For Michel Chasles (1793–1880), the geometry of old was "bristling with figures", and he already wished to break free from particular representations, which he felt imprisoned a problem. Many attempts have been made in this direction throughout the history of mathematics, and vector geometry is one of them.
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The allure of geometry without figures
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As early as the 17th century, although figures still loomed large in René Descartes's geometry, the mathematician and philosopher was already seeking to turn geometry into algebra by introducing coordinates. To solve a geometric problem, he recommended: "Of all the curved lines that are to be admitted into geometry, care must always be taken to choose the simplest one by which the problem can be solved." Pierre de Fermat took a similar approach, using loci to solve certain problems: "This gives the exact construction and the simplest possible solution of geometric problems by means of loci arising, as the case may be, from different kinds of curves suited to those problems." Soon afterwards, in a 1679 letter to Huygens, Gottfried Wilhelm Leibniz suggested "that we still need another, properly geometric, linear analysis, which expresses position directly as algebra expresses magnitude", and proposed a "geometry of position"; his idea, however, was not published until 1833.
Soon afterwards, Joseph-Louis Lagrange (1736–1813) stated in his treatise Mécanique analytique: "No figures will be found in this work. The methods I set out require neither constructions nor geometric or mechanical reasoning, but only algebraic operations." With his descriptive geometry, Gaspard Monge (1746–1818) likewise helped dispense with figures by transforming how they were represented, seeking, as he put it, to free himself "from that complexity of figures whose use distracts us from the attention owed to the substance of the ideas". Michel Chasles, though one of the greatest geometers of his time (see our feature in Tangente 160), needed no figure to write in his Traité de Géométrie Supérieure (1852): "Given three points a, b, c in any order on the same line […] we shall always have ab + bc + ca = 0, provided the segments are assigned the appropriate signs."
