# The Axiomatization of Projective Geometry

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The Axiomatization of Projective Geometry
Philippe NABONNAND

Objects + Relations between objects
At the end of 19th century, an axiomatized theory involves: Objects + Relations between objects It must be given primitive objects and primitive relations between objects. All other objects and relations have to be constructed or defined. “Le point de départ nécessaire d’une théorie déductive quelconque est un système de symboles non-définis et un système de propositions non-démontrées.” [Padoa 1901, 317]

Poncelet : How to obtain general theorems from a synthetic point of view. A geometry of transformation of figures (continuity principles, projection principles). The study of projective properties of figures. A main intention : to obtain a “good” (without using coordinates and which is an example of the study of projective properties of figures) presentation of theory of conics.

Steiner A geometry of fundamental forms (line, plane, bundle) and their projective correspondences. A projective correspondence between lines is defined as a correspondence which preserves the anharmonic ratio. Geometry as study of mutual dependence of figures. Main intentions : to obtain a “good” (which depends of the notion of projective correspondence) presentation of theory of conics and to organize the corpus of theorems of geometry.

To organize the geometrical corpus:
“Durch gehörige Aneignung der wenige Grundbeziehungen macht man sich zum Herrn des ganzen Gegenstandes; es tritt Ordnung in der Chaos ein, und man sieht, wie alle Theile naturgemäss in einander greifen, in schönster Ordnung sich in Reihen stellen, und verwandte zu wholbegrenzten Gruppen sich vereinigen.” [Steiner 1832, I-4]

“Die so eben aufgestellten neuen Sätze über den Kegel zweiten Grades und dessen Schnitte sind für die Untersuchung dieser Figuren wichtiger, als alle bisher bekannten Sätze über dieselben, denn sie sind die eigentlichen währen Fundamental sätze, weil sie nämlich so umfassend sind, dass fast alle übrigen Eigenschaften jener Figuren auf die leicheste und klarste Weise aus ihnen folgen, und weil auch die Methode, nach der sie daraus hergeleitet werden, jede bisherige Betrachtungweise an Einfachheit und Bequemlichkeit übertrifft.” [Steiner 1832, II-12-13]

B A

Main tool: Projective correspondences between forms defined from metrical considerations.
General theorems concern projective relations between forms and theorems about triangles, quadrangles or geometrical figures are corollaries. Theory of conics: Figures are given from considerations about projective correspondence. Other theories like duality, polarity, involution are subordinated to theory of conics.

Von Staudt: Systemizing Steiner’s point of view.
A systematic presentation of incidence properties of forms with the introduction of the expressions “at the infinite”. Main intentions: “Ich habe in dieser Schrift versucht, die Geometrie der Lage zu einer selbstständigen Wissenschaft zu machen, welche des Messens nicht bedarf.” [Staudt 1847, III]

The fundamental theorem of projective geometry as principal tool.
to obtain a “good” presentation of theory of conics (as a consequence of the theory of projective correspondences between forms without any reference to the notion of mesure), to organize the corpus of theorems of geometry to give a geometrical theory (die Geometrie der Lage) with no reference to mesure which is anterior to classical metric geometry. A purely incidental definition of projective correspondences between forms as bijective correspondences which preserve harmonicity. The fundamental theorem of projective geometry as principal tool. To study projective correspondences, in particular involutions.

Main objects: fundamental forms
Main tool: Involution defined as particular projective correspondence. A theory of projective correspondence between forms of same kind. Theory of polarity as an example of study of involutive correspondence between forms of second (or third) kind Conics defined as double line of polarities. In Beiträge, elementary forms as main objects.

Klein Discussion about parallel axiom and continuity axiom (Staudt’s Lücke). Limited extension of geometrical objects. A quite large discussion (Lüroth, Zeuthen, Darboux, Schur, …) about continuity axiom(s). Geometry meets Foundations of Analysis (Dedekind’s axiom of continuity). With this discussion, the importance of questions of order. (Some considerations about order of elements and separation in Staudt’s theory). The question of Archimedian axiom: Schur’s projective version of Archimedian axiom is sufficient to prove fundamental theorem.

The discussion of von Staudt’s proof
The density of harmonic sequences (Klein, Lüroth, Zeuthen, Thomae, Schur, Hölder, …) The non-periodicity of harmonic sequences (Fano). Incidence and order assumptions are not sufficient to assure the infinity of harmonic sequences Examples of Fano are interpreted by Veblen as finite geometries

M. Pasch The first rigorous (contemporary sense) presentation of foundations of projective geometry A first part to construct the “Stammbegriffe” and “Stammsätze” from “Grundbegriffe” and “Grundsätze”. An axiomatic presentation: “Die Lehrsätze werden aus den Grundsätzen deducirt, so dass alles, was zur Begründung der Lehrsätze gehört, ohne Ausnahme sich in der Grundsätzen niedergelegt finden muss.” [Pasch, 1882, 5]

A second part devoted to projective geometry
A second part devoted to projective geometry. A proof of fundamental theorem which use some considerations of congruence. Pasch put his development in Klein’s perspective to found projective geometry independently from metrical geometry. He introduced some axioms which assure the possibility of congruence. From Klein’s point of view, it is not so shocking. But Klein did not give his approbation.

M. Pieri To organize the corpus of theorems in an hypothetico-deductive system. To distinguish primitive propositions from derived propositions. To classify notions which are concerned by these propositions; “classer les notions sur lesquelles portent ces jugements, en discernant par là les idées-mères, primitives ou indécomposées.” [Pieri 1901, 378]

Many systems with more or less primitive objects
Many systems with more or less primitive objects. For example : in i principii della geometria di posizione composti in sistema logico deduttivo [1897], only two primitive objects: points and junction (congiungente) of two points as a set of points between these two points. From the axioms about these primitive objects, Pieri deduced the notion of alignment and lines and obtained with the theorem of complete quadrangular, the notion of harmonicity.

The notion of segment is derived:
“Premesso che a, b, c son punti d’una retta prj r l’un l’altro distinti, il segmento projettivo abc – rappresentato dal segno “(abc)” – non è altro che il luogo d’un punto prj x, a cui si può coordinar sulla r un qualche punto y diverso da a e da c, per modo che x sia l’armonico dopo y, Arm(a,c,y) e b.” [Pieri 1897, 24] A point by point transformation between two lines is “segmentaria” if it is bijective and if it respects segments.

With the 18th postulat which is an projective analogous to Dedekind’s axiom of continuity, Pieri proved that harmonic transformation are “segmentaria” and could prove fundamental theorem like Schur or Thomae. (The main point is that if two pairs of points do not separate themselves, there exist a pair of points that harmonically separate these two pairs – Darboux’s lemma)

Another way is to consider the notion of order: Enriques, Whitehead, Hilbert, …
Pieri argued that the notion of segment is more primitive than those of order : “[il] concetto di ordinamento naturale dei punti d’una retta […] implica di già una corrispondenza fra punti e classi di punti.” [Pieri 1894, 626]

Axiomatization of a field is also organization of this field.
The question of H. Wiener : to prove fundamental theorem of projective geometry from Desargues and Pappus Theorems and incidence axioms. Answer from F. Schur. G. Hessenberg : Desargues is a consequence of Pappus and incidence axioms.

When geometry meets Algebra.
Würfe : an addition and a multiplication Desargues theorem = The jets with their operations are a field Pappus theorem = Commutativity of the multiplication of jets.

Axiomatic of general projective geometry.
O.Veblen Axiomatic of general projective geometry. “What we call general projective geometry is, analytically, the geometry associated with a general number field.” [Veblen-Young, 1910] A very logical exposition: “All the relations are defined in general logical terms, mainly by means of the relation of belonging to a class and the notion of one-to-one correspondance.” [Veblen-Young, 1910] A first serie of axioms (alignment, extension (A,E)) which allows to define the notion of quadrangular set.

O.Veblen Q(ABC, DEF) is a quadrangular set (points in involution).
Harmonic sets are particular case of quadrangular set. Addition and Multiplication of jets can be defined with the notion of quadrangular sets. No need to define projective correspondence and to prove fundamental theorem. C D E B A F

Net of rationality (harmonic sequences, Möbius nets).
Fundamental theorem is evident in a net of rationality. The fundamental theorem as an axiom. (pedagogical way to develop geometry without asking question of foundations). Fundamental theorem is equivalent to commutativity of multiplication. Discussion about axioms from which fundamental theorem can be derived.