Visualizing geometry with pictures, symbols and words Jemma Lorenat | Université Pierre et Marie Curie Simon Fraser University! !!! [email protected] [email protected] ! “Just as the entire mode of existence of human collectives changes over long historical periods, so too does their mode of perception.” ! Walter Benjamin, Work of Art in the Age of Mechanical Reproduction (1936) “In a word, to translate into Algebra the questions of Geometry, and reciprocally, to translate into Geometry the results obtained by Algebra, that is the goal that we propose in the APPLICATION OF ALGEBRA TO GEOMETRY.” Pierre Louis Marie Bourdon, Application de l’algèbre à la géométrie (1825) Geometry in books Geometry in research Visualizing geometry with Poncelet’s figures pictures, symbols and words: Poncelet’s view Pure analytic geometry Representation and perception of geometry in Another form of purity France and Germany during the first third of the Conclusions nineteenth century Geometry in books Geometry in research Visualizing geometry with Poncelet’s figures pictures, symbols and words: Poncelet’s view Pure analytic geometry Representation and perception of geometry in Another form of purity France and Germany during the first third of the Conclusions nineteenth century Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions French publications, illustrations, definitions of geometry, textbooks Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Sylvestre François Lacroix, Application de l’algèbre à la géométrie (1807) Geometry in books “GEOMETRY takes its name from the principal Geometry in research usage for which it seems that this science was originally employed, that is, land measurement, Poncelet’s figures because this word signifies the art of land measuring. Poncelet’s view But then we have applied it to anything which Pure analytic geometry concerns bodily extension: so geometry has become Another form of purity the science of extension.” Conclusions G. F. Olivier, Géométrie usuelle (1835) Geometry in books “GEOMETRY takes its name from the principal Geometry in research usage for which it seems that this science was originally employed, that is, land measurement, Poncelet’s figures because this word signifies the art of land measuring. Poncelet’s view But then we have applied it to anything which Pure analytic geometry concerns bodily extension: so geometry has become Another form of purity the science of extension.” Conclusions G. F. Olivier, Géométrie usuelle (1835) “In each science there are things which cannot be Geometry in books taught, and that the student must acquire by himself; Geometry in research this is the habit of the procedures of science, or Poncelet’s figures otherwise the mechanism of the operations it Poncelet’s view prescribes: in arithmetic and in algebra these are Pure analytic geometry calculations, in geometry, constructions.” Another form of purity Sylvestre François Lacroix, Essai sur l’enseignement en général Conclusions et sur celui des mathématiques en particulier (1805) “In each science there are things which cannot be Geometry in books taught, and that the student must acquire by himself; Geometry in research this is the habit of the procedures of science, or Poncelet’s figures otherwise the mechanism of the operations it Poncelet’s view prescribes: in arithmetic and in algebra these are Pure analytic geometry calculations, in geometry, constructions.” Another form of purity Sylvestre François Lacroix, Essai sur l’enseignement en général Conclusions et sur celui des mathématiques en particulier (1805) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Jean-Victor Poncelet, Traité des propriétés projectives des figures (1822) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Pierre Louis Marie Bourdon, Application de l’algèbre à la géométrie (1825) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions M. H. Vernier, Géométrie élémentaire (1830) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Sylvestre François Lacroix, Élémens de géométrie (1799) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Journal articles, spiny geometry, generality of algebra Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions “Ancient geometry bristled with figures. The reason for this is simple. Because then they lacked general Geometry in books and abstract principles, each question could only be Geometry in research treated in a concrete manner, on the same figure which was the subject in question, and about which Poncelet’s figures the only view could discover the elements necessary Poncelet’s view for the proof or the solution sought.” Pure analytic geometry Another form of purity Michel Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837) Conclusions “Ancient geometry bristled with figures. The reason for this is simple. Because then they lacked general Geometry in books and abstract principles, each question could only be Geometry in research treated in a concrete manner, on the same figure which was the subject in question, and about which Poncelet’s figures the only view could discover the elements necessary Poncelet’s view for the proof or the solution sought.” Pure analytic geometry Another form of purity Michel Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837) Conclusions Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions L. M. P. Coste, Propriétés peu connues de la parabole, et construction de cette courbe au moyen de quatre conditions données (1818) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Jean Baptiste Durrande, “Théorie élémentaire des contacts des cercles, des sphères, des cylindres et des cônes” (1820) “We believe it superfluous to accompany this memoir with figures, often more overwhelming than useful in the geometry of space; figures that we Geometry in books could besides only present in a unique and Geometry in research individual aspect to the reader, who could, instead, Poncelet’s figures construct and fashion them to his taste, if he judges Poncelet’s view the assistance at all necessary. We are concerned, Pure analytic geometry indeed, only with logical deductions, always easy to Another form of purity follow when the notations are chosen in a Conclusions convenient manner.” Joseph-Diez Gergonne, “Considéraons philosophiques sur les élémens de la science de l’étendue” (1826) “We believe it superfluous to accompany this memoir with figures, often more overwhelming than useful in the geometry of space; figures that we Geometry in books could besides only present in a unique and Geometry in research individual aspect to the reader, who could, instead, Poncelet’s figures construct and fashion them to his taste, if he judges Poncelet’s view the assistance at all necessary. We are concerned, Pure analytic geometry indeed, only with logical deductions, always easy to Another form of purity follow when the notations are chosen in a Conclusions convenient manner.” Joseph-Diez Gergonne, “Considéraons philosophiques sur les élémens de la science de l’étendue” (1826) Geometry in books Geometry in research Poncelet’s figures Poncelet’s view Pure analytic geometry Another form of purity Conclusions Annales de mathématiques pures et appliquées (1820) Geometry in books “This fault of ancient Geometry was one of the relative advantages of analytic Geometry [...] One Geometry in research must ask, thus, if there was not also, in pure and Poncelet’s figures speculative Geometry, a manner of reasoning without Poncelet’s view the continual assistance of the figures, whose real Pure analytic geometry inconvenience, even when their construction is easy, is Another form of purity at least to tire the spirit and exhaust the mind.” Conclusions Michel Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837) Geometry in books “This fault of ancient Geometry was one of the relative advantages of analytic Geometry [...] One Geometry in research must ask, thus, if there was not also, in pure and Poncelet’s figures speculative Geometry, a manner of reasoning without Poncelet’s view the continual assistance of the figures, whose real Pure analytic geometry inconvenience, even when their construction is easy, is Another form of purity at least to tire the spirit and exhaust the mind.” Conclusions Michel Chasles, Aperçu historique sur l’origine et le développement des méthodes en géométrie (1837) “In geometry [...] the figure is described, one never Geometry in books loses it from view, one always reasons about Geometry in research magnitudes, real and existent forms, and never Poncelet’s figures reaches consequences that cannot be painted in the Poncelet’s view imagination or in sight, by sensible objects;