Os pontos imaginários nas obras de Poncelet, Chasles e Laguerre

Abstract

The goal of this work is to study elaboration of imaginary points concept from the point of view of three authors: Jean-Victor Poncelet (1788-1867), Michel Chasles (1793-1880) and Edmond Nicolas Laguerre (1834-1866). Imaginary points notion concept appears at Poncelet’s works into his Saratoff manuscripts. Later, it is found on his memoir Essai sur les propriétés projectives des sections coniques [Pon-celet, 1820], presented to the Académie des Sciences de Paris and, with more emphasis, on his Traité des propriétés projectives des figures [Poncelet, 1822]. Poncelet considers, during his work, that all circles’ circumferences of a plane intersect each other at two points, the so-called imaginary points (cyclic points). This notion is seen as one of the consequences of the principle of projection and the principle of continuity. Chasles’ idea, at his work Géométrie supérieure [Chasles, 1852], is to give an interpre-tation of imaginary elements through real elements. Laguerre, along three short articles [Laguerre, 1852, 1853a, 1853b], when dealing with imaginary elements, uses his predecessors’ works to present a brilliant solution for use of figures and angles’ metrical properties.