Skip to Main content Skip to Navigation
Theses

"Die Freude an der Gestalt" : methods, figures and practices in early nineteenth century geometry

Abstract : The standard history of nineteenth century geometry began with Jean Victor Poncelet's contributions which then spread to Germany alongside an opposition between Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. Our questions centre on how geometers distinguished methods, when opposition arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was "modern'' as claimed.We first examine Poncelet's argument that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. Our case study reveals visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coordinate equations. We also consider the debates involved as a medium for communicating geometry in which Poncelet and Gergonne in particular developed strategies for introducing new geometry to a conservative audience. We then turn to Plücker and Steiner. Through comparing their common research, we find that Plücker practiced a "pure analytic geometry'' that avoided calculation, while Steiner admired "synthetic geometry'' because of its organic unity. These qualities contradict usual descriptions of analytic geometry as computational or synthetic geometry as ad-hoc.Finally, we study contemporary French books on geometry and show that their methodological divide was grounded in student prerequisites, where "modern'' implied the use of algebra. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization.The standard history of nineteenth century geometry began with Jean Victor Poncelet's contributions which then spread to Germany alongside an opposition between Julius Plücker, an analytic geometer, and Jakob Steiner, a synthetic geometer. Our questions centre on how geometers distinguished methods, when opposition arose, in what ways geometry disseminated from Poncelet to Plücker and Steiner, and whether this geometry was "modern'' as claimed.We first examine Poncelet's argument that within pure geometry the figure was never lost from view, while it could be obscured by the calculations of algebra. Our case study reveals visual attention within constructive problem solving, regardless of method. Further, geometers manipulated and represented figures through textual descriptions and coordinate equations. We also consider the debates involved as a medium for communicating geometry in which Poncelet and Gergonne in particular developed strategies for introducing new geometry to a conservative audience. We then turn to Plücker and Steiner. Through comparing their common research, we find that Plücker practiced a "pure analytic geometry'' that avoided calculation, while Steiner admired "synthetic geometry'' because of its organic unity. These qualities contradict usual descriptions of analytic geometry as computational or synthetic geometry as ad-hoc.Finally, we study contemporary French books on geometry and show that their methodological divide was grounded in student prerequisites, where "modern'' implied the use of algebra. By contrast, research publications exhibited evolving forms of geometry that evaded dichotomous categorization.
Document type :
Theses
Complete list of metadata

https://tel.archives-ouvertes.fr/tel-01158895
Contributor : ABES STAR :  Contact
Submitted on : Tuesday, June 2, 2015 - 9:52:06 AM
Last modification on : Wednesday, January 12, 2022 - 3:47:17 AM
Long-term archiving on: : Tuesday, September 15, 2015 - 9:21:55 AM

File

these_archivage_3163795o.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01158895, version 1

Citation

Jemma Lorenat. "Die Freude an der Gestalt" : methods, figures and practices in early nineteenth century geometry. General Mathematics [math.GM]. Université Pierre et Marie Curie - Paris VI; Simon Fraser university (Burnaby, Canada), 2015. English. ⟨NNT : 2015PA066079⟩. ⟨tel-01158895⟩

Share

Metrics

Record views

351

Files downloads

1307