in the Géométrie, namely the algebraic criterion and the instrumental one, the latter being grounded on the use of instruments by which the curve could be traced. Nevertheless, Descartes was seemingly aware that the classification of curves according to the degree of their equations and the classification of geometrical problems according to the way they are built are not equivalent