Let us begin a "nostalgia" series with pencils of lines, a somewhat old-fashioned topic that is no longer taught or widely studied. We should not forget that this concept from affine geometry, which can be interpreted analytically, can yield constructive methods and welcome simplifications.
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Desargues: innovation and modernity
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Less well known than other geometers such as Pascal or Descartes, Girard Desargues (1591–1661) took an innovative approach to geometry. In 1639, in his Brouillon project d'une atteinte aux événemens des rencontres d'un cône avec un plan, a pamphlet of just thirty pages, he defined an "ordonnance de lignes droites" as a family of lines that were either all concurrent or all parallel: pencils of lines were born. Desargues even ventured quite far into modernity by introducing a point at infinity for each line and treating it no differently from points at finite distance, thereby establishing projective geometry de facto.
