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The Projective Geometry of Mario Pieri: a Legacy of Georg Karl Christian Von Staudt

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The Projective Geometry of Mario Pieri: a Legacy of Georg Karl Christian Von Staudt View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Historia Mathematica 33 (2006) 277–314 www.elsevier.com/locate/yhmat The projective geometry of Mario Pieri: A legacy of Georg Karl Christian von Staudt Elena Anne Corie Marchisotto ∗ Department of Mathematics, California State University, Northridge, CA 90265, USA Available online 21 October 2005 Abstract The research of Mario Pieri (1860–1913) can be classified into three main areas: metric differential and algebraic geometry and vector analysis; foundations of geometry and arithmetic; logic and the philosophy of science. In writing this article, I intend to reveal some important aspects of his contributions to the foundations of projective geometry, notably those that emanated from his intensive study of the works of Georg Karl Christian von Staudt (1798–1867). Pieri was the first geometer to successfully establish projective geometry as an independent subject (rigorous mathematical theory), freed from all ties to Euclidean geometry. The path to this achievement began with Staudt, and involved the reformulation of the classical ideas of cross ratio and projectivity in terms of harmonic sets, as well as a critical analysis of the proof of a fundamental theorem that connects these ideas. Included is a brief overview of Pieri’s life and work. © 2006 Elsevier Inc. All rights reserved. Sommario La ricerca di Mario Pieri (1860–1913) può essere classificata in tre zone principali: differenziale metrico e la geometria algebrica ed analisi di vettore; fondamenti della geometria e dell’aritmetica logica e la filosofia di scienza. Nel questo articolo, intendo rivelare alcune funzioni importanti dei suoi contributi ai fondamenti della geometria proiettiva, considerevolmente quelli che sono derivato dal suo studio intenso sugli impianti di Georg Karl Christian von Staudt (1798–1867). Pieri era il primo geometra per stabilire con successo la geometria proiettiva come oggetto indipendente, liberato da tutti i legami alla geometria euclidea. Il percorso a questo successo ha cominciato con Staudt ed ha coinvolto la nuova formulazione delle ideee classiche del rapporto e del projectivity trasversali in termini di insiemi armonici, così come un’analisi critica della prova di un teorema fondamentale che collega queste idee. Inclusa è una breve descrizione di vita e de lavoro del Pieri. * Fax: +1 818 677 3634. E-mail address: [email protected]. 0315-0860/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.hm.2005.08.002 278 E.A.C. Marchisotto / Historia Mathematica 33 (2006) 277–314 © 2006 Elsevier Inc. All rights reserved. MSC: 51-02; 51M35; 51J10; 01-02 Keywords: Mario Pieri; Georg Karl Christian von Staudt; Synthetic projective geometry; Harmonic set; Cross ratio; Projective transformation; The fundamental theorem of projective geometry; Pappus’s theorem 1. Introduction The research of Mario Pieri (1860–1913) can be classified into three main areas: (1) metric differential and algebraic geometry and vector analysis; (2) foundations of geometry and arithmetic; (3) logic and the philosophy of science. In writing this article, I intend to reveal some important aspects of his contributions to the foundations of projective geometry, notably those that emanated from his intensive study of the works of Georg Karl Christian von Staudt (1798–1867). Pieri was the first geometer to successfully establish projective geometry as an independent subject, freed from all ties to Euclidean geometry. The path to this achievement began with Staudt, whose famous Geometrie der Lage [1847] was intended to rid projective geometry of metric concepts. In this endeavor, the idea of a harmonic set played a pivotal role. Staudt appealed to it in reformulating two concepts of classical projective geometry: the cross ratio, which until his time had been defined metrically; and the projectivity, a transformation that had classically been defined in terms of sequences of correspondences called perspectivities. A fundamental theorem on projectivities (which ensures that the transformation is completely determined by assigning distinct collinear images to three distinct collinear points) intercedes in both of these ideas. This theorem is important not only to reconcile Staudt’s reformulated definitions of cross ratio and projectivity with their classical definitions, but also because it simplifies the proofs of other projective theorems, enabling them to be constructed without appealing to concepts extraneous to the theorems themselves (see, for example, [Coxeter, 1964, 34–35]). Staudt’s proof of the fundamental theorem, however, was one of the reasons that he did not achieve his goal of a metric-free projective geometry. Pieri would bring Staudt’s dream to fruition 50 years later with his axiomatization of projective geometry [Pieri, 1897–1898]. Before I trace the evolution of ideas beginning with [von Staudt, 1847] and culminating in [Pieri, 1897–1898] and its modifications, I give a brief overview in Section 2 of Pieri’s life and work.1 In Sec- tion 3, I discuss the classical concept of projection and section and relate this process to the construction of projectivities as finite sequences of perspectivities. In Section 4, I set the context for Staudt’s attempts to achieve a metric-free definition of projective geometry, by showing how he reformulated the classical definition of cross ratio into a purely projective one. To do this, I introduce the quadrangle, a funda- mental plane figure of projective geometry, and demonstrate how Staudt used it to construct harmonic sets. In Section 5, I show the role of harmonic sets in Staudt’s definition of projectivity. The fundamental theorem enters into the discussion in two ways: to ensure that Staudt’s cross ratio is well defined, and to equate Staudt’s definition of projectivity in terms of harmonic sets with the classical one in terms of 1 In-depth studies of Pieri’s life and research are currently being written. The following three books will be published by Birkhäuser, Boston: The Legacy of Mario Pieri: Geometry and Arithmetic [Marchisotto and Smith], The Legacy of Mario Pieri: Logic and Geometry [Marchisotto et al.],andThe Legacy of Mario Pieri: Differential and Algebraic Geometry [Marchisotto and Smith]. E.A.C. Marchisotto / Historia Mathematica 33 (2006) 277–314 279 perspectivities. In Section 6, I focus on gaps in Staudt’s reasoning, and in Section 7,IshowhowPieri attempted to fill them. In Section 8, I illustrate how Pieri eliminated the dependence on continuity to make Staudt’s arguments. An examination of Pieri’s [1897–1898] axiomatization in relation to certain other developments of n-dimensional projective geometry is the content of Section 9. 2. Overview of Pieri’s life and work Mario Pieri was born in Lucca, Italy, in the region of Tuscany. His father, Pellegrino, was a lawyer and recognized scholar. His mother was named Erminia Luporini. Pieri had four sisters and three brothers, one of whom, Silvio (1856–1936), was a noted professor of linguistics. Mario married late in life, at the age of 41, after he had moved to Catania in Sicily. He and his wife, Anelina Jannelli Anastasio, had no children. After pursuing his early studies in Lucca at elementary and technical schools, Pieri attended the Regio Istituto Tecnico di Bologna. This institute was established in 1862, one year after the foundation of the Regno d’Italia (Kingdom of Italy). Today it is known as the Istituto di Istruzione superiore Crescenzi Pacinotti. Pieri was enrolled in the Fisico-matematica section of the institute between 1876 and 1880, which made him eligible to study mathematics, physics, and engineering2 in any Italian university. Pieri studied mathematics at the University of Bologna beginning in 1880. His instructors included Salvatore Pincherle (1853–1936), one of the founders of functional analysis. In 1881, Pieri was awarded a scholarship to the leading mathematical center in Italy, the Scuola Reale Normale Superiore of Pisa. His instructors included Enrico Betti (1823–1892), who is credited with introducing the ideas of Bernhard Riemann (1826–1866) to Italy; Luigi Bianchi (1856–1928), who had been a student of Felix Klein (1849– 1925); and Riccardo de Paolis (1854–1892), who had been a student of Luigi Cremona (1830–1903). Pieri earned his doctorate in mathematics from the University of Pisa “with commendation” on June 27, 1884. His dissertation on the singularities of Jacobians of second-, third-, and fourth-degree polynomials was never published in any journal, but it was cited in an obituary [Levi, 1913, 1], written by Beppo Levi (1875–1961), who had been a student of Segre at Turin and later colleague of Pieri at Parma. Pieri wrote a second dissertation entitled Studi di Geometria Differenziali for the Scuola (dated September, 1884) which has, until now, never been reported in the literature.3 He also earned certification to teach in secondary school. Pieri taught briefly at a gymnasium (ginnasio) in Livorno and at a technical school in Pisa [Levi, 1913, 2; Rindi, 1919, 438]. During the same period (1885–1886), he gave a special course of lessons on polyhedra at his alma mater, the Scuola [Rindi, 1919, 438–439]. In 1886, Pieri won a competition to become professor of projective and descriptive geometry at the Regia Accademia Militare, which was located across the Via Verdi from the University of Turin. This Royal Military Academy was founded in the late 1600s by Carlo Emmanuelle II. It was the first institute of military instruction in the world. After World War II it was closed in Turin and became absorbed into the military academy in Modena, which exists today. 2 In Pieri’s time, in Italy, only those who studied at a liceo could study any subject they wanted at the university. Because Pieri was enrolled at an istituto, he was limited to university study of the indicated subjects. 3 Both dissertations are available at the University of Pisa library.
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