Mario Pieri in Loving Memory of Helen Cullura Corie, with Gratitude for Her Inspiration and Support Elena Anne Marchisotto James T

Home , Alessandro Padoa, Enrico Betti, Luigi Bianchi, Moritz Pasch, Salvatore Pincherle

Mario Pieri In loving memory of Helen Cullura Corie, with gratitude for her inspiration and support Elena Anne Marchisotto James T. Smith

The Legacy of Mario Pieri in Geometry and Arithmetic

Birkhauser¨ Boston • Basel • Berlin Elena Anne Marchisotto James T. Smith California State University, Northridge San Francisco State University Department of Mathematics Department of Mathematics Northridge, CA 91330 San Francisco, CA 94132 U.S.A. U.S.A. [email protected] [email protected]

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987654321 www.birkhauser.com (HP) Foreword

by Ivor Grattan-Guinness

One of the distortions in most kinds of history is an imbalance between the study devoted to major figures and to lesser ones, concerning both achievements and influence: the Great Ones may be studied to death while the others are overly ignored and thereby remain underrated. In my own work in the history of mathematics I have noted at least a score of outstanding candidates for neglect, of whom Mario Pieri (1860–1913) is one. A most able contributor to geometry, arithmetic and mathematical analysis, and mathematical logic during his rather short life, his work and its legacy are not well known. The main reason is that Pieri worked “in the shadow of giants,” to quote one of the authors of this volume.1

Born into a scholarly family in Lucca, Pieri was educated briefly at the University of Bologna and principally at the prestigious Scuola Normale Superiore, in Pisa; under the influence of Luigi Bianchi (1856–1928) he wrote there his doctoral dissertations on algebraic and differential geometry. During his twenties came appointments in Turin, first at the military academy and then also at the university, where he fell under the sway of Corrado Segre (1863–1924) in algebraic geometry, and Giuseppe Peano (1858–1932) in the foundations of arithmetic, mathematical analysis, and mathematical logic. From 1900 to 1908 he held a chair at the University of Catania before moving to Parma, where he died from cancer. During these last years Pieri continued and broadened his interests, especially in other parts of geometry and in related topics such as vector analysis.

As part of his contributions to geometry and logic Pieri took up the growing interest of that time in the axiomatisation of mathematical theories and the attendant reduction in the number of primitive notions. Among other noteworthy features, he pioneered talk of the “hypothetico-deductive” method in mathematics. David Hilbert (1862–1943) had been emphasising this method from the mid-1890s onwards, especially in geometry; and Pieri’s axiomatisation of geometry (especially its projective part) may equal in calibre Hilbert’s own axiom system. In addition, at that time Pieri’s work was noticed and praised in 1903 by another student of foundations, Bertrand Russell (1872–1970). How- ever, when the University of Kazan awarded its Lobachevsky Prize that year for recent contributions to geometry, Hilbert won, with Pieri receiving an honourable mention. After Pieri’s death in 1913, all of his work gradually disappeared from attention, until from the mid-1920s onwards some of his work on geometry gained the admiration of another giant, Alfred Tarski (1902–1983).

1 Marchisotto 1995. vi Foreword

Pieri has also not been completely ignored historically; there was a photoreprint edition of his writings on foundational matters, and an edition of letters addressed to him.2 Now the authors of this book and its two planned successor volumes have not only studied his life and work and their historical context in great detail but have also translated into English several of his principal papers. Thereby they exhibit his proper place in the histories of mathematics and of logic, and also clarify the context of some surrounding Great Ones. Their efforts supplement the recent revival of work on Peano himself that Clara Silvia Roero has led.3 One looks forward to further attention paid to other major “Peanists,” especially Cesaro Burali-Forti (1861–1931), who was a fellow student in Pisa, and Alessandro Padoa (1868–1937). But in the meantime my list of neglected figures requiring special consideration has definitely decreased by one.

2 Pieri 1980; Arrighi 1997. 3 Roero 2001; Roero, Nervo, and Armano 2002; Peano 2002; Roero 2003; Luciano and Roero 2005. Preface

The Italian mathematician Mario Pieri (1860–1913) played a central role in the research groups of Corrado Segre and Giuseppe Peano, and thus a major role in the development of algebraic geometry and foundations of mathematics in the years around the turn of the twentieth century.

In algebraic geometry, Pieri emphasized birational and enumerative geometry. The thread connecting much of his research is multidimensional projective geometry. Using birational transformations he investigated singular points of algebraic curves and surfaces, classified ruled surfaces, and explored their connections to higher-dimensional projective spaces in innovative ways. His research in enumerative geometry extended that of Hermann Schubert and others. Several of his results are now seen as precursors of modern intersection theory. Although the mainstream of algebraic geometry research departed markedly from Pieri’s approach after about 1920, his results nevertheless continue today to stimulate new developments in real enumerative geometry and control- systems theory.

In foundations of mathematics, Pieri created axiomatizations of the arithmetic of natural numbers, of real and complex projective geometry, of inversive geometry, of absolute (neutral) geometry based on the notions of point and motion, and of Euclidean geometry based on point and equidistance. His developments of geometry, born of his critical study of the underpinnings of G. K. C. von Staudt’s famous 1847 work Geometrie der Lage, differed greatly from other foundations research of Pieri’s time. Pieri broke new ground with novel choices of primitive terms. His incisive presentation of his hypothetical-deductive viewpoint, along with contemporary work of others of the Peano school, prepared the scene for deep studies of logical foundations of mathematical theories. The Italians’ precise formulations, and the practice of David Hilbert and his followers, established the abstract approach that soon became standard in foundations research and in the exposition of higher mathematics.

Pieri died young, in 1913. There immediately followed thirty-five years of turmoil and catastrophe in Italy and the rest of the world. During that period the work of the Segre and Peano schools lost much of its prominence, and recognition of Pieri’s individual contributions disappeared almost entirely from the mathematical literature.

A series of three books is in progress, to examine Pieri’s life and mathematical work. This first book, The Legacy of Mario Pieri in Geometry and Arithmetic, introduces Pieri and provides an overview of his results. It focuses on his studies in foundations, and provides English translations and analyses of two of his axiomatizations: one in arithmetic and one in geometry. Book two will continue examining Pieri’s research in foundations, and will include translations and analyses of two more of his axiomatizations, in absolute and projective geometry. It will provide a thorough analysis of the relationship viii Preface of Pieri’s philosophy of mathematics to that of other researchers in foundations, in particular Bertrand Russell. The third book will survey the background of Pieri’s work in algebraic and differential geometry, placing it in the context of the mathematics of his time. It will then describe his entire research opus in that field, highlighting his influence in such a way that those familiar with more recent work can readily confirm it. Near the end of his life, Pieri entered a different but related field. That final book will also describe his contribution to that effort—to develop Peano’s geometrical calculus into a form of vector analysis, with the aim of simplifying much of differential geometry.

The Legacy of Mario Pieri in Geometry and Arithmetic focuses on Pieri’s career and his work in foundations, from a viewpoint ninety years after his death. It places his life and research in context and traces his influence on the work of some contemporary and more recent mathematicians. It is addressed to scientists and historians with a general knowledge of logic and advanced mathematics, but with no specialized experience in mathematical logic or foundations of geometry.

Pieri’s contributions to foundational studies lie close to the common knowledge of mathematicians today. From this book’s concise overviews of Pieri’s work and of its relation to that of his contemporaries, readers should have little trouble understanding what Pieri did in this discipline. Moreover, the included translations of his 1907a and 1908a memoirs on the foundations of integer arithmetic and Euclidean geometry will provide an idea of the flavor of Pieri’s research and the style of his presentations.

After a first chapter devoted to Pieri’s biography and an overview of his entire research career, this book continues in chapter 2 with a survey of his works on foundations of geometry. Chapter 3 contains a complete translation of Pieri’s 1908a memoir, Elementary Geometry Based on the Notions of Point and Sphere. Next, chapter 4 presents his axiomatization of the arithmetic of natural numbers and its relations to work of Peano and others; section 4.2 is a complete translation of Pieri’s 1907a paper On the Axioms of Arithmetic.

Chapter 5 evaluates Pieri’s impact on some of his contemporaries and on later mathematicians in the foundations area. It explores how his results have been overshadowed, in particular by the research of Peano and Alfred Tarski, even though those two famous mathematicians held Pieri in high esteem. Peano was effusive with praise for Pieri’s work. He viewed Pieri’s 1907a axiomatization of arithmetic as relegating his own famous 1889 postulates to mere “historical value.” Moreover, Peano believed that Pieri’s axiomatizations of geometry constituted “an epoch” in such studies that would be a valuable resource for all who followed. Tarski adapted Pieri’s approach in the Point and Sphere memoir about twenty years later to fit into the elementary logic framework that had crystallized during the 1920s. The turmoil of the times delayed publication of Tarski’s system until 1959. Highly successful, Tarski’s paper led to a large number of later works in the area.

The authors believe that Pieri’s work is not as well known as it should be. Some important histories of mathematics fail to mention his work. Often, when recognized, Preface ix it is not discussed in any detail. Often, Pieri is lost among the Italians, with his work attributed merely to the Peano school, or to mathematicians working with Segre. In algebraic geometry, his formulas are well known, but the papers where they appeared are generally not cited, nor is their geometric motivation acknowledged. Pieri’s research in foundations has been eclipsed especially by that of Hilbert and his followers, even though Pieri explored some aspects of their work earlier and played a major role in formulating the abstract approach they followed. Finally, the work of various Italian algebraic geome- ters, including Pieri, has been criticized for generally lacking rigor. It is appropriate now for our project to counter that neglect, and that judgment, by describing and displaying Pieri’s work in detail.

The reasons for Pieri’s obscurity are complex, and we are careful in our attempts to reconstruct history to explain them. Historical events affected the reception of Pieri’s work. The world wars, totalitarianism, and economic depression probably detracted from its dissemination and general recognition. Particular examples are the death, on the first day of World War I, of Louis Couturat, one of Pieri’s major French proponents; and the delay of publication of Tarski’s [1957] 1959 paper What Is Elementary Geometry?,which acknowledged Pieri’s contribution, until a decade after the second war. One can also look to Pieri himself for reasons that his work is little known today. This book explores the negative impact of his close association with Peano, his use of Peano’s symbolic notation, and of the complexity of Pieri’s axiomatizations.

This book’s final chapter lists Pieri’s published works, lecture notes, and his surviving letters. Detailed annotations or translations describe his reviews, letters, and collected works. The remaining works will be annotated similarly in later books of this series.

There is no organized Nachlass of Mario Pieri. Arrighi 1997 is a collection of transcrip- tions of 132 letters to Pieri. The present authors have not been able to locate the originals of those letters. In sections 1.1, 6.6, and 6.7 we have described virtually everything we have found so far in possession of Pieri’s relatives and in archives of institutions and of other mathematicians. Section 6.6 describes all thirty-five of his surviving letters. For thirteen of them, and for one review in section 6.7, we have provided complete translations: these constitute their first publication. Material that we find after the present book is published will be reported in the later books of this series.

Style and organizational details. When necessary for clarity or precision, but sparingly, the narrative text in this book employs the concise mathematical terminology and symbolic notation now common in undergraduate courses in higher algebra. The translations in chapter 3 and section 4.2, however, adhere as closely as possible to Pieri’s original text; any deviations are noted in those sections. Throughout the book, parenthetical information relating to several sentences in a paragraph may be gathered into a single footnote. The huge bibliography lists all works referred to in the book. Each entry indicates where references occur. The author–date system is employed for citations: for example, “Pieri 1907a” is a citation for a paper Pieri published in 1907. (The present book mentions more than one author named Pieri; citations that include this surname only are references to Mario Pieri.) Sometimes the author is to be inferred from the context, x Preface so that a date alone may also serve as a citation of a work. Biographical sketches of ninety-six individuals closely involved with Pieri and his legacy as described in this book are provided in section 1.3. The book’s index lists both subjects and persons. The latter entries include personal dates when known.

Evolution of the project, and acknowledgments. This project to study and explain the life and legacy of Mario Pieri stemmed from Elena A. Marchisotto’s 1990 New York University doctoral research, and her correspondence with Francisco Rodriguez- Consuegra. The latter had become interested in Pieri through his 1988 University of Barcelona doctoral research and his 1991 book on Russell. They published a plan for the project in their joint 1993 paper. Over the intervening years, much research and many conversations—particularly with H. S. M. Coxeter, Steven R. Givant, Ivor Grattan- Guinness, Jeremy Gray, Steven L. Kleiman, David E. Rowe, James T. Smith, and Janusz Tarski—led to considerable expansion of the scope of the project. To make this complexity manageable, it was split. Its first part is this book. Subsequent books are in progress, one to consider Pieri’s philosophy of mathematics and further aspects of his work on foundations of geometry, and one to survey all his algebraic geometry work in its historical setting. Rodriguez-Consuegra will be a coauthor of the middle book, which is the kernel of the original project. J. T. Smith’s contribution to the present book reflects not only his collaboration with Marchisotto on its content, but his assistance with the overall execution of the plan, notably verification of all cited results and publication details, editing, and formatting. Smith is responsible for the annotated translation of Pieri 1908a in chapter 3. For the origin of his interest in Pieri, see a footnote in section 5.2. The authors particularly wish to acknowledge inspiration, assistance, or support by not only the mathematicians and historians of mathematics just mentioned, but also Marco Borga; Francesco, Marco, and Vittorio Campetti; Maria Grazia Ciampini; Salvatore Coen; Helen Cullura Corie; Mario and Mareno Da Collina; Philip J. Davis; Angelo Fabbi; Livia Gia- cardi; Haragauri N. Gupta; Reuben Hersh; Hubert G. Kennedy; Ann Kostant; Anneli and Peter D. Lax; Joseph A. Marchisotto, Jr.; Pier Daniele Napolitani; Franco Palladino; Patricia Cowan Pearson; Donald Potts; Raffaello Romagnoli; Thomas Sinsheimer; Helen M. Smith; Frank Sottile; Francesco Speranza; Alfred Tarski; and Michael- Markus Toepell. Finally, J. T. Smith is grateful to the Mathematics Department of the University of California at Berkeley for his appointment as visiting scholar, and both authors acknowledge the splendid interlibrary loan service of the California State University. Elena A. Marchisotto James T. Smith March 2007 Contents

Foreword, by Ivor Grattan-Guinness ...... v

Preface ...... vii

Illustrations ...... xv

History and Geography ...... xvii

1 Life and Works ...... 1 1.1 Biography ...... 4 1.1.1 Lucca ...... 4 1.1.2 Bologna: Studies ...... 7 1.1.3 Pisa ...... 10 1.1.4 Turin ...... 17 1.1.5 The Bologna Affair...... 25 1.1.6 Catania ...... 32 1.1.7 Parma ...... 44 1.1.8 Afterward ...... 47 1.2 Overview of Pieri’s Research ...... 50 1.2.1 Algebraic and Differential Geometry, Vector Analysis ...... 50 1.2.2 Foundations of Geometry ...... 54 1.2.3 Arithmetic, Logic, and Philosophy of Science ...... 58 1.2.4 Conclusion...... 61 1.3 Others ...... 62

2 Foundations of Geometry ...... 123 2.1 Historical Context ...... 125 2.2 Hypothetical-Deductive Systems ...... 126 2.3 Projective Geometry ...... 128 2.4 Inversive Geometry ...... 137 2.5 Absolute and Euclidean Geometry ...... 145 2.5.1 Point and Motion ...... 145 2.5.2 Point and Sphere ...... 153

3 Pieri’s Point and Sphere Memoir ...... 157 3.1 Point and Sphere...§I ...... 164 3.2 Orthogonality...§II ...... 178 3.3 Points Internal or External...§III ...... 192 3.4 Theorems on Rotations...§IV ...... 206 3.5 Relations “Smaller Than” and “Larger Than”...§V ...... 220 3.6 Parallelism...§VI ...... 229 xii Contents

3.7 Products of Isometries...§VII ...... 241 3.8 Ordering and Senses...§VIII ...... 249 3.9 Appendix ...... 265 3.10 Historical and Critical Remarks ...... 271 3.10.1 Pieri’s Point and Motion Monograph ...... 271 3.10.2 Hilbert’s Foundations of Geometry ...... 274 3.10.3 Veblen’s 1904 System of Axioms ...... 277 3.10.4 Pieri’s Point and Sphere Memoir ...... 278 3.10.5 The Definitions ...... 278 3.10.6 The Postulates ...... 282 3.10.7 Building Geometry ...... 283 3.10.8 Other Significant Features ...... 284 3.10.9 Questions Answered...... 286 3.10.10 New Questions ...... 287

4 Foundations of Arithmetic ...... 289 4.1 Historical Background ...... 290 4.1.1 The Real Number System ...... 291 4.1.2 The Natural Numbers ...... 294 4.1.3 Pieri’s Investigation of the Natural Number System ...... 305 4.2 Pieri’s 1907 Axiomatization ...... 308 4.3 Axiomatizing Natural Number Arithmetic ...... 313 4.3.1 Dedekind ...... 314 4.3.2 Peano...... 315 4.3.3 Padoa ...... 320 4.3.4 Pieri ...... 322 4.4 Reception of Pieri’s Axiomatization ...... 326

5 Pieri’s Impact ...... 331 5.1 Peano and Pieri ...... 331 5.1.1 Peano’s Background ...... 332 5.1.2 Peano’s Early Career ...... 333 5.1.3 Peano’s Ascent ...... 335 5.1.4 Pieri and the Peano School ...... 338 5.1.5 Peano’s Decline ...... 343 5.2 Pieri and Tarski ...... 347 5.2.1 Foundations of the Geometry of Solids ...... 349 5.2.2 Tarski’s System of Geometry...... 350 5.2.3 1929–1959 ...... 351 5.2.4 What Is Elementary Geometry? ...... 353 5.2.5 Basing Geometry on a Single Undefined Relation ...... 357 5.3 Pieri’s Legacy ...... 363 5.3.1 Peano and Pieri...... 363 5.3.2 Pieri and Tarski ...... 367 5.3.3 In the Shadow of Giants ...... 369 5.3.4 In the Future...... 370 Contents xiii

6 Pieri’s Works ...... 373 6.1 Differential Geometry ...... 374 6.2 Algebraic Geometry ...... 374 6.2.1 Beginnings ...... 375 6.2.2 Tangents and Normals ...... 375 6.2.3 Enumerative Geometry ...... 376 6.2.4 Birational Transformations...... 377 6.3 Vector Analysis ...... 378 6.4 Foundations of Geometry ...... 379 6.4.1 Projective Geometry...... 379 6.4.2 Elementary Geometry ...... 381 6.4.3 Inversive Geometry ...... 381 6.5 Arithmetic, Logic, and Philosophy of Science ...... 381 6.6 Letters ...... 382 6.7 Further Works ...... 392 6.7.1 Translations, Edited and Revised ...... 393 6.7.2 Reviews ...... 393 6.7.3 Lecture Notes...... 397 6.7.4 Collections ...... 398 6.7.5 Memorials to Pieri ...... 399

Bibliography ...... 401

Permissions ...... 459

Index ...... 463 Illustrations

Portraits Mario Pieri ...... frontispiece, 24, 130 Augusto Righi ...... 9 Silvio Pieri and his daughter ...... 9 Enrico Betti ...... 13 Luigi Bianchi ...... 13 Ulisse Dini ...... 13 Enrico D’Ovidio ...... 21 Federigo Enriques ...... 24 Mario Pieri ...... 24 Eugenio Bertini ...... 30 Luigi Cremona ...... 30 Salvatore Pincherle ...... 30 Vito Volterra ...... 30 Virginia Pieri and Paolo Anastasio ...... 37 Pieri’s sister Gemma and her sons ...... 39 Pieri and relatives ...... 39 Angiolina Pieri ...... 45 Beppo Levi ...... 45 Corrado Segre ...... 55 Cesare Burali-Forti ...... 55 Felix Klein ...... 55 Mario Pieri ...... 130 Moritz Pasch ...... 132 Gino Fano ...... 132 Giuseppe Veronese ...... 132 G. K. C. von Staudt ...... 136 Theodor Reye ...... 136 August F. Möbius ...... 144 B. L. van der Waerden ...... 144 Gino Loria ...... 152 David Hilbert ...... 275 Oswald Veblen ...... 279 Hermann Grassmann ...... 295 Richard Dedekind ...... 295 Giuseppe Peano ...... 303, 339 Bertrand Russell ...... 339 Alfred Tarski (2 portraits) ...... 359 Adolf Lindenbaum ...... 359 xvi Illustrations

Maps Some Italian cities ...... xviii Italy’s regions ...... xix Telescopic projections of the planet Mars: Mario Pieri, 1878 ...... 9

Figures Scuola Normale Superiore, Pisa ...... 13 Pieri’s doctoral thesis ...... 14 Military Academy, Turin ...... 19 Military Academy personnel, Turin ...... 19 Pieri’s first Turin Academy of Sciences paper ...... 21 University of Catania ...... 35 Pieri became an editor of the Gioenia Atti ...... 35 Announcement of Pieri’s promotion ...... 37 Pieri’s 1922 interment in Lucca ...... 48 Pieri’s tomb ...... 48 Pieri’s definition of cyclic order ...... 130 Pieri’s definitions of collinearity ...... 152 Pieri’s definition of betweenness ...... 152 Pieri’s Point and Sphere memoir, first page ...... 159 Hilbert’s Foundations of Geometry ...... 275 Veblen’s System of Axioms for Geometry ...... 279 Peano’s Principles of Arithmetic ...... 303 Pieri’s On the Axioms of Arithmetic ...... 309 Peano’s Formulario ...... 344 History and Geography xvii

HISTORY. After the Napoleonic wars, the European powers partially restored to Italy the status quo ante. Austria dominated its northeast, either by direct rule or through states headed by members of the Habsburg (Asburgo) family. In the northwest, the Savoy (Savoia) family provided kings for Sardinia, a buffer state between Austria and France. It consisted of that island and the Piedmont region of the mainland, with its capital at Turin. The Papal States in the middle of Italy, including Rome, were controlled by the Roman Catholic Church and numerous local dynasties. Austria was also the major influence in the south, the Bourbon (Borbone) Kingdom of the Two Sicilies—the island and the southern peninsula—with its capital at Naples. Here and there were enclaves whose political connections and history defied simple classification. LUCCA was one of those. Napoleon’s conquerors grafted onto Italian roots new social and political institutions. Many persisted through Pieri’s time, and some are still discernible. During the half-century after Napoleon, alliances, invasions, revolutions, industrialization, and social upheaval led unsteadily but inevitably to Italian unification and democratization. Before Napoleon, Lucca had long been an independent city-state. Afterward, the European powers awarded it to Maria Luisa, widow of the former Bourbon duke of Parma, on condition that her heir be his son, Carlo Lodovico. Parma, which had been ruled by Bourbons before Napoleon, was assigned to the Habsburg princess Maria Luigia, also known as Marie Louise, Napoleon’s second wife. Carlo Lodovico was designated her heir also. Maria Luisa died in 1824. Carlo Lodovico, profligate and despised, was forced out in 1847, just when Maria Luigia died. Lucca and Parma were then annexed to TUSCANY, the grand duchy of Maria Luigia’s cousin Leopoldo II. (Their paternal grandfather Leopoldo I had been a rather progressive ruler of Tuscany during 1765–1790, and emperor during 1790–1792.) In 1859 a constituent assembly forced Leopoldo II to abdicate. The next year it voted for annexation to Sardinia, and in 1861 the king of Sardinia was proclaimed King Vittorio Emmanuele II of a (partially) unified Italy. For more information on Parma, see page xli.

GEOGRAPHY. Today Italy is divided into twenty regions, listed here with their capitals: Abruzzo ...... L’Aquila Apulia (Puglia) ...... Bari Basilicata ...... Potenza Calabria ...... Catanzaro Campania ...... Naples (Napoli) Emilia–Romagna ...... Bologna Friuli–Venezia Giulia ...... Trieste Latium (Lazio) ...... Rome (Roma) Liguria ...... Genoa (Genova) Lombardy (Lombardia) ...... Milan (Milano) Molise ...... Campobasso Piedmont (Piemonte) ...... Turin (Torino) Sardinia (Sardegna) ...... Cagliari Sicily (Sicilia) ...... Palermo The Marches (Le Marche) ...... Ancona Trentino–South Tyrol (Trentino–Alto Adige) ...... Trent (Trento) Tuscany (Toscana) ...... Florence (Firenze) Umbria ...... Perugia Valley of Aosta (Valle d’Aosta) ...... Aosta Venice (Veneto) ...... Venice (Venezia) In turn, each region is composed of provinces; there are 103 altogether. For example, Tuscany is the region in central Italy between the Arno and Tiber valleys and the Tyrrhenian Sea. It was the locus of the sophisticated Etruscan culture, which flourished hundreds of years before the Roman. The Latin and Italian names for the region are Etruria and Toscana. Tuscany has ten provinces: Arezzo Grosseto Lucca Pisa Prato Florence (Firenze) Leghorn (Livorno) Massa Carrara Pistoia Siena xviii History and Geography

Switzerland Austria

Bergamo Milan " Venice " Padua Turin ! " ! !Pavia Cuneo Genoa !Parma France ! "!Modena ! Bologna Lucca " "Florence Pisa ! Leghorn" !Siena ! Macerata

!Rome

! Sassari !Naples

!Cagliari

Ustica "

!"!Messina Palermo Castro- reale !Catania

Some Italian cities

! Public universities in Pieri’s time " Other cities History and Geography xix

Switzerland Austria

Trentino– South Tyrol Friuli–Venezia Giulia Valley of Aosta Lombardy Venice Piedmont France Emilia– Romagna Liguria

Tuscany The Marches

Umbria

Abruzzo Latium Molise

Campania Apulia Basili- Sar- cata dinia

Calab- ria

Sicily

Italy’s regions 1 Life and Works

Mario Pieri was “a true bridge between the two most prestigious Italian schools of mathematics of [his] epoch: that of logic and that of algebraic geometry.”1 Yet his works are not as well known to today’s scholars as they should be. Pieri left a legacy of results in algebraic and differential geometry, vector analysis, foundations of mathematics, logic, and philosophy of science that are worth knowing, not just for their historical value, but as well for their mathematical and philosophical import.

This is the first of three books on Pieri’s life and work. It discusses his life, presents an overview of his research in foundations of geometry and arithmetic, and provides Eng- lish translations and analyses of two of his most important papers in those fields. The second book, The Legacy of Mario Pieri in Logic and Geometry, will present more detailed accounts of Pieri’s work in philosophy of science and foundations of Euclidean and projective geometry, and will include English translations of two more of his major papers on foundations of geometry. The third, The Legacy of Mario Pieri in Differential and Alge- braic Geometry, will place in historical context his results in those areas and in vector analysis, including the famous Pieri formulas, and will provide English summaries of all of his papers in those fields.

Pieri’s research was valued by his contemporaries and was widely known in his lifetime, even though most of his papers were published in Italian academic journals. Distribution of these was often limited, but their authors generally circulated reprints privately to the mathematical community. Around 1900, Pieri was elected to membership in the acade- mies of science of his home city Lucca, in Tuscany, and of Catania, in Sicily, where he spent eight years as professor.2

At the turn of the twentieth century Pieri’s recognition extended well beyond Italy. For example, because of his contributions to logic and foundations of geometry he was invited to speak at the August 1900 International Congress of Philosophy, in Paris. Unfortunately, he did not attend. Pieri had recently settled in Catania and perhaps was not yet ready to travel so far. He did submit a paper, Pieri [1900]1901, which was summarized in Paris by Louis Couturat, one of the conference organizers.

1 Brigaglia and Masotto 1982, 135. 2 Rindi [1913] 1919, 437, 439. 2 1 Life and Works

For his research on foundations of projective and absolute geometry, Pieri earned honorable mention at the 1904 awarding of the Lobachevsky Prize by the Physico- Mathematical Society of Kazan. Several years later, Pieri was named Knight of the Crown of Italy; and on behalf the Imperial Library of Berlin, historian Ludwig Darmstaedter invited him to contribute to a collection of short handwritten notes by “the most distinguished scholars, inventors, and initiators...since the end of the fifteenth century.”3

Pieri’s competence in enumerative geometry, demonstrated through his publications, was so widely recognized that he was entrusted with translation and revision, for the French edition of the Encyklopädie der mathematischen Wissenschaften, of the major article that H. G. Zeuthen had written on that subject for the 1905 German edition.4

Early in 1912, Bertrand Russell invited Pieri to address the International Congress of Mathematicians that summer. Russell wrote, As secretary of the philosophy section of the congress of mathematicians that will be held during the month of August in Cambridge, and as an admirer of your works, I have the honor of earnestly asking you to give a talk and take part in our discussions. The works of Whitehead and me, as you know, are based on the works of the Italian school, and I deeply desire that it be well represented here in the philosophy section. Have faith, dear sir, in the assurance of my highest esteem ... .5 But Pieri was near the end of his life by then, and too ill to travel. He died of cancer on 1 March 1913.

In the years following Pieri’s death, Italian scholars noted the value of his research and praised his character. His younger colleague Beppo Levi wrote, The work of Pieri was distinguished by carefulness of method, of order, of rigor. And such were the signs of his character: he was on every occasion sincere, exact, and honest without possible compromise. In 1925, the mathematician Eugenio Maccaferri published a letter Pieri had written him in 1912, commenting that it would certainly be read “with interest by those who know the high quality of the mind and heart of Mario Pieri.”6

Such praise continued, awarded by scholars who knew Pieri’s work. For example, Ugo Cassina, a student of Giuseppe Peano, Pieri’s senior colleague at Turin, asserted, more than forty-seven years have passed since the death of Pieri, so that we can better evaluate—from the perspective of time—his collective works and recognize those of major importance which have contributed to place the name of Mario Pieri in the restricted circle of Italian mathematicians well known in Italy and abroad at the turn

3 David Hilbert won the prize, for the 1903 edition of Foundations of Geometry, [1899] 1971. See Pieri’s 1908b letter, translated in section 6.6, and Arrighi 1997, letter 43 (15 October 1910). 4 Rindi [1913] 1919, 440; Meyer and Mohrmann 1907–1927. Pieri’s revised translation, [1915] 1991, was published posthumously. 5 Arrighi 1997, letter 106 (16 March 1912). 6 Levi 1913, 69. Maccaferri 1925, 49; see the discussion of Pieri [1912] 1925 in section 6.6. Introduction 3

of the century.... [These] works of Pieri have ... major originality [and have] brilliantly withstood the passage of decades and the fashion of the moment. Pieri’s boyhood friend and later colleague Scipione Rindi perhaps best expressed why the present authors undertook this project. In a speech before the Royal Academy of Science, Letters, and Arts in Lucca, Rindi said, “[Pieri] leaves a precious legacy to Science, the fruits of his untiring and earnest study; to the city and to his family, the honor of his name; to all, the memory and the example of a life nobly spent in the search for truth.” This book explores that legacy.7

Section 1.1 of the present chapter is a biography of Pieri. The mathematical historian Thomas Hawkins has noted, The problem of assessing the influence of a particular work on the course of history can be difficult. A historical event, such as the publication of a mathematical work, does not exist in isolation, but as part of a “collage” of events linked together by the institutions through which mathematics is cultivated, communicated, and evaluated. The more we know about the entire collage the easier it is to appreciate the signifi- cance of one particular piece.8 The present study provides some of the pieces of Pieri’s collage. It is really a report of work in progress. Secondary sources on Pieri’s life and family are very rare, and primary sources difficult to access. The present authors are uncovering additional information in their continued research on Pieri and his work. Facts that come to light after publication of this book will be reported in later books of this series.

Section 1.2 is a brief overview of Pieri’s research. Later chapters expand on selected aspects of that work, as follows. Pieri’s contributions to foundations of geometry are surveyed in more detail in chapter 2. Chapter 3 is entirely devoted to a translation and analysis of his axiomatization of Euclidean geometry, Pieri 1908a, based on the notions of point and equidistance. Pieri’s work on foundations of arithmetic, mentioned briefly in the present chapter, is considered in more detail in chapter 4, which includes a translation of his axiomatization of the arithmetic of natural numbers, Pieri 1907a. Chapter 5 is a discussion of the impact of Pieri’s research in those areas. Chapter 6 presents a list of all of Pieri’s works, organized by subject. Detailed annotations or translations are provided for some of them, including all his reviews, surviving letters, and collected works. The remaining annotations will appear in subsequent books of this series. Biographical sketches of ninety-six individuals closely involved with Pieri and his legacy as described in this book are provided in section 1.3. A typical sketch is generally one paragraph in length, provides the most basic biodata, and relates the individual to Pieri’s context. Some sketches are more extensive. Thus the subsequent books of this series will depend on the present chapter and chapter 6.

7 Cassina 1961, 191; Rindi [1913] 1919, 452. 8 Hawkins 1984, 443. 4 1 Life and Works

The present book is completed by a bibliography listing all works cited in it. Author–date citations such as “Pieri 1907a” refer to that bibliography.9 The book’s index lists both subjects and persons. The latter entries include personal dates when known.

1.1 Biography

This section presents a biography of Mario Pieri. It concentrates on his family and career, but avoids the details of the mathematics he created and taught. It is subdivided according to phases of his life: childhood and family in Lucca, studies in Bologna and in Pisa, his early academic career in Turin, his prolonged and unsuccessful competition for a better position in Bologna, the later stages of his career in Catania and in Parma, and some concluding observations. Pieri’s mathematical work is surveyed in section 1.2. The lives of many individuals mentioned in these first two sections are sketched in section 1.3.10

1.1.1 Lucca

Mario Pieri was born on 22 June 1860 in Lucca, a city in the region of Tuscany, just when Italy was coalescing into a unified country. See pages xvii–xix for relevant Italian political history and geography. Lucca’s famous sons included the composers Boccherini, Catalani, Geminiani, and Puccini. Pieri’s mother was Erminia Luporini. His father, Pellegrino, belonged to an old family from Vellano, a village about twenty kilometers east of Lucca. A respected lawyer and scholar, Pellegrino Pieri became a member of the Royal Lucca Academy of Sciences, Letters, and Arts. See page 11 for information about the academy. Pellegrino wrote a history of the life and work of Domenico Barsocchini, an influential Lucca church administrator during and following the Napoleonic occupation, and later a noted medieval historian. The Pieris resided in the eastern part of the walled central city, at 1164 Piazza San Giusto, with a domestic servant, Petronilla Antichi.11

Mario was the third of eight children12 of Erminia and Pellegrino, all born in Lucca:

9 “Pieri 1907a” is a citation for a paper Pieri published in 1907. Sometimes the author is to be inferred from the context, so that a date alone may also serve as a citation of a work. This book mentions more than one author named Pieri; citations that include this surname only are references to Mario Pieri. 10 Throughout this book, parenthetical information relating to several sentences in a single paragraph is often gathered into a single footnote cited near the last of those sentences. 11 Lucca 2004; Rindi [1913] 1919, 438; Bonfante 1969, v; P. Pieri 1872. Vellano is now in the municipality of Pescia in the province of Pistoia. Erminia was the daughter of Luigi Luporini and Marianna Davini; Pellegrino, the son of Romano Pieri and Teresa Ricci. Five generations of Puccini musicians enlivened Lucca. There is a Puccini museum in Lucca today. 12 Some data in this paragraph are from Lucca 2004. Others were provided by Vittorio Campetti and Maria Grazia Ciampini in 2004 and have not been confirmed by other sources. (These relatives of Mario are identified in a later paragraph.) Further sources are mentioned in later paragraphs about Mario’s family. 1.1 Biography 5

dates married Teresa 1853–1942? Giuseppe Brancoli 1897 Silvio Dante 1856–1936 Enrica Montanari pre-1890 Mario 1860–1913 Angiolina Anastasio Janelli 1901 Gemma 1863–1955? Umberto Campetti 1887 Ferruccio Fabio 1864–1933? Maria Dal Poggetto 1895 Paolina Livia 1865–1959? Geminiano Pellegrini 1884 Felice Ettore Pacifico Giovanni 1866–1920? Beppina Bastian ???? Virginia 1867–1929? Paolo Anastasio 1894 Three of Mario’s siblings had great influence on his life: his elder brother Silvio and younger sisters Virginia and Gemma. Their personal stories are closely intertwined with Mario’s. Moreover, most of the present authors’ knowledge of Mario’s personal life and his family has come via Gemma’s descendants. To clarify these relationships, and provide a general picture of the Pieri family’s cohesion and of their emphasis on individuality, initiative, and intellectual accomplishment, the next paragraphs sketch some aspects of the stories of Pieri’s siblings.

Silvio Pieri earned the doctorate (laureate) and did postgraduate work at the University of Bologna. During that time, the Pieri family arranged for him to act as guardian and mentor for his younger brother Mario, who attended school in that city from the age of sixteen and then studied for a year at the same university. Silvio taught in middle schools until 1905 and then held temporary positions at several universities and with the Reale Accademia dei Lincei before becoming professor of comparative history of classical and Romance languages at the University of Naples in 1915. See page 11 for information about the academy. Silvio was an authority on the origin of place names in central Italy. His posthumously published study S. Pieri 1969 is commonly referred to today in that field. Silvio was also an accomplished and well-published poet. The memorial pamphlet Parducci 1936 contains a list of his works. Silvio’s sons Piero and Pellegrino (nicknamed Rinuccio) both served as military officers, and Piero became a noted professor of military history. Silvio’s daughter Gemma also earned the laureate, became a teacher, and married a diplomat named Meriano. Silvio died of throat cancer, the same disease that had killed Mario, at her house in 1936. A portrait of Silvio with Gemma is on page 9; see also his biographical sketch in section 1.3.13

Mario’s father died in 1882. In 1886 Mario moved to Turin to begin teaching at the military academy there. Sometime after that, his mother and youngest sister, Virginia, moved there so that he could care for them. They all became citizens of Turin in 1894.

That same year, on the remote island of Ustica, Virginia married Paolo Anastasio, a Sicilian from Castroreale—perhaps they eloped! Their portrait, taken after thirteen years of marriage, is on page 37. In 1900 Mario became a professor at the University of Catania,

13 Bonfante 1969; Parducci 1936. S. Pieri 1933 is a book of Silvio’s poems. Arrighi 1997, letter 95 (18 Feb- ruary 1910) to Mario from Rinuccio, then a student at Bologna, confirms some information about the family. Letter 96 (27 October 1911) to Mario from Silvio also mentioned Rinuccio. Gemma Pieri Meri- ano’s husband died around 1935; he was minister to Afghanistan. Silvio had a another son, Alfonso, who lived only a month during 1890 (Lucca 2004). Information about Silvio’s children is from Campetti 2005. 6 1 Life and Works in Sicily, not far from Castroreale. Shortly after that, he married Paolo’s sister Angiolina Anastasio Janelli. Her use of a second surname suggests that she was a widow. A supposed portrait of Angiolina is on page 45.14 Mario’s sister Gemma Pieri Campetti lived in the village of Sant’Andrea di Compito near Lucca, at 92 Via di Sant’Andrea. Her husband, Umberto, was a lawyer. They had three sons: Gaetano, Pellegrino, and Ottorino. Sometime in the 1890s, Umberto emigrated to Brazil, leaving Gemma behind to care for the family. In Brazil he worked as a shopkeeper. On page 39 is a portrait of Gemma and her sons, taken in 1896 or 1897. Around 1900, Gemma joined her husband abroad for several years, leaving Pellegrino and Ottorino in the care of Mario. Gemma and Umberto had returned to Sant’Andrea by 1913, for Mario died there that year, of throat cancer, in Gemma’s care. Eventually Pellegrino Campetti also departed for Brazil and started a branch of the family there. Gaetano became a professor of accounting at a technical institute in Pisa; Ottorino, professor of mathematics and physics at a technical institute in Lucca. With his wife, Beatrice Giusfredi, Ottorino had a daughter and two sons: Anarosa, Vittorio, and Marco. Vittorio and Marco Campetti maintain summer homes in Sant’Andrea today. Marco and his son Francesco, and Vittorio and his wife Maria Grazia Ciampini, have provided the present authors much information about the Pieri and Campetti families. Gaetano outlived Ottorino, and as custodian of Mario Pieri’s Nachlass presented much of Mario’s surviving correspondence to Gino Arrighi, professor of mathematics at Pisa, who long before had been Ottorino’s student in Lucca. That is the basis of the publications Arrighi 1981 and 1997, which are cited frequently in this section.15 Mario Pieri’s remaining four siblings play little role in the story told in this book. Of Teresa Pieri Brancoli, the present authors know nothing further. Ferruccio Pieri, a postal worker in Sant’Andrea di Compito, was evidently beloved by its citizens for his poetry. According to the Campetti family, Felice Pieri emigrated to Brazil, married a widow, and died rather young, of an accident. Paolina Pieri Pellegrini’s husband, Geminiano, was a professor. Their son Aldo became a famous aviator: the general in operational com- mand of the seaplanes that Italo Balbo, the minister of aviation, sent across the Atlantic in the 1930s to advertize Italy’s aircraft industry and flaunt its air power.16

14 Lucca 2004; Ustica 1894. Rindi ([1913] 1919, 452) reported that Pieri married in 1906, but his employ- ment record (Parma 1908) and his 1901a letter, translated in section 6.6, confirm the 1901 date. Arrighi (1997, iv) identified Mario’s wife. Her name is sometimes spelled Angelina. When first interviewed for the present book, Mario’s surviving relatives seemed unaware of her. Angiolina lived until at least 1950: the Campetti family recently found a postcard from her to Gemma Pieri Campetti dated that year (Ciampini 2004). Nothing further is known about Angiolina. The portrait on page 45 was identified by Francesco Campetti (2005) only indirectly as hers. 15 Ciampini 2004; Campetti 2005; Arrighi 1997, v. Ottorino Pieri, nicknamed Ghigo, is mentioned in Arrighi 1997, letter 95 (8 February 1910). The present authors do not know the current location of the material given Arrighi. Ottorino’s grandson Francesco Campetti recalled (2005) that Ottorino credited Mario Pieri for inspiring him to study mathematics. 16 Ciampini 2004; Campetti 2005. Ferruccio Pieri 1901 and 1931 are books of poems. Ferruccio’s wife Maria Dal Poggetto’s mother was Nisedi Pieri; perhaps Ferruccio and Maria were cousins. Arrighi 1997, letter 55, identifies Geminiano Pellegrini as a professor. According to Lucca 2004, Ferruccio became a citizen of Leghorn, in Tuscany, in 1896 and Paolina became a citizen of Messina, in Sicily, in 1904. Ferruccio evidently returned to Sant’Andrea di Compito. 1.1 Biography 7

The previous paragraphs have emphasized the structure of the large and closely knit Pieri family. It is now time to turn to Mario’s own story.

During Mario Pieri’s time, schooling was compulsory only from ages six to nine, and was not well enforced. As a result, among industrializing countries, Italy’s rate of literacy was lowest: 68% in the north, 30% in the south. There was a single elementary-school system, serving ages six through twelve. About 30% of elementary teachers were priests. There were two tracks of middle schools. The classical track consisted of lower-secondary ginnasio and upper-secondary liceo—five plus three years. The utilitarian track consisted of four years in technical school, followed by three years in normal school or four in a technical institute. Normal schools trained elementary teachers. A student in the last two years of a technical institute would select a curriculum in accounting, agriculture, commerce, industry, or physics-mathematics. The liceo provided full access to university; the technical-institute physics-mathematics curriculum, access to university science faculties. Only rarely were graduates of other curricula allowed to attend university. Most Italian mathematicians of Mario Pieri’s time attended technical institutes, not licei. A typical university program of study in mathematics led to the doctoral degree, or laureate, after four years. Italian standards concerning doctoral dissertations apparently varied during the time covered by the present book.17

Mario Pieri completed elementary and technical school in Lucca. During elementary school he studied music with Gaetano Luporini and learned to play the piano. He loved both music and mathematics, and would for long be uncertain which he should pursue as a career.18

1.1.2 Bologna: Studies

In 1876, at age sixteen, Mario was enrolled in the Royal Technical Institute in Bologna. He lived in Bologna with his older brother and mentor Silvio, who was studying at the university there. See page 11 for information about Bologna and its region, Emilia– Romagna. Mario’s curriculum consisted of Italian ...... 4 years French, German ...... 3 years each Social studies (geography, history, economics, ethics) ...... two courses each year for 4 years Mathematics and design courses, sometimes two or more at once ...... 4 years

17 Barbagli [1974] 1982, 13, 48–70, 329–330: literacy statistics for 1901, teacher statistics for 1868. Cur- ricula at technical institutes had been lengthened from three to four years in 1871. The ginnasio and liceo were together similar to the German Gymnasium, while the technical school and institute were analogous to the Realschule. In the present book the term middle school is used for schools at this level in all countries. That facilitates comparison, but readers should realize that in some countries, particularly Germany and the United States, middle school now has a different meaning. 18 Arrighi 1997, v–vi; Rindi [1913] 1919, 438. 8 1 Life and Works

Physics ...... 2 years Chemistry and natural history ...... 1 year each The science courses were all in the last two years. Mario’s performance improved each year: eighth of 19 students in the first year, then sixth of 28, fourth of 16, and finally, first of 20. The only subject that seemed to give him trouble was German. Mario particularly impressed his physics instructor, Augusto Righi, who had recently earned doctorates in mathematics and physics from the University of Bologna. Righi was beginning a pro- lific career, with substantial research and publication in several areas of physics. A paper on composition of oscillations in three mutually orthogonal directions referred to a method for illustrating them stereographically, and two other papers dealt with an application of projective geometry to that problem. Righi was Pieri’s physics teacher during 1879–1880, and presumably they were acquainted during the preceding three years, too. Six astronomical drawings by Mario survive, evidently crafted for physics classes taught by Righi. One is reproduced on page 9. Perhaps it was Righi who first introduced Pieri to advanced geometry and provided him an inspiring example of scientific discipline. Their careers would intersect again, as reported later in this section. By the end of the century Righi would become Italy’s leading physicist. Mario’s brother Silvio must have been a major intellectual influence, too. During their years together, Silvio would have been engrossed in his [1880–1882] 1983 compilation and analysis of Tuscan folk lyrics, and in further studies in literature and philology.19

In 1880, at age twenty, Mario was ready for university. A physical examination revealed that he was too slight in stature (chest measurement) for military service. The family hoped and planned for him eventually to attend the Scuola Reale Normale Super- iore in Pisa. See page 11 for information about the Scuola. But finances were strained and Silvio needed one more year of research at Bologna, so it was decided that Mario should enter university there, and compete for a scholarship to transfer the following year to the Scuola in Pisa. The family applied for and was granted a waiver of fees at Bologna. Mario attended university there for the academic year 1880–1881. The brothers lived at 11 Via Santo Stefano.20

During the previous decade, the mathematics faculty at the University of Bologna had lost vigor. Earlier stars had departed, and it evidently did not resume offering a full program of study until the year that Pieri entered. He would have encountered senior professors—ordinari—Pietro Boschi, who lectured on projective and descriptive geometry, and Giovanni Capellini, Matteo Fiorini, Ferdinando Ruffini, Domenico Santagata,

19 Rindi [1913] 1919, 438; Bonfante 1969, v; Bottazzini, Conte, and Gario 1996, 40; Bologna 1876–1880. In Pieri’s time, the Royal Technical Institute of Bologna was located near the Piazza San Domenico in buildings that had formerly belonged to the Church. It is now called Istituto di Istruzione Superiore Crescenzi–Pacinotti and located at a different site (Breveglieri 1995). Righi 1873, 1875, 1877. Pieri 1876–1880. For further information about Righi, see his biographical sketch in section 1.3. 20 P. Pieri 1881; Bologna [no date]. On the fee-waiver application, Mario’s father listed his income for the preceding year as £937, which, according to historian Salvatore Coen (personal communication), was very modest. Mario’s student number at Bologna was 2395. 1.1 Biography 9

Augusto Righi

Telescopic projections of the planet Mars Mario Pieri, 1878

Silvio Pieri, around 1890, with daughter Gemma 10 1 Life and Works and Emilio Villari, who taught geology, geodesy, mechanics, chemistry, and physics, respectively. Two younger professors had recently arrived as straordinari—Luigi Donati in 1877 and Cesare Arzelà in 1880—and Salvatore Pincherle would come in 1881. All with degrees from the Scuola Reale Normale Superiore in Pisa, they formed a renaissance. During 1880–1881, Arzelà lectured on algebra with analytic geometry, and on calculus; and Donati lectured on mathematical physics. See page 11 for information about descriptive geometry and about Italian academic ranks in Pieri’s time.

Pieri attended six courses at Bologna that year: Algebra and analytic geometry 6 hours Arzelà Projective geometry 2 hours Boschi Design for projective geometry 6 hours Boschi Experimental physics 3 hours Villari Chemistry 3 hours Santagata Mineralogy 3 hours Santagata Mineralogy was his only elective course; the others were all required in his program. In June 1881 Pieri passed three examinations: examiners Analytic and projective geometry Boschi, Fiorini, Pincherle Physics and inorganic chemistry Boschi, Santagata, Villari Algebra Arzelà, Pincherle Pieri’s examination scores were all perfect, and he received honors in the first two. Pincherle, who had just arrived in Bologna, “quickly recognized [Pieri’s] worthiness and held for him an esteem and affection that [Pieri] recalled fondly.” Both Arzelà and Pincherle would play further roles in Pieri’s developing career, as reported later in this section.21

1.1.3 Pisa

In October 1881, after only a year at the University of Bologna, Pieri took the entrance examination for the Scuola Reale Normale Superiore, the more distinguished institution that his Bologna teachers had attended. He scored 24 of 30, won admission as a second- year student, number 196, and began studies the next month. The Scuola was extremely selective: during 1879–1883 the number of students admitted averaged only thirteen, about half in science and half in liberal arts. They took courses there and at the nearby University of Pisa and received degrees from both. During his years at the Scuola, Pieri would have met these mathematics students, admitted in the indicated years: 1877 Scipione Rindi 1880 Rodolfo Bettazzi 1879 Geminiano Pirondini 1881 Enrico Boggio-Lera 1879 Carlo Somigliana 1882 Angelo Andreini 1879 Vito Volterra

21 Coen 1991a, v–vi; Francesconi 1991, 420–422, 444–445; Bologna [no date]. B. Levi 1913–1914, 65. 1.1 Biography 11

LUCCA ACADEMY. The Accademia degli Oscuri existed in Lucca for more than two centuries during the city’s independence. Its name referred to scholars’ overcoming darkness (oscurità). After the Napoleonic turmoil it acquired royal sponsorship and became the Reale Accademia Lucchese di Scienze, Lettere, ed Arti. Its journals, the Memorie and Atti, date from 1813 and 1821.

The ACCADEMIA DEI LINCEI was founded in 1603 in Rome, to investigate nature with perception equal to that of lynxes (lincei). It did not survive long, but was briefly revived several times. Finally, reestablished in 1847 as the Accademia Pontifica dei Nuovi Lincei, it began publishing its journal, the Atti. When Rome became the capital of Italy in 1872, the academy split into two parts, one retain- ing ecclesiastical sponsorship and name. The other, government sponsored, became the Reale Acca- demia dei Lincei. During the next fifteen years it was organized into two classes: moral, historical, and philological sciences, and physical and mathematical sciences. Membership was greatly increased, and its list of journals expanded. Of the ninety-five individuals whose lives are sketched in section 1.3, twenty-nine Italians and nine others became members of the Accademia.

The EMILIA–ROMAGNA region is bordered on the east by the Adriatic Sea, on the north by the regions of Venice and Lombardy, on the west by bits of Piedmont and Liguria, and on the south by Tuscany and the Marches. It was formed after Italian unification from the papal states Emilia and Romagna and the former duchies of Parma and Modena. With capital at Bologna, it consists of nine provinces: Bologna Forlì Parma Ravenna Rimini Ferrara Modena Placentia (Piacenza) Reggio nell’Emilia Bologna was a very important center of medieval Europe: a center of transportation and of the silk industry, and the capital of Emilia. Its university, founded about 1100, is Europe’s oldest. Until Italian unification, the city was governed jointly by a representative of the pope and by a senate of citizens. The Papal States were absorbed slowly into the unified state, with Rome becoming its capital only in 1872. During that period of turmoil, the university suffered neglect; it began to emerge from disfunction only around 1880. During the 1800s Bologna became a major center for the production of food products. For more information on Parma, see page 41.

DESCRIPTIVE GEOMETRY is the study of (1) techniques and theorems used by engineers, designers, and artists to draw things, and (2) consequences of a set of incidence postulates usually sufficient for a model to be embeddable in a projective geometry. Professors of descriptive geometry developed the first aspect of this subject and taught design courses. They sometimes researched its second aspect.

FACULTY RANKS. In Pieri’s time, Italian university faculties were endowed with chairs responsible for instruction and research in specific academic subdisciplines, such as descriptive geometry. A professor holding a chair could have higher or lower rank—ordinario or straordinario. The ordinari of the university constituted its collegium, the voting faculty; they were also called professori colle- giati. Often, research opportunities or instructional needs required adjustments to this basic organi- zation. An ordinario visiting from another institution was called noncollegiato. An occupant of a local chair teaching a different subject would be called professore insegnante. Temporary faculty hired to teach, but without a chair, were called professori incaricati. The laureate, or doctoral degree, qualified a scholar only to be a professor’s assistant. A position as an independent teacher required prior certification by the faculty as libero docente in a particular subdiscipline. Systems of faculty ranks at lower-level institutions differed substantially from this university system.*

PISA was an independent republic during medieval times—a formidable marine power. Its university dates from 1338. Pisa fell to Tuscany in 1406 and since then has functioned mainly as its intellectual center. The Scuola Reale Normale Superiore in Pisa was founded by a decree of Napoleon in 1810 as a branch of l’École Normale Supérieure in Paris. It functioned as such for only two years. It was reconstituted by Leopoldo II of Tuscany in 1846; after unification in 1860 it assumed the role of an institute of advanced instruction and research, affiliated with the university.

*Roero 2004. The term libero docente is analogous to the German Privatdozent. See also De Santis 1988, chapter 4. 12 1 Life and Works

Pieri’s classmates at the University included Edgardo Ciani and Cesare Burali-Forti. Bettazzi and Volterra joined the Pisa faculty during Pieri’s years there; they and Burali- Forti would be his colleagues later in Turin. Pieri’s instructors at Pisa were quite prominent. The following table summarizes his official record. In 1882–1883 Pieri earned certification (licenza) to teach in middle schools.22

Pieri’s curriculum, instructors, and scores at Pisa23

1881–1882 Scuola German Sigismundo Friedmann 30/30 University Calculus Ulisse Dini 30/30 University Projective and descriptive Angiolo Nardi Dei 30/30 geometry with design 1882–1883 Scuola German Girolamo Curto 27/30 Scuola Algebraic functions Luigi Bianchi 30/30 Scuola Mechanics exercises Enrico Betti 28/30 University Mechanics Betti 30/30 University Higher analysis Dini 30/30 1883–1884 Scuola Differential geometry Bianchi 30/30 University Mathematical physics Betti 30/30 University Higher geometry Riccardo De Paolis 30/30 University Celestial mechanics Betti 30/30

Pieri evidently wrote his University of Pisa doctoral dissertation, On the Singularities of the Jacobian of Four, of Three, of Two Surfaces, under Bianchi’s supervision. In its introduction, Pieri referred to Cremona 1870 for background information. In particular, the Jacobian of four given surfaces is the surface defined by the determinant of partial derivatives of the polynomials that define the given surfaces. It is the locus of points whose polar planes with respect to those surfaces are linearly dependent. The previous sentence can be modified to obtain definitions of the Jacobian curve of three given surfaces, and the Jacobian (finite) point set of two. Pieri employed analytic methods to study these objects. The thesis was never published in any journal; it will be discussed further in the third book of the present series. Beppo Levi reported that Pieri may have intended to publish an abstract, “On the singularities of the Jacobian of points and of multiple

22 Campedelli 1959, 5; Pisa 1999, 18–19; Tricomi 1962. Pisa 1884. Pisa 1881–1882, 12–15, 20, 60, 62; 1882–1883, 13, 20, 42, 58; 1883–1884, 62, 71. Pieri 1881–1882 is a set of handwritten lecture notes, apparently from Dini’s calculus course that Pieri attended. Pieri’s licenza award was termed a “full pass” (approvato con pieni voti assoluti). 23 Pieri’s record lists an 1881–1882 course merely as descriptive geometry, which was offered by the architect Guglielmo Martolini. But Beppo Levi reported (1913–1914, 65) that Pieri’s teachers included Nardi Dei, who gave the course listed here at that time. While all these courses are listed on Pieri’s record as obligatory, university records show Nardi Dei’s course and Dini’s calculus as elective. Levi also identified Cesare Finzi as one of Pieri’s teachers. Pieri’s record lists an 1882–1883 course as calculus exercises, but the only exercises course offered that year was the one listed here. The subject of Bianchi’s course that year is listed on Pieri’s record as algebraic equations, but on university records as algebraic functions, as shown here. During that time, Betti was the director of the Scuola.