1 Most Internet sources are free, but some—for example JSTOR —are free only through libraries with paidsubscriptions.

Bibliography

This bibliography includes all, and only, works referred to in the present volume. It includes thepublications that Pieri cited in the papers whose translations are found in chapter 3 and section4.2. Its entries for Pieri’s own works duplicate those in chapter 6. The present book mentionsmore than one author named Pieri; citations that include this surname only are references toMario Pieri. Alphabetization of the entries ignored diacritical marks, punctuation, and spaceswithin names.

Some cited works have appeared in several editions. Information about the first often has his-torical interest, even when reference to a later one is more appropriate. In such cases, both datesare given, as in

Amaldi, Ugo. [1938] 1954.

The remaining data in such citations refer to the later edition unless otherwise specified.

Some journals cited here identify their volumes with year ranges rather than single years. Anarticle in such a journal is identified by its year of publication; the range is given in parenthesesfollowing the volume number.

Book citations often include an International Standard Book Number (ISBN) and a Libraryof Congress (LC) catalog number. Caution! A book may have similar editions with differentISBNs. Moreover, the LC number is not standard, unless the book was cataloged upon publicationand its number included on the copyright page. From one library to the next, the LC number mayvary slightly—particularly its last digits. Many cited publications are now available on the Inter-net,1 but no standard means of locating them has yet evolved. Notations such as Internet: Cornellare provided to help readers locate publications in academic Internet repositories. The word afterthe colon (:) identifies an Internet http:// address in the following list:

Cornell . . . . . . . . . cdl.library.cornell.eduEBSCO . . . . . . . . . ejournals.ebsco.comGöttingen . . . . . . . gdz.sub.uni-goettingen.deGutenberg . . . . . . www.gutenberg.orgJSTOR . . . . . . . . . www.jstor.orgMichigan . . . . . www.hti.umich.edu/u/umhistmathNUMDAM . . . . . . archive.numdam.orgParis . . . . . . . . . . . gallica.bnf.frPoland . . . . . . . . . matwbn.icm.edu.plScienceDirect . . . . www.sciencedirect.com

Many citations contain brief annotations that explain their origin, import, or relationship toother items. An annotation may include a code in the format

JFM: 12.0123.01 Surname

402 Bibliography

2 JFM reviewers are identified because many played other roles in the life and legacy of Mario Pieri. Someindicated JFM and MR reviews refer to earlier editions of the work cited. Some early MR reviews werenumbered in a different format.

3 Chicago 1993, 15.208–209; Mann 1998.4 The serials list in May 1973, Appendix 2, may help with this problem.

to indicate, for example, that the work cited was reviewed in the Jahrbuch über die Fortschritteder Mathematik, volume 12, page 0123, item 01, by the named reviewer. This journal may haveLC: QA1.J3, and is available on the Internet at http://www.emis.de/MATH/JFM/JFM.html.Similarly, a code in the format

BSM2: 12(date)0123

indicates that the work cited was reviewed in Bulletin des sciences mathématiques, seconde partie,volume 12(date), page 0123. This journal may have LC: QA1.B8. A code in the format

JSL: 1234

indicates that the work cited was listed as author 123, work 4, in volume 1 or 3 of the Journal ofSymbolic Logic (Church 1936). That bibliography is available on the Internet via JSTOR; thejournal may have LC: BC1.J6. A code in the format

MR: 1234567(date:12345)

indicates that the work cited was reviewed in Mathematical Reviews, available on the Internetat http://www.ams.org/mathscinet.2

Each citation concludes with a list [in brackets] of the sections of this book and other itemsin this bibliography that contain references to it.

As much as possible, journal and book titles appear in the present bibliography as they did onthe published title pages. This practice, advocated by current documentation authorities,3 isconsistent with major catalogs now accessible and searchable electronically. But readers mayencounter difficulty matching titles, particularly of journals, with nomenclature in past literature,or in catalogs based on it, which often differs considerably from that of title pages. For example,the Giornale di matematiche was often referred to as the Giornale di Battaglini, after the originaleditor; and the journal published by the organization known as the Dorpater Naturforscher-gesellschaft may have been listed by an institutional name—Universität Tartu—or in thelanguage—Estonian—more politically correct at the time the catalog was constructed.4

Agazzi, Evandro. 1985. Prefazione. In Borga, Freguglia, and Palladino 1985, 7–10. [Cited in 5.3.]Amaldi, Ugo. [1938] 1954. Della vita e delle opere di Salvatore Pincherle. In Pincherle 1954,

2–16. Commemoration originally presented to the Reale Accademia dei Lincei in 1938. [Citedin 1.3 and item Pincherle 1954.]

Amato, Vincenzo. 1946. Sebastiano Catania. Annuario dell’Università di Catania per l’annoaccademico 1945–1946: 244–245. [Cited in 1.3.]

Anellis, Irving H. 1991. The first Russell paradox. In Drucker 1991, 33–46. [Cited in 4.1 anditem Drucker 1991.]

Aprile, Giorgio. 1946. Giuseppe Marletta. Bollettino della Unione Matematica Italiana (series3) 8: 55–56. LC: QA1.U6. [Cited in 1.3.]

Bibliography 403

Archimedes. [1897] 2002a. On the Sphere and Cylinder, book I. In Archimedes [1897] 2002b,1–55. Pieri cited this work, but no specific edition, in 1908a. [Cited in 3.4 and items Archi-medes [1897] 2002b and Pieri 1908a.]

———. [1897] 2002b. The Works of Archimedes, edited by Thomas L. Heath. Mineola, New York:Dover Publications. LC: QA31.A692. ISBN: 0-486-42084-1. MR: 2000800. Originally pub-lished in Cambridge, England, in 1897 by Cambridge University Press, with a supplement in1912. [Cited in item Archimedes [1897] 2002a.]

Arrighi, Gino. 1981. L’archivio di Mario Pieri. Accademia Lucchese di Scienze, Lettere e Arti,Studi e testi, 15. Lucca: Maria Pacini Fazzi Editore. LC: Z6616.P546.A77. This describes Pieri[no date]. [Cited in 6.6, 6.7, and items Pieri 1898a, 1905a, [no date].]

———, editor. 1997. Lettere a Mario Pieri (1884–1913). Quaderni P.RI.ST.EM 6 per l’archiviodella corrispondenza dei matematici italiani. Milan: ELEUSI, Sezione P.RI.ST.EM. The seriestitle is “Booklets of the Project on Historical and Methodological Research (Progetto RicercheStoriche e Metodologiche) for archiving the correspondence of Italian mathematicians.” In thepresent book, these letters are cited under this publication, not under their authors’ names.[Cited in 1.3, 2.5, 4.1, 5.3, 6.6, 6.7, permissions, and item Pieri [1901] 1997.]

Aspray, William. 1991. Oswald Veblen and the origins of mathematical logic at Princeton. InDrucker 1991, 54–70. MR: 1234045 (94g:03004). [Cited in 1.3 and item Drucker 1991.]

Associazione per l’Insegnamento della Fisica, Sezione di Catania. 2005. Internet website http://www.ct.infn.it/~aif/boggio_lera.htm. [Cited in 1.3.]

Avellone, Maurizio, and Marco Borga. 1997. Mario Pieri e i fondamenti della geometria. Letteramatematica P.RI.ST.EM. 26: 26–51. The acronym stands for Progetto Ricerche Storiche eMetodologiche. [Cited in 2.2.]

Avellone, Maurizio, Aldo Brigaglia, and Carmela Zappulla. 2002. The foundations of projectivegeometry in Italy from De Paolis to Pieri. Archive for History of Exact Sciences 56: 363–425.LC: Q125.A1.A75. Internet: EBSCO. MR: 1918541 (2003g: 01028). [Cited in 2.3.]

Bachmann, Friedrich. 1973. Aufbau der Geometrie aus dem Spiegelungsbegriff. Second edition.Die Grundlehren der mathematischen Wissenschaften, 96. Berlin: Springer-Verlag. LC:QA473. B3. ISBN: 3-540-06136-3. MR: 0107835 (21 #6557), 0346643 (49 #11368). [Cited in3.10.]

Baldwin, James M., editor. 1901–1905. Dictionary of Philosophy and Psychology; IncludingMany of the Principal Conceptions of Ethics, Logic, Aesthetics, Philosophy of Religion, MentalPathology, Anthropology, Biology, Neurology, Physiology, Economics, Political and SocialPhilosophy, Philology, Physical Science, and Education; and Giving a Terminology in English,French, German and Italian. Three volumes in four. London: Macmillan and company. LC:B41.B3. [Cited in the chapter 2 introduction.]

Baltzer, Richard. 1885. August Ferdinand Möbius. In Möbius 1885–1887, volume 2, v–xx. [Citedin 1.3 and item Möbius 1885–1887.]

Barbagli, Marzio. [1974] 1982. Educating for Unemployment: Politics, Labor Markets, and theSchool System—Italy, 1859–1973. Translated by Robert H. Ross. New York: ColumbiaUniversity Press. LC: LA791.B3213. ISBN: 0-231-05284-7. First published in Italian asDisoccupazione intellectuale e sistema scolastico in Italia, Bologna: Società Editrice Il Mulino,1974. [Cited in 1.1.]

Barbarin, Paul. 1902. La géométrie non-Euclidienne. Scientia, série physico-mathématique, 15.Paris: C. Naud. LC: QA685.B3. Internet: Michigan. [Cited in 1.3.]

Barberis, Bruno. 1999. Francesco Siacci, Carlo Somigliana. In Roero 1999b, 487–489, 511–514.[Cited in 1.3 and item Roero 1999b.]

Bar-Hillel, Yehoshua, editor. 1965. Logic, Methodology and Philosophy of Science: Proceedingsof the 1964 International Conference. Amsterdam: North-Holland Publishing Company. LC:BC135.I52. [Cited in item Schwabhäuser 1965.]

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Barker, William A. et al. 1976. Alfred North Whitehead. In Gillispie 1970–, volume 14, 302–310.[Cited in 1.3 and item Gillispie 1970–.]

Bayerische Akademie der Wissenschaften. Historische Kommission. 1954–. Neue deutscheBiographie. Twenty-two volumes to date. Berlin: Duncker & Humblot. LC: DD85.N48. [Citedin item Fritsch 2001.]

Beaufront, Louis de. 1914. Doktoro Louis Couturat, 1868–1914. Frankfurt am Main: Universi-tätsdruckerei Werner und Winter. Available on the Internet at http://www.literaturo.ido.li/louiscouturat.pdf . Written in Ido. [Cited in 1.3.]

Bekemeier, Bernd. 1987. Martin Ohm (1792–1872): Universitäts- und Schulmathematik in derneuhumanistischen Bildungsreform. Foreword by Michael Otte. Studien zur Wissenschafts-,Sozial-, und Bildungsgeschichte der Mathematik, 4. Göttingen: Vandenhoeck & Ruprecht.LC: QA29.O35.B4. ISBN: 3-525-40311-9. MR: 0919944 (89e:01026). [Cited in 4.1.]

Bellavitis, Giusta. 1836. Teoria delle figure inverse, e loro uso nella geometria elementare.Annali delle scienze del Regno Lombardo-Veneto 6: 126–142. LC: Q4.A48. [Cited in 2.4.]

Bertini, Eugenio. 1907. Introduzione alla geometria proiettiva degli iperspazi, con appendice sullecurve algebriche e loro singolarità. Pisa: Enrico Spoerri. LC: QA691.B4. Internet: Cornell.JFM: 38.0582.02 Steinitz. [Cited in 1.3.]

Bertini, Eugenio, et al. [1896] 1897. Relazione della Commissione esaminatrice del concorso allaCattedra di professore straordinario di geometria projettiva e descrittiva vacante presso la R.Università di Bologna. Bollettino ufficiale del Ministero della Pubblica Instruzione 24(1,11):529–535. The report is dated 19 October 1896. [Cited in 1.1.]

Berzolari, Luigi, Giulio Vivanti, and Duilio Gigli, editors. 1930–1953. Enciclopedia delle matema-tiche elementari. Three volumes, in two, two, and three parts. Milan: Ulrico Hoepli. LC: QA37.E57. JFM: 56.0061.03, 63.1151.01 Feigl; 64.1255.01 Zacharias. MR: 0034291 (11,570c). [Citedin 1.3, 4.4, and items Burali-Forti 1938, Cassina 1934, Fano 1938, Gigli 1930, and Padoa 1930.]

Beth, Evert W. 1959. The Foundations of Mathematics: A Study in the Philosophy of Science.Amsterdam: North-Holland Publishing Company. LC: QA9.B47. MR: 0118674 (22 #9445).[Cited in 4.1.]

Bettazzi, Rodolfo. 1890. Teoria delle grandezze. Opera premiata dalla Reale Accademia deiLincei. Pisa: Enrico Spoerri. JFM: 22.0071.01 Vivanti. [Cited in 1.3.]

Betti, Enrico. 1860. Sopra la teorica generale delle superficie curve. Annali di matematica puraed applicata 3: 336–339. LC: QA1.A62. Reprinted in Betti 1903–1913, volume 2, 11–15. [Citedin 6.1 and items Betti 1903–1913 and Pieri 1886a.]

———. 1871. Sopra gli spazi di un numero qualunque di dimensioni. Annali di matematica puraed applicata 11: 140–158. LC: QA1.A62. JFM: 03.0301.01. Reprinted in Betti 1903–1913,volume 2, 273–290. [Cited in 1.3, 6.1, and item Betti 1903–1913.]

———. 1903–1913. Opere matematiche. Two volumes. Milan: Ulrico Hoepli. LC: QA3.B56.Internet: Michigan. [Cited in items Betti 1860, 1871.]

Bézout, Etienne. 1779. Théorie générale des équations algébriques. Paris: Imprimerie de Ph.-D.Pierres. LC: QA211.B49.T5. Internet: Paris. [Cited in 1.2.]

Biermann, Kurt-Reinhold. 1971. Julius Wilhelm Richard Dedekind. In Gillispie 1970–, volume4, 1–5. [Cited in 1.3 and item Gillispie 1970–.]

Black, Max, editor. 1965. Philosophy in America. Ithaca, New York: Cornell University Press.LC: B934.B56. [Cited in item Parsons 1965.]

Blair, David E. 2000. Inversion Theory and Conformal Mapping. Student mathematical library,9. Providence: American Mathematical Society. LC: QA473.B57. ISBN: 0-821-82636-0. MR:1779832 (2001m:30025). Contains a reprint of Carathéodory 1937. [Cited in 2.4 and itemCarathéodory 1937.]

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Bologna, Provincia di. Istituto Tecnico “Pier Crescenzi.” 1995. Una scuola nel tempo: Documentie collezioni dell’Istituto Tecnico “Pier Crescenzi.” Bologna: Edizioni Grafis. ISBN: 8-880-81037-5. When Mario Pieri attended this school it was called the Reale Istituto Tecnico di Bologna.[Cited in items Bresadola 1995 and Breveglieri 1995.]

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Bolondi, Giorgio. 1998. I matematici italiani e i problemi della scuola: Alcuni spunti storici.Notiziario del Laboratorio LRM3D2: Supplemento 1 al notiziario dell’Università degli Studidi Trento 5(3, 20 January): 2–3. Available on the Internet at www-math.science.unitn.it/LRM3D2. The acronym stands for the Laboratorio di Ricerca sui Materiali e i Metodi per laDidactica e la Divulgazione della Matematica at the University of Trent. [Cited in 1.1.]

Bolyai Farkas. [1851] 1913. Kurzer Grundriss eines Versuches. In Bolyai and Bolyai 1913,volume 2, 118–179. The second part of this work, “Von den Gründen der Geometrie,” startson page 149. [Cited in 3.4 and item Bolyai and Bolyai 1913.]

Bolyai Farkas and Bolyai János. 1913. Geometrische Untersuchungen. Edited by Paul Stäckel.Two volumes. Leipzig: B. G. Teubner. LC: QA685.S78. Internet: Michigan. JFM: 44.0015.03Salkowski. The title page lists the authors as Wolfgang and Johann Bolyai. [Cited in itemBolyai [1851] 1913.]

Bolzano, Bernard. [1817] 1996. Purely analytic proof of the theorem that between any two valueswhich give results of opposite sign there lies at least one real root of the equation. Translatedby Steven Russ. In Ewald 1996, volume 1, section 6.C, 225–248. The original German editionwas published in Prague by Gottlieb Haase. [Cited in 4.1 and item Ewald 1996.]

———. [1851] 1950. Paradoxes of the Infinite. Edited by Frantisek P ihonsky. Translated, witha historical introduction, by Donald A. Steele. New Haven: Yale University Press. LC: QA9.B662. Partially reprinted in Ewald 1996, volume 1, 249–292. The original book, published inLeipzig by Reclam, was reprinted in Berlin by Mayer & Müller in 1889. [Cited in 4.1 and itemEwald 1996.]

Bompiani, Enrico. 1950. Commemorazione del Socio Luigi Berzolari. Accademia Nazionale deiLincei, Classe di Scienze Fisiche, Matematiche e Naturali: Rendiconti (series 8) 9(2): 396–410.LC: AS222.A24. [Cited in 1.3.]

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Borga, Marco, and Dario Palladino. 1992. Logic and foundations of mathematics in Peano’sschool. Modern Logic 3: 18–44. LC: BC1.M63. MR: 1187731 (94b:03012). [Cited in 1.2, 2.5,and 4.1.]

Borodin, A. I., and N. I. Lavrenko. 1993. Winners of N. I. Lobachevskii International Prize forMathematics. Russian Mathematics (Iz. VUZ) 37(7): 68–76. LC: QA1.S658. MR: 1287780(95d:01026). [Cited in 1.1.]

Borsuk, Karol, and Wanda Szmielew. 1960. Foundations of Geometry. Revised English transla-tion. Amsterdam: North-Holland Publishing Company. LC: QA681.B632. MR: 0071791 (17,184f), 0143072 (26 #638). The original 1955 edition was published in Warsaw by Pa stwoweWydawnictwo Naukowe. [Cited in 5.2.]

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———. 1903–1904. Sulla teoria generale delle grandezze e dei numeri. Atti della Reale Acca-demia delle Scienze di Torino 39: 256–272. LC: AS222.A29. JFM: 35.0088.01 Vivanti. JSL:8617. [Cited in 4.4.]

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———. 1925. A proposito di «Una lettera di Mario Pieri». Bollettino di matematica 21: 136–137. LC: QA1.B6. JFM: 51.0049.02 Freudenthal. JSL: 8626. Response to Maccaferri 1925.[Cited in 6.6 and item Maccaferri 1925.]

———. 1938. Elementi di calcolo vettoriale. In Berzolari, Vivanti, and Gigli 1930–1953, volume2, part 2, 105–139. JFM: 64.1255.01 Zacharias. [Cited in 1.3 and item Berzolari, Vivanti, andGigli 1930–1953.]

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———. [1900] 1901. Essai d’une théorie algébrique des nombres entiers, précédé d’une intro-duction logique à une théorie déductive quelconque. In International Congress of Philosophy1900–1903, volume 3 (1901), 309–365. The introduction, “Logical introduction to any deduc-tive theory,” translated by Jean van Heijenoort, is in Heijenoort [1967] 1970, 118–123. [Citedin 4.3 and items International Congress of Philosophy 1900–1903 and Lovett 1900–1901.]

———. [1900] 1902. Un nouveau système de définitions pour la géométrie euclidienne. InInternational Congress of Mathematicians [1900] 1902, 353–363. JFM: 32.0485.07 Wallenberg.[Cited in 2.5, the chapter 3 introduction, 3.10, and item International Congress of Mathe-maticians [1900] 1902.]

———. 1901. Numeri interi relativi. Revue de mathématiques (Rivista di matematica) 7: 73–84.LC: QA1.R5. JFM: 32.0174.03. JSL: 986. Reissued in Roero 2003. [Cited in 4.3 and itemRoero 2003.]

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———. [1912] 1967. La valeur et les rôles du principe d’induction mathématique. In Inter-national Congress of Mathematicians [1912] 1967, 471–479. JFM: 44.0074.04 Lampe. JSL:9818. [Cited in 4.3 and item International Congress of Mathematicians [1912] 1967.]

———. 1930. Logica. In Berzolari, Vivanti, and Gigli 1930–1953, volume 1, part 1 (1930),chapter 1, 1–79. [Cited in 4.4 and item Berzolari, Vivanti, and Gigli 1930–1953.]

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Palladino, Franco. 2000a. La corrispondenza epistolare di Federico Amodeo: Catalogo ragionatocon regesti degli scritti. Publicazioni dell’Università degli Studi di Salerno, Sezione di Studidi Scienze Naturali, Fisiche, Matematiche e Tecnologiche, 5. Naples: Edizioni ScientificheItaliane. LC: Z6616.A575.P35. ISBN: 8-849-30203-6. [Cited in 1.3 and item Pieri 1890c.]

———, editor. 2000b. Le corrispondenze epistolari tra Peano e Cesàro e Peano e Amodeo.Quaderni P.RI.ST.EM, 13. Salerno: Università Bocconi. The acronym stands for ProgettoRicerche Storiche e Metodologiche. [Cited in 1.1.]

Pambuccian, Victor. 1988. Simplicity. Notre Dame Journal of Formal Logic 29: 396–411. LC:BC1. N6. Available on the Internet at http://projecteuclid.org. MR: 0953707 (89f:03007).[Cited in 3.10.]

———. 1990–1991. Unit distance as single binary predicate for plane Euclidean geometry.Zeszyty Naukowe—Geometria 18: 5–8, 19: 87. LC: QA440.Z4. MR: 1067923 (91j: 51025).[Cited in 5.2.]

Panetti, Modesto. 1920. Nicodemo Jadanza. Atti della Reale Accademia delle Scienze di Torino55(1919–1920): 633–649. LC: AS222.A29. [Cited in 1.3.]

Parducci, Amos. 1936. Silvio Pieri. Bollettino storico Lucchese 8: 43–47. LC: DG975.L9.A12.Reprinted in 1936 by the Scuola Tipographica Artigianelli in Lucca. [Cited in 1.1 and 1.3.]

Parma, Archivio Storico e Generale dell’Università degli Studi di. [no date]. Registro delpersonale universitario. [Cited in 1.3 and items Parma 1908 and Pieri 1911a.]

Parma, University of. 1908. Professore Mario Pieri. In Parma [no date] 4, 146 (16 October).[Cited in 1.1 and item Parma [no date].]

———. 1908–1914. Annuario della Reale Università di Parma. One volume for each academicyear. [Cited in 1.1, 1.3, and items B. Levi 1913–1914 and Tartufari 1913.]

Parsons, Charles D. 1965. Frege’s theory of number. In Black 1965, chapter 9, 180–203. Reprint-ed in Parsons 1983, chapter 6, 150–175. [Cited in 4.1 and items Black 1965 and Parsons 1983.]

———. 1983. Mathematics in Philosophy: Selected Essays. Ithaca, New York: Cornell UniversityPress. LC: QA8.6.P37. ISBN: 0-801-41471-7. MR: 0728948 (86a:01050). [Cited in itemParsons 1965.]

———. 1987. Developing arithmetic in set theory without infinity: Some historical remarks.History and Philosophy of Logic 8: 201–213. LC: BC1.H58. MR: 923650 (89a: 03088). [Citedin 4.1.]

Pasch, Moritz. 1882a. Einleitung in die Differential- und Integralrechnung. Leipzig: Verlagvon B. G. Teubner. LC: QA303.P35. JFM: 14.0197.02 Hoppe. [Cited in 1.3, the chapter 4introduction, and 4.1.]

———. 1882b. Vorlesungen über neuere Geometrie. Leipzig: B. G. Teubner. LC: QA471.P3.JFM: 14.0498.01 Stahl, 43.1113.03 and 43.0612.05 Salkowski. The 1912 printing is availablevia Internet: Michigan. [Cited in 1.2, 1.3, 2.5, and 3.3.]

———. 1894. Ueber den Bildungswerth der Mathematik: Akademische Festrede zur Feier desJahresfestes der Grossherzoglich Hessischen Ludewigs-Universität am 2. Juli 1894. Giessen:

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Peano, Giuseppe. [1886] 1973. On the integrability of first-order differential equations. In Peano1973, 51–57. Translation of “Sull’integrabilità delle equazioni differenziali di primo ordine,”Atti della Reale Accademia delle Scienze di Torino 21(1885–1886): 677–685, LC: AS222.A29,JFM: 18.0284.02 Hamburger, which was reprinted in Peano 1957–1959, volume 1, 74–81, andincluded in Peano 2002 as file 1885a.pdf. [Cited in 5.1 and items Peano 1957–1959, 1973, and2002.]

———. 1887. Applicazioni geometriche del calcolo infinitesimale. Torino: Fratelli Bocca Editore.Internet: Cornell. JFM: 19.0248.01 Vivanti. Included in Peano 2002 as file 1887b.pdf.Excerpts from chapter 5 are translated in Peano 1973, 67–74. [Cited in 5.1 and items Peano1973, 2002.]

———. [1887] 1973. Integration by series of linear differential equations. In Peano 1973, 58–66.Translation of “Integrazione per serie delle equazioni differenziali lineari,” Atti della RealeAccademia delle Scienze di Torino 22(1886–1887): 437–446, LC: AS222.A29, JFM: 19.0308.02Hamburger, which is included in Peano 2002 as file 1887a.pdf. [Cited in 5.1 and items Peano1973, 2002.]

———. [1888] 2000. Geometric Calculus According to the Ausdehnungslehre of H. Grassmann.Translated by Lloyd C. Kannenberg. Boston: Birkhäuser. LC: QA205.P4313. ISBN: 0-817-64126-2. Translation of Calcolo geometrico secondo l’Ausdehnungslehre di H. Grassmann,preceduto dalle operazioni della logica deduttiva, Torino: Fratelli Bocca Editori, LC:QA259.P38, JFM: 20.0689.04 Loria, which is included in Peano 2002 as file 1888a.pdf. Thetitle refers to Grassmann [1844] 1994. [Cited in 1.2, 1.3, 5.1, and items Grassmann [1844] 1994and Peano 2002.]

———. 1889. I principii di geometria logicamente esposti. Torino: Fratelli Bocca Editore. LC:QA681.P4. Internet: Michigan. JFM: 21.0524.01 Loria. JSL: 713. Reprinted in Peano 1957–1959, volume 2, 56–91. Included in Peano 2002 as file 1889d.pdf. [Cited in 1.2, 2.3, 5.1, anditems Peano 1957–1959, 2002.]

———. [1889] 1973. The Principles of Arithmetic, Presented by a New Method. Translated byHubert C. Kennedy. In Peano 1973, 101–134. Translation of Arithmetices Principia, NovaMethodo Exposita, Turin: Fratres Bocca, JFM: 21.0051.02 Loria, JSL: 712, which was reprintedin Peano 1957–1959, volume 2, 20–55, and included in Peano 2002 as file 1889a.pdf. Theoriginal title page is reproduced on page 303 of the present book. [Cited in 1.3, 4.1, 4.2, 4.3,5.1, and items Peano 1957–1959, 1973, 2002.]

———. 1890. Démonstration de l’intégrabilité des équations differentielles ordinaires. Mathe-matische Annalen 37: 182–228. LC: QA1.M29. Internet: Göttingen. JFM: 22.0302.01Hamburger. JSL: 714. Reprinted in Peano 1957–1959, volume 1, 119–170. Included in Peano2002 as file 1890f.pdf. [Cited in 4.1 and items Peano 1957–1959, 2002.]

———. [1890] 1973a. A space-filling curve. In Peano 1973, 143–148. Translation of “Sur unecourbe, qui remplit toute une aire plane,” Mathematische Annalen 36: 157–160, LC: QA1.M29,Internet: Göttingen, JFM: 22.0405.01 Hilbert, which was reprinted in Peano 1957–1959,volume 1, 110–114, and included in Peano 2002 as file 1890b.pdf. [Cited in 5.1 and itemsPeano 1957–1959, 1973, 2002.]

———. [1890] 1973b. On some singular curves. In Peano 1973, 150–152. Translation of “Sopraalcune curve singolari,” Atti della Reale Accademia delle Scienze di Torino 26(1890–1891):299–302, LC: AS222.A29, JFM: 22.0701.01 Hilbert, which was reprinted in Peano 1957–1959,volume 1, 201–203, and included in Peano 2002 as file 1891a.pdf. [Cited in 5.1 and itemsPeano 1957–1959, 1973, 2002.]

———. 1891a. Formole di logica matematica. Rivista di matematica 1: 24–31. LC: QA1.R5.JFM: 23.0051.04 Schlegel. JSL: 716. Reprinted in Peano 1957–1959, volume 2, 102–113.

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Included in Peano 2002 as file 1891g.pdf. Reissued in Roero 2003. [Cited in 4.3 and itemsPeano 1957–1959, 2002 and Roero 2003.]

———. 1891b. Osservazioni del Direttore sull’articolo precedente. Rivista di matematica 1:66–69. LC: QA1.R5. Included in Peano 2002 as file 1891h.pdf. Reissued in Roero 2003.Comments on C. Segre [1891] 1903–1904; C. Segre 1891 is a response. [Cited in 1.3, 5.1, anditems Peano 2002, Roero 2003, and C. Segre 1891, [1891] 1903–1904.]

———. 1891c. Risposta. Rivista di matematica 1: 156–159. LC: QA1.R5. Included in Peano2002 as file 1891k.pdf. Reissued in Roero 2003. Response to C. Segre 1891. [Cited in 1.3, 5.1and items Peano 2002, Roero 2003, and C. Segre 1891.]

———. 1891d. Sul concetto di numero. Rivista di matematica 1: 87–102,256–267. LC: QA1.R5.JFM: 23.0052.01 Schlegel. JSL: 717. Reprinted in Peano 1957–1959, volume 3, 80–109.Included in Peano 2002 as files 1891i.pdf and 1891o.pdf. Reissued in Roero 2003. The partsare subtitled “Nota 1” and “Nota 2.” [Cited in 1.2, 4.2, 4.3, and items Peano 1957–1959, 2002and Roero 2003.]

———. [1891] 1973. The principles of mathematical logic. In Peano 1973, 153–161. Translationof “Principii di logica matematica,” Rivista di matematica 1: 1–10. LC: QA1.R5. JSL: 715.Reprinted in Peano 1957–1959, volume 2, 92–101. Included in Peano 2002 as file 1891c.PDF. Reissued in Roero 2003. [Cited in 5.1 and items Peano 1957–1959, 1973, 2002, and Roero2003.]

———. 1892. Review of Veronese 1891. Rivista di matematica 2: 143–144. LC: QA1.R5.Included in Peano 2002 as file 1892n.pdf. Reissued in Roero 2003. [Cited in 2.3 and itemsPeano 2002 and Roero 2003.]

———. 1894. Sui fondamenti della Geometria. Rivista di matematica 4: 51–90. LC: QA1.R5.JFM: 25.0854.01 Kötter. JSL: 7112. Reprinted in Peano 1957–1959, volume 3, 115–157.Included in Peano 2002 as file 1894c.pdf. Reissued in Roero 2003. [Cited in 1.2, 2.5, 5.1, 6.7,and items Peano 1957–1959,2002, and Roero 2003.]

———. 1895. Review of Frege 1893–1903, volume 1. Rivista di matematica 5: 122–128. LC:QA1.R5. JSL: 7119. Reprinted in Peano 1957– 1959, volume 3, 189–195. Partially translatedin Niddich 1963. Included in Peano 2002 as file 1895 .PDF. Reissued in Roero 2003. [Citedin 4.1 and items Frege 1893–1903, Nidditch 1963, Peano 1957–1959, 2002, and Roero 2003.]

———. 1898. Sul §2 del Formulario, t. II: Aritmetica. Rivista di matematica 6: 75–89. LC:QA1.R5. JFM: 29.0047.04 Michaelis. Reprinted in Peano 1957–1959, volume 3, 232–248.Included in Peano 2002 as file 1898e.pdf. Reissued in Roero 2003. [Cited in 4.3 and itemsPeano 1957–1959, 2002, and Roero 2003.]

———. [1900] 1901. Les définitions mathématiques. In International Congress of Philosophy1900–1903, volume 3 (1901), 279–288. Reprinted in Peano 1957–1959, volume 2, 362–368.Included in Peano 2002 as file 1901a.pdf. [Cited in 5.1 and items International Congress ofPhilosophy 1900–1903 and Peano 1957–1959, 2002.]

———. 1902. Aritmetica generale e algebra elementare. Turin: G. G. Paravia e C. LC:QA248.P35. JSL: 7139. Included in Peano 2002 as file 1902b.pdf. Reviewed in Pieri 1903c.[Cited in 6.7 and items Peano 2002 and Pieri 1903c.]

———. 1903. La geometria basata sulle idee di punto e distanza. Atti della Reale Accademia delleScienze di Torino 38(1902–1903): 6–10. LC: AS222.A29. JFM: 34.0530.02 Dehn. JSL: 7141.Reprinted in Peano 1957–1959, volume 3, 268–272. Included in Peano 2002 as file 1903a.pdf. [Cited in 2.5, the chapter 3 introduction, 3.10, 5.1, and items Peano 1957–1959, 2002.]

———. [1903] 1902–1906. De latino sine flexione: Lingua auxiliare internationale. Revista demathematica 8: 74–83. LC: QA1.R5. JFM: 34.0078.03 Vivanti. Included in Peano 2002 as file1903d.pdf. Reissued in Roero 2003. This specification starts in Scholastic Latin; as it proposessyntax simplifications, it implements them, so that it ends in Uninflected Latin. [Cited in 5.1and items Peano 2002 and Roero 2003.]

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———. 1905. Sur les principes de la géométrie selon M. Pieri. Bulletin de la Société Physico-Mathématique de Kasan: - -

’ (series 2)14: 1–4. LC: QA1.K38. Included in Peano2002 as file 1904b.pdf. [Cited in 1.1 and item Peano 2002.]

———. 1906. Super theorema de Cantor-Bernstein. Rendiconti del Circolo Matematico diPalermo 8: 360–366. LC: QA1.C52. Also in Revista de mathematica 8(1906): 136 –143, LC:QA1.R5. JSL: 7143. Reprinted in Peano 1957–1959, volume 1, 337–344. Included in Peano2002 as file 1906b. Reissued in Roero 2003. Written in Uninflected Latin. [Cited in 1.2 anditems Peano 1957–1959, 2002 and Roero 2003.]

———. 1912. Contra gli esame. Torino nuova (17 August): 2. Newspaper article. Included inPeano 2002 as file 1912o.pdf. [Cited in 5.3 and item Peano 2002.]

———. 1913. Mario Pieri. Academia pro Interlingua: Discussiones 4: 31–34. Included in Peano2002 as file 1913f.pdf. Reissued in Roero 2003. Not signed. Written in Uninflected Latin.[Cited in 1.1, 5.1, 6.7, and items Peano 2002 and Roero 2003.]

———. 1915. Vocabulario commune ad Latino-Italiano-Français-English-Deutsch pro usu deinterlinguistas. Second edition. Cavoretto: Academia pro Interlingua. LC: PM8401.P3.Included in Peano 2002 as file 1915b.pdf. Written in Uninflected Latin. [Cited in 5.1 and itemPeano 2002.]

———. [1915] 1973. The importance of symbols in mathematics. Translated by Hubert C.Kennedy. In Peano 1973, 227–234. Translation of “Importanza dei simboli in matematica,”Scientia (Rivista di scienza) 18: 165–173, LC: Q4.S4, JSL: 7151, which was reprinted in Peano1957–1959, volume 3, 389–396, and included in Peano 2002 as file 1915j.pdf. [Cited in thechapter 2 introduction and items Peano 1957–1959, 1973, 2002.]

———. [1916] 2002. Pubblicazioni di Giuseppe Peano. Unpublished memorandum. Includedin Peano 2002 as file 1916e.pdf. [Cited in 4.3 and item Peano 2002.]

———. [1921] 1973. Definitions in mathematics. Translated by Hubert C. Kennedy. In Peano1973, 235–246. Translation of “Le definizioni in matematica,” Periodico di matematica perl’insegnamento secondario (series 4)1: 175–189, LC: QA1.P5, JSL: 7152.1, which was reprintedin Peano 1957–1959, volume 2, 423–438, and included in Peano 2002 as file 1921d.pdf. [Citedin 4.1 and items Peano 1957–1959, 1973, 2002.]

———. 1926. Sinense. Academia pro Interlingua 6: 129–130. Included in Peano 2002 as file1926i.pdf. Reissued in Roero 2003. Written in Uninflected Latin. [Cited in 5.3 and itemsPeano 2002 and Roero 2003.]

———. 1957–1959. Opere scelte. Volume 1, Analisi matematica—calcolo numerico. Volume 2,Logica matematica—Interlingua ed algebra della grammatica. Volume 3, Geometria e fonda-menti meccanica razionale—varie. Rome: Edizioni Cremonese. LC: QA3.P43. MR: 0090539(19,827a), 0099903 (20 #6339), 0104549 (21 #3302). [Cited in 1.3 and items Peano [1886] 1973,1889, [1889] 1973, 1890, [1890] 1973a, [1890] 1973b, 1891a, 1891d, [1891] 1973, 1894, 1895,1898, [1900] 1901, 1903, 1906, [1915] 1973, [1921] 1973.]

———. 1973. Selected works of Giuseppe Peano. Translated and edited, with a biographicalsketch and bibliography, by Hubert C. Kennedy. Toronto: University of Toronto Press. LC:QA3.43131. ISBN: 0-802-05267-3. MR: 0465764 (57 #5653). [Cited in items Genocchi 1884and Peano [1886] 1973, 1887, [1887] 1973, [1889] 1973, [1890] 1973a, [1890] 1973b, [1891]1973, [1915] 1973, [1921] 1973.]

———. 2002. L’opera omnia di Giuseppe Peano. Edited by Clara S. Roero. Turin: Dipartimentodi Matematica, Università di Torino. CD-ROM. ISBN: 8-890-08762-5. [Cited in items D’Ovi-dio, Segre, and Peano 1898; Genocchi 1884; Peano [1886] 1973, 1887, [1887] 1973, [1888]2000, 1889, [1889] 1973, 1890, [1890] 1973a, [1890] 1973b, 1891a, 1891b, 1891c, 1891d, [1891]1973, 1892, 1894, 1895, 1898, [1900] 1901, 1902, 1903, [1903] 1902–1906, 1905, 1906, 1912,1913, 1915, [1915] 1973, [1916] 2002, [1921] 1973, 1926; and Peano et al. 1895–1908.]

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Peano, Giuseppe, and Corrado Segre. 1905. Relazione sulla memoria del prof. Mario Pieri:“Nuovi principî di Geometria projettiva complessa.” Atti della Reale Accademia delle Scienzedi Torino 40(1904–1905): 378–379. LC: AS222.A29. JFM: 36.0613.01 Loria. Review of Pieri1905c. [Cited in 1.2, 6.4, and item Pieri 1905c.]

Peano, Giuseppe, et al. 1895–1908. Formulaire mathématique. Five volumes, the second in threeparts and the fifth in two. Turin: Bocca Frères, Ch. Clausen; Paris: Carré et Naud (volume3). LC: QA41.F6. JFM: 28.0063.03, 30.0073.02, 32.0069.02, 33.0073.04, 36.0087.03, 39.0084.01Vivanti. JSL: 7117, 21,25,30,35,40, 45. Volume five is titled Formulario mathematico. Thematerial by Peano in volume one is included in Peano 2002 as files 1895d.pdf to 1895h.pdf.The entire volume two is in files 1897b.pdf, 1898f.pdf, and 1899b.pdf. Volumes three andfour are in files 1901b.pdf and 1903f.pdf. The 1906 and 1908 editions of volume five are in1906g.pdf and 1908a.pdf. The last was reprinted, with introduction and notes by Ugo Cassina,and with the support of the city of Cuneo, Rome: Edizioni Cremonese, 1960, LC: QA41.P42,MR: 0114745 (22 #5564). [Cited in 3.9, 4.1, and item Peano 2002.]

Pearson, Patricia Cowan. 1949. Hilbert’s Geometric Postulates as Theorems. Senior thesis.Portland, Oregon: Reed College. Mrs. Pearson’s maiden name, Patricia M. Cowan, is on thetitle page. [Cited in 2.5.]

———. 2005. Private communication to James T. Smith about Pearson 1949. [Cited in 5.3.]Pedrazzi, Maino. 1978. Le isometrie: da Eduard Study a Mario Pieri. Cultura e scuola 17:

204–215. LC: AS221.C8. [Cited in 2.5.]Peirce, Charles Sanders. 1881. On the logic of number. American Journal of Mathematics 4:

85–95. LC: QA1.A5. Internet: JSTOR. JFM: 13.0055.02 Michaelis. Reprinted in Ewald 1996,volume 1, 596–608. [Cited in 4.1 and item Ewald 1996.]

———. 1885. On the algebra of logic: A contribution to the philosophy of notation. AmericanJournal of Mathematics 7: 180–202. LC: QA1.A5. Internet: JSTOR. JFM: 17.0044.02Michaelis. Reprinted in Ewald 1996, volume 1, 608–632. [Cited in 4.1 and item Ewald 1996.]

Pejlare, Johanna. 2004. Torsten Brodén and the Principles of Geometry. Licentiate Dissertation.Göteborg: Department of Mathematical Sciences, Chalmers University of Technology andGöteborg University. Review of Brodén 1890. [Cited in 1.3, 2.5, and item Brodén 1890.]

Pensa, Angelo. 1912. Elementi di geometria. Turin: G. B. Petrini. [Cited in 3.10.]Petkantschin, Boyan Lazarov. 1940.

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———. 1931. Versioncelle da Fedro. Lucca: Scuola Tipografia Artigianelli. Poetry by MarioPieri’s bother. Fedro refers to Plato’s dialogue Phaedrus. [Cited in 1.1.]

Pieri, Mario. 1876–1880. Six astronomical drawings. In possession of Marco Campetti. Anno-tated in section 6.7. One of these—views of Mars—is also reproduced on page 9 of the presentbook. [Cited in 1.1 and 6.7.]

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———. 1884a. Questions for oral defense of the laureate. One signed handwritten sheetpreserved in Pieri 1884c. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and itemPieri 1884c.]

———. 1884b. Studi di geometria differenziale. Dissertation 6144A, library of the Universityof Pisa. Thesis completed in September 1884, evidently under the supervision of Luigi Bianchifor the Scuola Reale Normale Superiore. [Cited in 1.1, 1.2, 1.3, and 6.1.]

———. 1884c. Sulle singolarità della Jacobiana di quattro, di tre, di due superficie. Dissertation6144, library of the University of Pisa. Thesis for the laureate, completed in June 1884,evidently under the supervision of Luigi Bianchi for the University of Pisa. Contains Pieri1884a. The title page and first paragraph are reproduced on page 14 of the present book. [Citedin 1.1, 1.2, 1.3, 6.2, 6.6, permissions, and item Pieri 1884a.]

———. 1886a. Intorno ad un teorema dei sigg. Betti e Weingarten. Giornale di matematiche 24:290–308. LC: QA1.G5. JFM: 18.0723.01 August. Completed in 1885. This refers to Betti 1860and Weingarten 1863. [Cited in 1.2, 6.1, and items Betti 1860 and Weingarten 1863.]

———. 1886b. Sopra alcuni problemi riguardanti i fasci di curve e di superficie algebriche.Giornale di matematiche 24: 13–22. LC: QA1.G5. JFM: 18.0636.01 Loria. Completed in 1884.[Cited in 1.1 and 6.2.]

———. 1886c. Sulle normali doppie di una curva gobba algebrica. Atti della Reale Accademiadei Lincei: Rendiconti (series 4) 2: 327–329. LC: AS222.A23. JFM: 18.0638.01 Loria. BSM2:12(1888)91–92. Presented to the Academy by Riccardo De Paolis. [Cited in 1.2 and 6.2.]

———. 1886d. Sulle normali doppie di una superficie algebrica. Atti della Reale Accademia deiLincei: Rendiconti (series 4) 2: 40–42. LC: AS222.A23. JFM: 18.0639.01 Loria. BSM2: 12(1888)94. Presented to the Academy by Riccardo De Paolis. [Cited in 1.2 and 6.2.]

———. 1887a. Intorno alle superficie elicoidali. Giornale della Società di Letture e ConversazioniScientifiche di Genova 1–15. JFM: 19.0820.01 Loria. Completed in 1884. [Cited in 1.2 and6.1.]

———. 1887b. Sul principio di corrispondenza in uno spazio lineare qualunque ad n dimensioni.Atti della Reale Accademia dei Lincei: Rendiconti (series 4) 3: 196–199. LC: AS222.A23. JFM:19.0668.02 Loria. BSM2: 13(1889)168. Presented to the Academy by Riccardo De Paolis.[Cited in 6.2.]

———. 1888. Sopra un teorema di geometria ad n dimensioni. Giornale di matematiche 26:251–254. LC: QA1.G5. JFM: 20.0675.01 Loria. [Cited in 1.2 and 6.2.]

———, editor and translator. 1889a. Geometria di posizione, by G. K. C. von Staudt. Precededby a study of the life and works of Staudt by Corrado Segre. Biblioteca matematica, 4. Turin:Fratelli Bocca Editori. JFM: 21.0592.03 Loria. Edited translation of Staudt 1847. [Cited in1.1, 1.3, the chapter 2 introduction, 2.3, 5.1, 6.4, 6.7, and item Staudt 1847.]

———. 1889b. Sulle tangenti triple di alcune superficie del sest’ordine. Atti della Reale Acca-demia delle Scienze di Torino 24(1888–1889): 514–526. LC: AS222.A29. JFM: 21.0663.01Kötter. BSM2: 21(1897)196. Presented to the Academy by Enrico D’Ovidio. The opening isreproduced on page 21 of the present book. [Cited in 1.1, 1.2, and 6.2.]

———. 1890a. Sulla corrispondenza algebrica fra due spazi rigati. Atti della Reale Accademiadelle Scienze di Torino 25(1889–1890): 365–371. LC: AS222.A29. JFM: 22.0687. 01 Loria.Presented to the Academy by Corrado Segre. [Cited in 1.2 and 6.2.]

———. 1890b. Sulla geometria projettiva delle forme di 4a specie. Giornale di matematiche 28:209–218. LC: QA1.G5. JFM: 22.0682.02 Kötter. [Cited in 1.2 and 6.2.]

438 Bibliography

———. 1890c. Two letters to Federico Amodeo, 11 and 13 November. Amodeo papers B70.University of Salerno. Summarized in Palladino 2000a, 199; under the number B70. Thepresent authors’ copies of these letters bear the number B69. Annotated in section 6.6. [Citedin 1.1, 6.6, and item Palladino 2000a.]

———. 1891a. A proposito della nota del sig. Rindi «Sulle normali comuni a due superficie». Ren-diconti del Circolo Matematico di Palermo 5: 323. LC: QA1.C52. JFM: 23.0828.03 Braunmühl.BSM2: 37(1913)85. The cited note is Rindi 1891. [Cited in 1.2, 1.3, 6.2, and item Rindi 1891.]

———. 1891b. Formule di coincidenza per le serie algebriche ∞n di coppie di punti dello spazioa n dimensioni. Rendiconti del Circolo Matematico di Palermo 5: 252–268. LC: QA1.C52.JFM: 23.0700.02 Kötter. BSM2: 37(1913)82. [Cited in 1.2 and 6.2.]

———. 1891c. Geometria proiettiva: Lezioni per gli allievi nella Reale Accademia Militare diTorino. Turin: Tipografia Candelotti. JFM: 23.0618.01 Loria. [Cited in 6.4 and 6.7.]

———. 1892a. Osservazioni geometriche intorno alle linee diurne di un orologio solare. Bollet-tino della Società di Letture e Conversazioni Scientifiche di Genova 15: 83–95. JFM: 24.0560.01Loria. Completed in 1891. [Cited in 1.2 and 6.2.]

———. 1892b. Sopra le linee uniformemente illuminate di una superficie qualunque. Atti dellaReale Accademia delle Scienze di Torino 27(1891–1892): 347–353. LC: AS222.A29. JFM:24.0779.03 Vivanti. BSM2: 22(1898)190. Presented to the Academy by Giuseppe Bruno. [Citedin 1.2, 6.1, and 6.2.]

———. 1892c. Sopra un problema di geometria enumerativa. Giornale di matematiche 30:133–140. LC: QA1.G5. JFM: 24.0629.02 Loria. Completed in 1891. [Cited in 6.2.]

———. 1892d. Sulle trasformazioni birazionali dello spazio inerenti a un complesso linearespeciale. Rendiconti del Circolo Matematico di Palermo 6: 234–244. LC: QA1.C52. JFM: 24.0786.03 Loria. BSM2: 37(1913)89–90. [Cited in 1.2 and 6.2.]

———. 1892e. Sulle trasformazioni involutorie dello spazio determinate da un complesso Hirst-iano di rette. Reale Istituto Lombardo di Scienze e Lettere: Rendiconti (series 2) 25: 1037–1060.LC: AS222.I42. JFM: 24.0788.01 Haussner. BSM2: 28(1904)18. [Cited in 1.2 and 6.2.]

———. 1893a. Sopra alcune congruenze di coniche. Atti della Reale Accademia delle Scienzedi Torino 28(1892–1893): 289–303. LC: AS222.A29. JFM: 25.1119.01 Waelsch. BSM2:22(1898) 192. Presented to the Academy by Enrico D’Ovidio. [Cited in 1.2 and 6.2.]

———. 1893b. Sui sistemi lineari di coni. Rivista di matematica 3: 44–47. LC: QA1.R5. JFM:25.1236.01 Färber. Reissued in Roero 2003. [Cited in 1.2, 5.1, 6.2, and item Roero 2003.]

———. 1893c. Sui sistemi lineari di monoidi. Giornale di matematiche 31: 151–155. LC:QA1.G5. JFM: 25.1236.03 Wallenberg. [Cited in 6.2.]

———. 1893d. Sul problema degli spazi secanti. Reale Istituto Lombardo di Scienze e Lettere:Rendiconti (series 2) 26: 534–546. LC: AS222.I42. JFM: 25.1038.02 Schönflies. BSM2: 28(1904)245–246. This is the first note of a series with Pieri 1894c, 1895b. [Cited in 1.2, 6.2, anditems Pieri 1894c, 1895b.]

———. 1893e. Teoremi da dimostrare. Giornale di matematiche 31: 368–369. LC: QA1.G5.[Cited in 6.2.]

———. 1893f. Le trasformazioni razionali dello spazio inerenti ad una conica. Rendiconti delCircolo Matematico di Palermo 7: 296–306. LC: QA1.C52. JFM: 25.1027.01 Loria. BSM2: 37(1913)107. [Cited in 1.2 and 6.2.]

———. [1893] 1914. Di due proprietà caratteristiche per superficie elicoidali. Periodico di mate-matica per l’insegnamento secondario (series 3) 11: 229–233. LC: QA1.P5. JFM: 45.0875.05Loria. First published in 1893 as a pamphlet, Lucca: Tipografia Giusta, JFM: 25.1277.02Ostrowski. Republished in 1914 posthumously. [Cited in 1.2 and 6.1.]

———. 1894a. Per trovare graficamente i raggi di massima curvatura nelle superficie quadriche.El progresso matemático 4: 257–260. LC: QA1.P7. JFM: 25.1958.01 Teixeira. This journalwas published in Zaragoza, Spain. [Cited in 1.2, 6.1, and 6.2.]

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———. 1894b. Review of Thomae 1894. Rivista di matematica 4: 36–39. LC: QA1.R5. Reissuedin Roero 2003. Annotated in section 6.7. [Cited in 5.1, 6.7, and items Thomae 1894 and Roero2003.]

———. 1894c. Sul problema degli spazi secanti: Nota 2a. Reale Istituto Lombardo di Scienzee Lettere: Rendiconti (series 2) 27: 258–273. LC: AS222.I42. JFM: 25.1038.02 Schönflies.BSM2: 28(1904)247. This note is in a series with Pieri 1893d and 1895b. [Cited in 1.2, 6.2, anditems Pieri 1893d, 1895b.]

———. 1894d. Trasformazione di ogni curva algebrica in altra priva di punti multipli. Rivistadi matematica 4: 40–42. LC: QA1.R5. JFM: 25.1096.04 Hilbert. Reissued in Roero 2003.[Cited in 1.2, 5.1, 6.2, and item Roero 2003.]

———. 1895a. Sui principî che reggono la geometria di posizione. Atti della Reale Accademiadelle Scienze di Torino 30(1894–1895): 607–641. LC: AS222.A29. JFM: 27.0420.03 Kötter.BSM2: 22(1898)199. JSL: 921. Presented to the Academy by Giuseppe Peano. Reprinted inPieri 1980, 13–48. This is the first note of a series with Pieri 1896a and 1896b. [Cited in 1.2,the chapter 2 introduction, 2.3, 4.1, 5.1, 6.4, and items Pieri 1896a, 1896b, 1980.]

———. 1895b. Sul problema degli spazi secanti: Nota 3a. Reale Istituto Lombardo di Scienzee Lettere: Rendiconti (series 2) 28: 441–454. LC: AS222.I42. JFM: 27.0453.02 Kötter. BSM2:29(1905)6. This is the last note of a series with Pieri 1893d, 1894c. [Cited in 1.2, 6.2, and itemsPieri 1893d, 1894c.]

———. 1895c. Sulle trasformazioni razionali dello spazio che individuano complessi di tangenti.Giornale di matematiche 33: 167–178. LC: QA1.G5. JFM: 26.0767.01 Waelsch. Completedin 1894. [Cited in 1.2 and 6.2.]

———. 1896a. Sui principî che reggono la geometria di posizione: Nota 2a. Atti della Reale Acca-demia delle Scienze di Torino 31(1895–1896): 381–399. LC: AS222.A29. JFM: 27.0420.03Kötter. BSM2: 22(1898)203. JSL: 922. Presented to the Academy by Giuseppe Peano.Reprinted in Pieri 1980, 49–68. This is the second note of a series with Pieri 1895a and 1896b.[Cited in 1.2, the chapter 2 introduction, 2.3, 4.1, 5.1, 6.4, and items Pieri 1895a, 1896b, 1980.]

———. 1896b. Sui principî che reggono la geometria di posizione: Nota 3a. Atti della Reale Acca-demia delle Scienze di Torino 31(1895–1896): 457–470. LC: AS222.A29. JFM: 27.0420.03Kötter. BSM2: 22(1898)203. JSL: 923. Presented to the Academy by Giuseppe Peano.Reprinted in Pieri 1980, 69–82. This is the last note of a series with Pieri 1895a and 1896a.[Cited in 1.2, 2.3, 5.1, 6.4, and items Pieri 1895a, 1896a, 1980.]

———. 1896c. Un sistema di postulati per la geometria projettiva astratta degli iperspazî. Rivistadi matematica 6: 9–16. LC: QA1.R5. JFM: 27.0431.01 Kötter. JSL: 924. Reprinted in Pieri1980, 83–90. Reissued in Roero 2003. [Cited in 1.2, 2.3, 4.1, 5.1, 6.4, and items Pieri 1980 andRoero 2003.]

———. 1897a. Di alcune questioni metriche circa le superficie algebriche. Giornale di mate-matiche 35: 75–80. LC: QA1.G5. JFM: 28.0562.02 Toeplitz. Completed in 1896. [Cited in 1.2and 6.2.]

———. 1897b. Intermezzo. Periodico di matematica per l’insegnamento secondario (series 3)12: 151–153. LC: QA1.P45. JFM: 28.0470.02 Lampe. [Cited in 1.3, 6.4, and 6.7.]

———. 1897c. Sugli enti primitivi della geometria projettiva astratta. Atti della Reale Accademiadelle Scienze di Torino 32(1896–1897): 343–351. LC: AS222.A29. JFM: 28.0469.03. BSM2:25(1901)228. JSL: 925. Presented to the Academy by Giuseppe Peano. Reprinted in Pieri1980, 91–100. [Cited in 1.2, 2.3, 5.1, 6.4, and item 1980.]

———. 1897d. Sull’ordine della varietà generata di più sistemi lineari omografici. Rendicontidel Circolo Matematico di Palermo 11: 58–63. LC: QA1.C52. JFM: 28.0585. 02 Loria. BSM2:39(1915)123–124. Completed in 1896. [Cited in 6.2.]

440 Bibliography

———. 1897e. Two letters to Felix Klein, 18 March and 19 April. Klein papers 22F, 96–99. Staats- und Universitätsbibliothek, Göttingen. Annotated in section 6.6. [Cited in 6.6 andpermissions.]

———. 1898a. Draft of a letter to Corrado Segre. In Pieri [no date]. Listed in Arrighi 1981, 11.Annotated in section 6.6. [Cited in 6.6 and items Arrighi 1981 and Pieri [no date].]

———. 1898b. Nuovo modo di svolgere deduttivamente la geometria projettiva. Reale IstitutoLombardo di Scienze e Lettere: Rendiconti (series 2) 31: 780–798. LC: AS222.I42. JFM:29.0454.02 Loria. BSM2: 29(1905)18. Reprinted in Pieri 1980, 163–182. [Cited in 1.2, 2.3,5.1, 5.3, 6.4, and item Pieri 1980.]

———. 1898c. I principii della geometria di posizione composti in sistema logico deduttivo.Memorie della Reale Accademia delle Scienze di Torino (series 2) 48: 1–62. LC: AS222.A3.JFM: 29.0407.01 Schlegel. BSM2: 31(1907)190. JSL: 926. Also reviewed in D’Ovidio, Segre,and Peano 1898. Reprinted in Pieri 1980, 101–162. Completed in 1897. [Cited in 1.2, 1.3, 2.3,the chapter 3 introduction, 4.1, 4.4, 5.1, 5.3, 6.4, and items D’Ovidio, Segre, and Peano 1898and Pieri 1980.]

———. 1899a. Letter to the president of the faculty of the University of Catania, 9 November.In Catania [no date], fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6,permissions, and item Catania [no date].]

———. 1899b. Review of Ingrami 1899. Revue de mathématiques (Rivista di matematica) 6:178–182. LC: QA1.R5. Reissued in Roero 2003. Annotated in section 6.6. [Cited in 5.1, 6.7,and items Ingrami 1899 and Roero 2003.]

———. 1900a. Della geometria elementare come sistema ipotetico deduttivo: Monografia delpunto e del moto. Memorie della Reale Accademia delle Scienze di Torino (series 2) 49: 173–222. LC: AS222.A3. JFM: 30.0426.02 Loria. BSM2: 31(1907)192. Presented to the Academyby Enrico D’Ovidio. Also reviewed in D’Ovidio and Segre 1899. Reprinted in Pieri 1980,183–234. Completed in 1899. [Cited in 1.1–1.3, the chapter 2 introduction, 2.2, 2.3, 2.5, thechapter 3 introduction, 3.3, 3.10, 4.4, 5.1–5.3, 6.4, 6.6, and items D’Ovidio and Segre 1899, Loria1899, and Pieri 1980.]

———. 1900b. Letter to the rector of the University of Catania, 31 January. In Catania [nodate], fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and itemCatania [no date].]

———. 1900c. Sopra i sistemi di congruenze lineari, che generano semplicemente lo spazio rigato.Atti della Accademia Gioenia di Scienze Naturali in Catania (series 4) 14: 1–7. LC: QH7.A3.JFM: 33.0700.03. [Cited in 1.2 and 6.2.]

———. [1900] 1901. Sur la géométrie envisagée comme un système purement logique. In Inter-national Congress of Philosophy 1900–1903, volume 3 (1901), 367–404. Delivered by LouisCouturat. Reprinted in Pieri 1980, 235–272. Reviewed in Boutroux 1900, 593–594; Couturat[1900] 1899, 404–405; and Lovett 1900–1901, 171–172. [Cited in the chapter 1 introduction,1.1–1.3, 2.2, 2.5, the chapter 3 introduction, 3.10, 4.1, 4.3, 4.4, 5.1, 5.2, 6.4, 6.5, and itemsBoutroux 1900, Couturat [1900] 1899, International Congress of Philosophy 1900–1903, Lovett1900–1901, and Pieri 1980.]

———. 1901a. Letter to the rector of the University of Catania, 30 September. In Catania [nodate], fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and itemCatania [no date].]

———. 1901b. Sui principî che reggono la geometria delle rette. Atti della Reale Accademia delleScienze di Torino 36(1900–1901): 335–350. LC: AS222.A29. JFM: 32.0661. 01 Sommer.BSM2: 33(1909)92. Presented to the Academy by Giuseppe Peano. Reprinted in Pieri 1980,273–288. Completed in 1900. [Cited in 1.2, the chapter 2 introduction, 2.3, 5.1, 5.3, 6.4, anditem Pieri 1980.]

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———. [1901] 1997. Draft of a letter to Benjamin Kagan, 20 September 1901. In Arrighi 1997,69. Annotated in section 6.6. [Cited in 6.6 and item Arrighi 1997.]

———. 1902a. Letter to the rector of the University of Catania, 28 May. In Catania [no date],fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and itemCatania [no date].]

———. 1902b. Sul complesso cubico di rette che contiene una stella di raggi e un piano rigato.Atti della Accademia Gioenia di Scienze Naturali in Catania (series 4) 15: 1– 30. LC: QH7.A3.JFM: 34.0706.01 Lampe. [Cited in 1.2 and 6.2.]

———. 1903a. Letter to Bertrand Russell, 22 May. In the Bertrand Russell Archives, McMasterUniversity, Hamilton, Ontario. Listed in Feinberg et al. 1967, section VI.1, 178. Translatedin section 6.6. [Cited in 4.1, 6.6, permissions, and item Feinberg et al. 1967.]

———. 1903b. Letter to the rector of the University of Catania, 17 July. In Catania [no date],fascicule Mario Pieri. The rector was Pietro Delogu. Translated in section 6.6. [Cited in 1.1,permissions, and item Catania [no date].]

———. 1903c. Review of Peano 1902. Periodico di matematica per l’insegnamento secondario(series 2) 5: 293–295. LC: QA1.P45. JSL: 926.1. Translated in section 6.7. [Cited in 6.7 anditem Peano 1902.]

———. 1904a. Circa il teorema fondamentale di Staudt e i principî della geometria projettiva.Atti della Reale Accademia delle Scienze di Torino 39(1903–1904): 313–331. LC: AS222.A29.JFM: 35.0557.01 Loria. BSM2: 33(1909)184. Presented to the Academy by Giuseppe Peano.Reprinted in Pieri 1980, 289–308. Completed in 1903. [Cited in 1.2, 2.3, 3.4, 4.1, 5.1, 6.4, anditem Pieri 1980.]

———. 1904b. Letter to the rector of the University of Catania, 19 October. In Catania [no date],fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and itemCatania [no date].]

———. [1904] 1984. Letter to Giuseppe Vitali, 5 January. In Vitali 1984, letter 23, 438–439. Annotated in section 6.6. [Cited in 1.1, 6.6, and item Vitali 1984.]

———. 1905a. Draft of a letter to Louis Couturat. In Pieri [no date]. Listed in Arrighi 1981,16. Annotated in section 6.6. [Cited in 6.6 and items Arrighi 1981 and Pieri [no date].]

———. 1905b. Letter to Oswald Veblen, 2 July. In the Veblen papers, section P: miscellany,box 10. Manuscript Division, Library of Congress, Washington. Annotated in section 6.6.[Cited in 1.1, 6.6, and permissions.]

———. 1905c. Nuovi principii di geometria projettiva complessa. Memorie della Reale Accademiadelle Scienze di Torino (series 2) 55: 189–235. LC: AS222.A3. JFM: 36. 0613.02 Dehn. BSM2:46(1922)12. Presented to the Academy by Giuseppe Peano. Also reviewed in Peano and Segre1905. Reprinted in Pieri 1980, 309–356. [Cited in 1.2, 1.3, 2.3, 5.1, 5.3, 6.4, and items Peanoand Segre 1905 and Pieri 1980.]

———. 1905d. Review of S. Catania 1904. Periodico di matematica per l’insegnamento secon-dario (series 3) 2: 47–48. LC: QA1.P45. [Cited in 1.1, 1.3, 6.4, 6.7, and item Catania 1904.]

———. 1905e. Two letters to the rector of the University of Catania, 7 May and 26 October. InCatania [no date], fascicule Mario Pieri. Annotated in section 6.6. [Cited in 6.6 and itemCatania [no date].]

———. 1906a. Breve aggiunta alla memoria: Nuovi principî di Geometria projettiva complessa.Atti della Reale Accademia delle Scienze di Torino 41(1905–1906): 339–342. LC: AS222.A29.JFM: 37.0488.02 Dehn. BSM2: 34(1910)102. Presented to the Academy by Giuseppe Peano.Reprinted in Pieri 1980, 357–360. [Cited in 1.2, 1.3, 6.4 and item Pieri 1980.]

———. 1906b. Letter to Giuseppe Grassi Cristaldi, 30 July. In Catania [no date], fascicule MarioPieri. Annotated in section 6.6. [Cited in 1.1, 6.6, and item Catania [no date].]

———. 1906c. Letter to the director, 24 January. In Catania [no date], fascicule Mario Pieri.Annotated in section 6.6. [Cited in 1.1, 6.6, and Catania [no date].]

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———. 1906d. Uno sguardo al nuovo indirizzo logico-matematico delle scienze deduttive: Dis-corso per l’inaugurazione dell’anno accademico 1906–1907 nella Reale Università di Catania.Annuario della Reale Università di Catania 1906–1907: 21–82. JSL: 928. Reprinted in Pieri1980, 389–448. [Cited in 1.1, the chapter 4 introduction, 4.1, 5.1, 6.5, and item Pieri 1980.]

———. 1906e. Sopra una definizione aritmetica degli irrazionali. Bollettino delle sedute dellaAccademia Gioenia di Scienze Naturali in Catania 87: 14–22. LC: QH7.A33. JFM: 37.0198.05Vivanti. Reprinted in Pieri 1980, 367–376. Completed in 1905. [Cited in the chapter 4introduction, 4.1, 4.4, 5.1, 6.5, and item Pieri 1980.]

———. 1906f. Sulla definizione Staudtiana dell’omografia tra forme semplici reali. Periodicodi matematica per l’insegnamento secondario (series 3) 3: 1–5. LC: QA1.P45. JFM: 36.0581.03Lampe. Reprinted in Pieri 1980, 361–366. Completed in 1905. [Cited in 2.3, 6.4, and item Pieri1980.]

———. 1906g. Sur la compatibilité des axiomes de l’arithmétique. Revue de métaphysique et demorale, 13: 196–207. LC: B2.R37. JFM: 37.0062.03 Fehr. JSL: 927. Also reviewed in Brown1906. Reprinted in Pieri 1980, 377–388. [Cited in 1.2, 1.3, the chapter 4 introduction, 4.1, 4.4,5.1, 6.5, and items Brown 1905 and Pieri 1980.]

———. 1906h. Three letters to the rector of the University of Catania: 17 February, 6 July, and6 November. In Catania [no date], fascicule Mario Pieri. Annotated in section 6.6. [Cited in1.1, 6.6, and Catania [no date].]

———. 1907a. Sopra gli assiomi aritmetici. Bollettino delle sedute della Accademia Gioenia diScienze Naturali in Catania (series 2) 1–2(1907–1908): 26–30. LC: QH7.A33. JSL: 928.1.Reprinted in Pieri 1980, 449–454. Section 4.2 of the present book is an English translationof this paper; its original first page is reproduced on page 309. [Cited in the preface, chapter1 introduction, 1.2, the chapter 4 introduction, 4.1–4.4, 5.1, 5.3, 6.5, and item Pieri 1980.]

———. 1907b. Two letters to the rector of the University of Catania, 30 April and 25 May. InCatania [no date], fascicule Mario Pieri. Annotated and translated, respectively, in section 6.6.[Cited in 1.1, 4.4, 6.6, permissions, and item Catania [no date].]

———. 1908a. La Geometria Elementare istituita sulle nozioni di “punto” e “sfera.” Memoriedi matematica e di fisica della Società Italiana delle Scienze (series 3) 15: 345–450. LC:Q54.R65. JFM: 39.0545.08. Presented to the Society by Guido Castelnuovo and approved byCorrado Segre. Reprinted in Pieri 1980, 455–560. Chapter 3 of the present book contains anEnglish translation of this memoir; its original first page is reproduced on page 159. Pieri 1914is a Polish translation. [Cited in the preface, chapter 1 introduction, 1.2, 1.3, 2.5, chapter 3introduction, 3.10, 4.1, 5.1–5.3, 6.4, and items Archimedes [1897] 2002a, Cauchy [1833] 1974,Euclid [1908] 1956, Lazzeri 1900, Legendre [1794] 1849, Méray [1874] 1903, and Pieri 1914,1980.]

———. 1908b. Two letters to the rector of the University of Catania, 15 April and 4 July. InCatania [no date], fascicule Mario Pieri. Translated and annotated, respectively, in section 6.6.[Cited in the chapter 1 introduction, 1.1, 6.6, permissions, and item Catania [no date].]

———. [1909] 1984. Letter to Giuseppe Vitali, 25 November. In Vitali 1984, letter 52, 471.Annotated in section 6.6. [Cited in 6.6 and item Vitali 1984.]

———. 1910. Lezioni di geometria proiettiva. Edited by Mario Camivi. Parma: LitògrafoAnghinetti & Giaroli. Approximately 900 pages. Lithographed handwritten course notesfrom the academic year 1909–1910. [Cited in 6.4 and 6.7.]

———. 1911a. Letter to the rector of the University of Parma, 22 March. In Parma [no date].Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and item Parma [no date].]

———. 1911b. Letter to Gaetano Campetti, 2 May. In possession of Marco Campetti. Translatedin section 6.6. [Cited in 1.1, 6.6, and permissions.]

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Veblen, Oswald, and John Wesley Young. 1908. A set of assumptions for projective geometry.American Journal of Mathematics 30: 347–380. LC: QA1.A5. Internet: JSTOR. JFM: 39.0606.01 Steinitz. [Cited in 2.3.]

———. 1910–1918. Projective Geometry. Two volumes. New York: Ginn and Company. LC:QA471.V4. Internet: Michigan. JFM: 41.0606.06 Beck. [Cited in 1.3.]

Veronese, Giuseppe. 1882. Behandlung der projectivischen Verhältnisse der Räume von ver-schiedenen Dimensionen durch das Princip des Projicirens und Schneidens. Mathematische

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Permissions

The authors, confident that they speak for the mathematical community, are greatly indebtedto Francesco Campetti and Maria Grazia Ciampini, in Sant’Andrea di Compito, Tuscany, Italy,for providing access to family photographs and archival documents, and to Marco and VittorioCampetti for permission to publish them. These include translations of Pieri’s letters 1911b and1912a in section 6.6, and the following photographs in the indicated sections:

Mario Pieri . . . . . . FrontispieceTelescopic projections Pieri & Campetti familiesSilvio Pieri Virginia & Paolo AnastasioMilitary academy personnel Angiolina PieriMario Pieri Pieri’s tombGemma Campetti & sons

. . . . . . . . . . . . . . 1.1

Pieri’s portrait of Klein . . . . . . . . . . . . . . 1.2Mario Pieri . . . . . . . . . . . . . . 2.3

The authors acknowledge gratefully the cooperation of several archival institutions that providedpersonal access to their resources and permission to publish images or translations of variousitems in the indicated sections of the present book:

Istituto di Istruzione SuperioreCrescenzi Pacinotti, in Bologna, Italy

Pieri’s student records:Bologna 1876–1890

. . . . 1.1

Archivio Storico dell’Università degliStudi di Bologna, in Bologna, Italy

Pieri’s student records:Bologna [no date]

Photo: Salvatore Pincherle

. . . . 1.1

. . . . 1.1

Biblioteca dell’Università di Pisa,in Pisa, Italy

Doctoral thesis: Pieri 1884cQuestions for oral defense of the

laureate: Pieri 1884a

. . . . 1.1

. . . . 6.6

Archivio Storico della ScuolaNormale Superiore, in Pisa, Italy

Pieri’s student records:Pisa 1884

. . . . 1.1

Archivio Storico dell’Università degliStudi di Catania, in Catania, Italy

Announcement of Pieri’s promo-tion: from Catania 1903b

Letters: Pieri 1899a, 1900b,1901a, 1902a, 1903b, 1904b,1907b, 1908b

Review: Pieri, Lauricella, andCatania 1903

. . . . 1.1

. . . . 6.6

. . . . 6.7

Archivio Centrale dello Stato,in Rome, Italy

Documents concerning Pieri . . . . 1.1

Bertrand Russell Archives,at McMaster University,in Hamilton, Ontario, Canada

Photo: Bertrand RussellLetter: Pieri 1903a

. . . . 5.1

. . . . 6.6

460 Permissions

Staats- und Universitätsbibliothek,in Göttingen, Germany

Letters: Pieri 1897e . . . . 6.6

Manuscript Division, Libraryof Congress, in Washington DC,United States

Letter: Pieri 1905b . . . . 6.6

Archivio Storico dell’Università degliStudi di Parma, in Parma, Italy

Letter: Pieri 1911a . . . . 6.6

Several individuals, publishers, and other organizations graciously extended permission fortranslating or reprinting images or text from their Internet websites or earlier publications:

Mother House of the Salesians ofDon Bosco, in Turin, Italy

Photo: exterior of the formermilitary academy in Turin

. . . . 1.1

Archivio Storico dell’Accademia delleScienze di Torino, in Turin, Italy

Photos: Enrico D’Ovidio, Pieri,and Luigi Cremona

Photos: Corrado Segre andCesare Burali-Forti

Photo: Gino FanoPhoto: Gino LoriaPhoto: Giuseppe Peano

. . . . 1.1

. . . . 1.2

. . . . 2.3

. . . . 2.5

. . . . 5.1

Queen’s University, in Kingston,Ontario, Canada

Photo: Federigo Enriques, fromBrigaglia and Ciliberto 1995

. . . . 1.1

Pietro Nastasi, in Palermo, Italy Photo: Vito Volterra fromGuerraggio and Nastasi 2005

. . . . 1.1

Section P.RI.ST.EM (ProgettoRicerche Storiche e Metodologiche) ofthe center ELEUSI (L’ElaborazioneLogica e l’Utilizzazione Sistematicadell’Informazione) at the BocconiUniversity, in Milan, Italy

Translation: FrancescoChizzoni’s 1900 letter to Pieri,from Arrighi 1997

. . . . 1.1

Emilia Levi Resta, in Argentina Photo: Beppo Levi, fromLevi 1999

. . . . 1.1

Historische Kommission für Hessen,in Marburg, Germany

Photo: Moritz Pasch, fromPickert 1982

. . . . 2.3

Archivio dell’Istituto Venetodi Scienze, Lettere ed Arti,in Venice, Italy

Photo: Giuseppe Veronese . . . . 2.3

Image Archive ETH-Bibliothek,in Zürich, Switzerland

Photo: B. L. van der Waerden . . . . 2.4

Springer-Verlag, in Heidelberg,Germany

Photo: David Hilbert,from Reid 1970

Photos: Alfred Tarski andAdolf Lindenbaum,from Givant 1991

. . . 3.10

. . . . 5.2

Bollati Boringhieri Editore,in Turin, Italy

Photo: Giuseppe Peano,from Kennedy 1983

. . . . 4.1

Permissions 461

The authors also included images from the Internet websites of three other organizations:

Comitato Guglielmo MarconiInternational, in Bologna, Italy

Photo: Augusto Righi . . . . 1.1

Archivio Fotografico dell’Ateneodell’Università di Pisa, in Pisa, Italy

Photos: Luigi Bianchi, UlisseDini, and Eugenio Bertini

. . . . 1.1

Princeton University, in Princeton,New Jersey, United States

Photo: Oswald Veblen . . . 3.10

Index

This index lists both persons and subjects. For persons, dates are supplied if possible. Section1.3 includes biographical sketches of many of them; those names and the page numbers of thesketches are italicized. Page numbers of illustrations are boldfaced. Most entries under“symbol” include examples of usage. For persons surnamed Pieri, the relationship to Mario Pieriis indicated. Italian names with particles—for example De Paolis—are listed with the particlefirst. Alphabetization ignores spaces, punctuation, and diacritical marks within names. Section1.3, chapter 6, and the bibliography contain many further cross-references.

1848 revolutions, 66, 75, 82–83, 85, 101

abscissa, 257, 285absolute geometry. See under geometryabstraction

definition by (see under definition)See also under symbol

Academia pro Interlingua. See underjournals; Giuseppe PEANO

ACKERMAN, Michael (19??–), 426acknowledgments, x, 459ADAMSON, Iain T. (19??–), 421addition. See sumaffinity, circular. See under transformationAfghanistan, 5AGAZZI, Evandro (1934–), 365, 402, 406air power, 6, 119algebraic curve. See under curvealgebraic geometry. See under geometryalgebraic number theory, 76, 84algebraic surface. See under surfaceALIGHIERI, Dante (1265–1321), 395ALMAGIÀ, Virginia. See Virginia VOLTERRA

AMALDI, Ugo (1875–1957), 65, 98, 402work with Enriques, 368, 415–16

AMATO, Vincenzo (1881–1963), 72, 402AMATURO, Enrico (1863–19??), 42American Mathematical Society, 43, 388AMODEO, Federico (1859–1946), 20, 62, 107

correspondence, 383, 432, 438rivalry with Pieri, 22–23, 29, 32, 36

analysisarithmetization, 291–92functional (see functional analysis)Peano’s work, 301, 334, 337, 365use of set theory, 296, 298

ANASTASIO, Angiolina. See Angiolina PIERIANASTASIO, Paolo (18??–19??), 37 (see also

Virginia ANASTASIO; Angiolina PIERI)ANASTASIO, Virginia Pieri (1867–1929?),

5, 17, 20, 36, 37, 459marriage, 5, 27, 454

ANDREINI, Angelo (18??–19??), 10, 62ANELLIS, Irving H. (19??– ), 302, 402ANGELELLI, Ignacio (19??–), 417angle, 146, 149, 272

bisector, 220concave, convex, or right, 204of triangle, 205See also under symbol

Annuario scientifico ed industriale. See under journals

ANTICHI, Petronilla (18??–????), 4anticongruence. See under transformationanticongruent figures, 150antihomography. See under transformationanti-inversion. See under transformationantinomies

Burali-Forti’s, 69responses to, 302, 304See also Russell’s antinomy

antipolar plane, 238antirotation. See reflection: rotaryantitranslation. See reflection: glideapplied mathematics, 87, 365APPOLONIUS of Perga (225–190 B.C.), 121,139APRILE, Giorgio Antonio (1884–1943),

40, 42, 63, 92, 398, 402, 410arc, 252ARCHIMEDES (287–212 B.C.), 41, 121, 403, 421

principle (see under postulate)archival institutions, ix, 40, 459

464 Index

archive. See Mario PIERI: Nachlassarea, 263ARISTOTLE (384–322 B.C.), 266, 305, 347arithmetic

cardinal, 296–97complex, 125, 299integer, modulo n, 320ordinal, 297–98, 387rational, 77, 289, 292, 294, 299, 334, 392signed-integer, 77, 299, 334See also natural-number arithmetic;

real-number arithmeticARMANO, Tiziana (19??–), vi, 446ARRIGHI, Gino (1906–2001), vi, ix, 6–7, 36,

42–44, 63, 382, 388, 398–9, 403, 443, 460. Citations of individual letters in Arrighi1997 are indexed by their authors.

ARZELÀ, Cesare (1847–1912),10, 63, 98, 109, 398, 426

Bologna affair, 25–29ASBURGO. See HABSBURG

ascending. See progressionASCHIERI, Ferdinando (1844–1907),

22–23, 25, 29ÀSCOLI, Graziadio Isaia (1829–1907), 97ASKEY, Richard Allen (1933–), 415ASPRAY, William (1952–), 116, 403associativity, 270, 294, 395AUGUST, Friedrich Wilhelm Oscar

(18??–????), 374Ausdehnungslehre, 83–4, 334AUSSANT-CARÀ, P. (18??–19??), 426AUSTIN, John Langshaw (1911–1960), 417Austria, xvii–xixautomorphism, 358, 360AVELLONE, Maurizio (19??–), 127, 129, 403axiom, 54 (see also postulate)

of choice, 315, 360of determinacy, 360of foundation, 323of infinity, 300, 307

axiomatic methodin general, v, vii, 87, 95, 291, 350, 370 in Hilbert, 84, 277in Peano, 300–301, 334–35in Pieri, 54, 124–27See also hypothetical-deductive system

axisof rotation, 183of symmetry, 176

BACCARINI, Pasquale (1858–1919), 32BACCELLI, Guido (1832–1916),

26–27, 29, 63–64, 456BACHI, Albina. See Albina LEVI.BACHMANN, Friedrich (1909–1982),

284, 403, 427BALBO, Italo (1896–1940), 6BALDWIN, James Mark (1861–1934),

124, 343, 403BALTZER, Richard (1818–1887), 93, 403, 430BALZAC, Honoré de (1799–1850), 97BARBAGLI, Marzio (1938–), 7, 15–17, 403BARBARIN, Paul (1855–1931), 41, 64, 403, 407BARBERIS, Bruno (1953?–), 109, 110, 403BAR-HILLEL, Yehoshua (1915–1975), 403BARIDON, Paolo (18??–19??), 408BARKER, William A. (????–????), 120, 404BARSOCCHINI, Domenico (1777–1862),

4, 96, 443BARTORELLI, Antonio (18??–19??), 46, 64barycentric calculus. See under calculusBASSO, Giuseppe (1842–1895),

20, 64, 332, 416BASTIAN, Beppina. See Beppina PIERI.BATTAGLINI, Giuseppe (1826–1894),

78–79, 398Giornale di (see journals: Giornale di

matematiche)BAUER-MENGELBERG, Stefan (1927–1996),

417, 456BEAUFRONT, Louis de (1855–1935), 74, 404BECK, Hans (1876–1942), 381, 443, 454Begriffsschrift. See under FREGE

BEKEMEIER, Bernd (19??– ), 292, 404BELLAVITIS, Giusta (1803–1880),

117, 139, 404BELTRAMI, Eugenio (1835–1900), 65, 97BEMAN, Wooster Woodruff (1850–1922), 413BENZ, Walter (1931–), 143Berkeley. See University of CaliforniaBerlin, Imperial Library, 2BERNAYS, Paul Isaac (1888–1977), 421–22BERNSTEIN, Felix (1878–1956)

See under theoremBERTINI, Eugenio (1846–1933),

30, 65, 78, 398, 404as referee, 22–23, 26, 28–29, 31, 42, 404

BERTRAND, Joseph Louis François(1822–1900), 78

Index 465

BERZOLARI, Luigi (1863–1949),22, 65–66, 107, 398, 405

Encyclopedia, 69, 71, 81, 95, 327, 404at Pavia, 31–32, 36, 94at Turin, 20, 23, 26, 89,

BETH, Evert Willem (1909–1964),300, 304–305, 314–15, 319, 404

BETTAZZI, Rodolfo (1861–1941),10, 12, 18, 20, 66, 337, 404

BETTI, Enrico (1823–1892),12, 13, 66, 406, 409, 437

works by, 167, 374, 398–99, 404, 416betweenness, 146, 334–35, 354, 358, 360–62

on arc, 252Pieri’s definition, 152, 195, 272, 357undefined, 149, 274, 277, 288, 351, 353See also under order

BÉZOUT, Étienne (1739–1783), 53, 404See also under theorem

BIAGINI, Roderigo (18??–19??), 398BIANCHI, Francesco Saverio (1827–1908), 67BIANCHI, Luigi (1856–1928),

13, 34, 67, 398, 409, 461Pieri’s teacher, v, 12, 50, 374–75, 437

bibliographic citations, ix, 4, 373, 401–03in chapter 3, 62, 157, 167, 416

BIEBERBACH, Ludwig Georg Elias Moses(1886–1982), 412

BIERMANN, Kurt-Reinhold (1919–2002),77, 404

bijection. See under functionbinary relations. See under primitive notionsbiographical sketches, x, 3, 4, 62birational

equivalence (see under surface)See also under geometry; transformation

bisector. See under angleBLACK, Max (1909–1988), 404, 432BLAIKLOCK, Mary (18??–19??), 446BLAIR, David E. (1940–), 138–41, 404BLANC, Charles (19??–), 416BLANCHET, Marie Parfait Alphonse

(1813–????), 426BLASCHKE, Wilhelm Johann Eugen

(1885–1962), 425BLUMBERG, Albert Emanuel (????–19??),

127, 405BLUMENTHAL, Otto (1876–1944), 60, 405BOCCHERINI, Luigi (1743–1805), 4BODEWIG, Ewald (1902–????), 112, 405

BOGGIO, Tomasso (1877–1963), 31, 34, 398BOGGIO-LERA, Enrico (1862–1956), 10, 67–68BOHLMANN, Georg (1869–1928), 418BOI, Luciano (1957–), 129, 133–34, 405Bollati Boringhieri Editore, 460Bollettino di bibliografia e storia della

scienza. See under journalsBologna, xvii, xviii, 7–8, 11

affair, 25–28Royal Technical Institute, 7–8, 459See also Bologna, university of

Bologna, university of, 10–11, 16, 459faculty renaissance, 8, 63, 70, 75, 98Pieri’s chair quest, 25–28

BOLONDI, Giorgio (19??–), 17, 405BOLYAI, Farkas Wolfgang (1775–1856),

209, 405See also under parallel postulate

BOLYAI, János (1802–1860), 125, 155, 405BOLZANO, Bernard Placidus Johann Nepomuk

(1781–1848), 291–92, 300, 405, 450BOMPIANI, Enrico (1889–1975), 66, 405BONACCORSO, G. (18??–19??), 33BONAPARTE, Napoléon (1769–1821),

xvii, 11, 18, 41BONFANTE, Giuliano (1904–1983),

4, 5, 8, 20, 49, 97, 405, 444BONOLA, Roberto (1874–1911), 327BOOLE, George (1815–1864), 316, 334BOOLOS, George Stephen (1940–1996),

299, 318, 405BORBÓN, BORBONE. See BOURBON

BORDIGA, Giovanni (1854–1933), 42BORDONI, Antonio (1789–1860), 75BOREL, Félix Edouard Justin Émile Borel

(1871–1956). See under theoremBORGA, Marco (????– ), x, 60, 147, 156, 403,

405–06Borga & Palladino 1992, 54, 58,

153, 301, 316, 326, 335, 406BORKOWSKI, Ludwik Stefan (1914–1993), 428BORODIN, Alexandr Ivanovich ( ,

, 19??–), 41, 406BORSUK, Karol (1905–1982), 361, 406BOSCHI, Pietro (1833–1918), 8, 10, 68BOSIO (cav. uff., 18??–19??), 48BOTTAZZINI, Umberto (1947–), 92, 131, 406

Enriques & Castelnuovo, 70, 72–73, 80Enriques & Castelnuovo, Bologna affair,

23, 25–28, 31–32

466 Index

BOTTAZZINI (continued)Enriques & Castelnuovo, otherwise

involving Pieri, 8, 22, 36, 38, 42boundary of triangle, 205BOURBAKI, Nicolas (pseudonym),

115, 294, 406BOURBON, Carlo Lodovico of (1799–1883),

xvii, 41Bourbon dynasty, xvii, 41BOURBON, Ferdinando II of (1810–1859), 41BOURBON, Maria Luisa of (1782–1824),

xvii, 41BOUTROUX, Émile (1845–1921),

381, 406, 423, 440BRANCOLI, Giuseppe (18??–????), 5BRANCOLI, Teresa Pieri (1853–1942?),

5–6, 390–391BRAUNMÜHL, Anton, Edler von (1853–1908),

375, 438, 446Brazil, 6, 17, 36, 44BREDOW, Ilse (????–????), 448BRESADOLA, Marco (1969–), 100, 406BREVEGLIERI, Bruno (1942–), 406BRIGAGLIA, Aldo (1945–),

53, 129, 403, 406, 460Brigaglia & Masotto 1982, 1, 42, 131, 406Brigaglia & Ciliberto 1995, 52, 80, 406Brigaglia, Ciliberto, & Sernesi 2005,

74, 92, 106, 406BRILL, Alexander Wilhelm von

(1842–1935), 65BRIOSCHI, Francesco (1824–1897),

66, 75, 167, 416BROADBENT, Thomas Arthur Alan

(1903–1973), 103, 406BRODÉN, Torsten (1857–1931),

68, 124, 153–154, 407, 436BROWDER, Felix Earl (1928–), 407, 424BROWN, Harold Chapman (1879–1943),

290, 382, 407, 442BRUNO, Giuseppe (1828–1893), 20, 22–23,

68–69, 107, 332, 375, 398, 438, 446BUCCA, Lorenzo (18??–19??), 32, 37BUEK, Otto (1873–1966), 47, 398BUHL, Adolphe (1878–1949), 64, 407Bulletin des sciences mathématiques

See under journalsBURALI-FORTI, Cesare (1861–1931),

vi, 55, 460 geometry, 52, 58, 156

BURALI-FORTI (continued)life, 12, 20, 69, 398, 429logic, 301, 306–307, 316, 321, 326–28Peano school, 22, 337, 341–42vector analysis, 46–47, 53, 378, 392, 428See also under antinomies

BURAU, Werner (1906–1994), 84, 87,105, 111, 112, 408, 448

BURKHARDT, Heinrich (1861–1914), 430BURSTALL, Rod M. (1934–), 313, 408BUSSEY, William Henry (1879–19??), 116BÜTZBERGER, Friedrich (18??–19??), 139, 408

CAEMMERER, Hanna vonSee Hanna NEUMANN

CAIO, 268CAJORI, Florian (1859–1930), 313, 315, 408calculus

barycentric, 93geometric (see under Giuseppe PEANO)mathematics as uninterpreted, 59of variations, 123See also differential and integral calculus

and under SCHUBERT

CALAPSO, Renato (1901–1976), 92, 408CALDARERA, Gaetano (1865–19??),

34, 40, 69–70CALDARERA, Grazia Macrina (1880–19??),

40, 70, 408, 410California State University, xCalifornia, University of. See University

of CaliforniaCAMBRIA, Maria (1923–), 114CAMIVI, Mario (18??–19??), 397, 442CAMPEDELLI, Luigi (1903–1978),

12, 74, 101, 408CAMPETTI, Annarosa (1920–2004), 6CAMPETTI, Beatrice Giusfredi (1894–1995), 6Campetti family, 6, 36, 39, 44, 459CAMPETTI, Francesco (1969–), x, 5, 6, 17, 36,

43, 97, 408, 412, 444, 454CAMPETTI, Gaetano Enrico (1890–1980),

6, 39, 42, 44, 48, 390, 442CAMPETTI, Gemma Pieri (1863–1955?),

5–6, 36, 39, 44, 47, 390, 408, 411, 459CAMPETTI, Ghigo. See Ottorino CAMPETTICAMPETTI, Marco (1924–), x, 6, 47, 390–91,

397, 436–37, 442–43, 459CAMPETTI, Ottorino (1888–1947), 6, 36, 39,

42–44, 46, 48, 63, 390–91, 443

Index 467

CAMPETTI, Pellegrino (1892–1974),6, 36, 39, 43

CAMPETTI, Umberto (1863–1929),5–6, 17, 36, 44

CAMPETTI, Vittorio (1930–), x, 4, 6, 411, 459cancer. See throat cancerCANDIDO, Giacomo (1871–1941), 147, 408CANTOR, Georg Ferdinand Ludwig Philipp

(1845–1918), 408–9, 413, 420infinite sets, 69, 117, 296–98, 300, 305–6logic, 299, 302, 306, 313real numbers, 254, 263, 289, 290, 292–94See also under natural-number arithmetic;

real-number arithmetic; theoremCANTOR, Moritz Benedikt (1829–1920),

281, 409CAPELLINI, Giovanni (1833–1922),

8, 29, 70, 414, 454CARATHÉODORY, Constantin (1873–1950),

140, 404, 409CARDANI, Pietro (1858–1925), 100, 391, 409cardinal number, 296, 315

See also under arithmeticCARDUCCI, Giosuè (1835–1907), 96–97CARLO EMANUELE II. See Carlo

Emanuele II of SAVOYCARLO LODOVICO. See Carlo

Lodovico of BOURBON

CARRUCCIO, Ettore (1908–1980), 67, 92, 409Cartesian geometry. See under geometryCASATI, Gabrio (1798–1873). See under lawsCASORATI, Felice (1835–1890), 65–66, 91, 97CASSINA, Ugo (1897–1964), 71, 139,

319, 409, 436, 450on Peano, 337, 345–46, 365on Pieri, 2–3, 22, 31, 34, 42,

284–85, 307, 326, 399CASTELLANO, Filiberto (1860–1919),

20, 71, 79, 334, 337, 410CASTELNUOVO, Enrico (1839–1915), 71CASTELNUOVO, Guido (1865–1952),

71–72, 119, 263, 410–11, 422algebraic geometry, 52, 107–08,concerning Enriques, 25–29, 406, concerning Pieri, 44, 49, 157,

160, 381, 398, 399, 442concerning Turin, 20, 22–23, 36, 42,See also under BOTTAZZINI; ENRIQUES

CATALANI, Alfredo (1854–1893), 4

CATANIA, Sebastiano (1853–1946), 40, 72, 82,395–96, 402, 410, 419, 441, 443, 459

Catania, xviii, 1, 32–33, 41university, 16, 32–33, 35, 41, 88, 410, 459

categorical. See under theoryCAUCHY, Augustin Louis (1789–1857),

167, 410convergence criterion, 292

CAVINESS, Bob F. (19??–), 352CAYLEY, Arthur (1821–1895), 52, 125center

of anti-inversion, 238of arc, 252of circle, 173of circular ordering, 253of homothety, 233of semicircle, 252of symmetry, 175

centered figure, 160central point. See midpointCEPPI, Carlo Giulio (1829–1921), 332CESÀRO, Ernesto (1859–1906), 20, 73, 432chain

in complex projective line, 56, 134–35, 145in foundations of arithmetic, 314

CHASLES, Michel (1793–1880),56, 72–73, 133, 411, 426

on enumerative geometry, 52, 104, 121 Chicago, University of, 87, 115CHINI, Mineo (1866–1933), 20, 73, 337, 411CHIÒ, Felice (1813–1871), 82chirality, 150CHIZZONI, Francesco (1848–1904), 22, 32–34,

40, 73, 384, 398, 459–60choice. See under axiomCHURCH, Alonzo (1903–1995), 116, 348,

402, 411CHWIA KOWSKI, Zygmunt (????–????),

348, 452CHWISTEK, Leon (1884–1944), 451CIACCIO, Francesco Paolo (1821–1901), 28CIAMPINI, Maria Grazia (1942–),

x, 4, 6, 411, 459CIANI, Edgardo (1864–1942),

12, 42, 74, 114, 408rivalry with Pieri, 22–23, 29, 32, 36

CIESIELSKI, Krzystof (1957– ), 350, 411CIGNA, Gianfrancesco (1734–1790), 18CILIBERTO, Ciro (1950–). See BRIGAGLIACIPOLLA, Michele (1880–1947), 40, 42, 94, 411

468 Index

circlein inversive geometry, 57, 138, 141in Point and Sphere, 160, 173See also under postulate

circular. See under transformationclass, 163–64, 249, 269

enumerable, 254CLAUSIUS, Rudolf Julius Emmanuel

(1822–1888), 99CLEBSCH, Rudolf Friedrich Alfred

(1833–1872), 86, 444COEN, Salvatore (19??–), x, 8,

10, 42, 90, 91, 411coherent. See under fundamental sequenceCOHN-VOSSON, Stefan (1902–1936), 426coincidence formulas, 376, 396COLAJANNI, Napoleone (1847–1921), 43, 398collegium, 11, 16collimation, 169collinearity, definition

using betweenness, 277, 361using equidistance, 152, 154, 169, 280, 357using motion, 146, 152, 272

collineation. See under transformationcolonialism. See under Italycommissione giudicatrice, 16communication among mathematicians,

1, 87, 98, 107, 121, 154commutativity, 143, 226, 270, 294, 298compass, geometry of, 263complete quadrilateral, 131completeness

of real numbers, 151, 293See also under postulate; theory

complexanalysis, 125, 140, 142Hirstian, 378number (see under arithmetic)of lines, 377–78

complexity of axiom systems, 156, 273–74,276–77, 283, 285, 287–88, 350, 353

concave. See under angleconcorso, 16cones, system of, 377confocal varieties, 51conformal. See under transformationcongruence

in algebraic geometry, 53, 376–78of angles, 149, 274, 278, 280–81, 338

congruence (continued)of figures, 146, 148, 150, 161,

249, 272, 278, 338, 348of point pairs, 155, 216SAS (see under postulate)of segments, 149, 206, 274, 278,

280–81, 338, 351, 353, 357, 394of triangles, 220See also under equidistance; postulate;

symbol; transformationconic section, 52, 121, 376, 378, 393conjugation. See harmonic conjugationconjunction. See under symbolconsequence, 267conservation of number, 104, 112consistency. See under natural-number

arithmetic; projective geometry; theorycontact point, 167, 200CONTE, Alberto (1942–), 81, 411, 427

See also BOTTAZZINI

continuity, 278, 289Dedekind, 137, 141, 146, 149,

155, 273, 276, 336, 355elementary, 355–56Weierstrass, 285See also under postulate; transformation

continuous magnitude, 291–92CONTRO, Walter Salvator (1937–),

68, 128, 153–54, 407, 411control-systems theory, vii, 52convergence criterion. See under CAUCHY

convexangle (see under angle)figure, 206

COOLIDGE, Julian Lowell (1873–1954),133, 139–140, 142, 412

coordinatesCartesian, 57, 276, 278, 282, 288, 354homogeneous, 60, 93, 128, 277pentaspherical, 57tetracyclic, 57, 140

coplanarity, 146, 154, 173, 277, 280CORCORAN, John (1937– ), 305, 347, 412, 452CORIE, Helen Cullura (1911–2005), xCORRY, Leo (1956– ), 351, 412correspondence, 269

algebraic, 52principle, 73, 104, 121, 376

COURANT, Richard (1888–1972), 426

Index 469

COUTURAT, Louis (1868–1914)1900 Paris congress, 1, 36, 281,

321, 341, 381, 423, 440 arithmetic, 307correspondence, 388, 398, 428, 448geometry, 58, 145, 327 Leibniz, 281, 343, 427 life, ix, 74, 218, 404, 412, 415 logic, 304, 316

covariant, 200, 250COWAN, Patricia M. See PEARSONCOXETER, Harold Scott MacDonald (1907–

2003), x, 129, 131, 142–43, 145, 361, 412CRELLE, August Leopold (1780–1855)

See journals: Journal für die reineund angewandte Mathematik

CREMONA, Antonio Luigi Gaudenzio Giuseppe(1830–1903), 30, 33, 75, 412–13, 420, 460

algebraic geometry, 12, 14–15Pieri’s chair quest, 27–29See also under transformation

CREMONA, Tranquillo (1837–1878), 75Crescenzi–Pacinotti, Istituto, 8, 100CRISPI, Francesco (1818–1901), 26CROCE, Benedetto (1866–1952), 90, 119, 366CROWE, Michael J. (19??–), 53, 413CULMANN, Karl (1821–1881), 99CURTO, Girolamo (1852–1917), 12CURTZE, Maximilian (1837–1903), 412–13curvature, 51, 375curve

algebraic, vii, 51, 65, 75, 375–76, 378, 383of constant illumination, 51, 375space-filling, 334

cut, Dedekind, 76, 293CUTRONE (Procuratore Generale

comm., 18??–19??), 48cyclic order

in Euclidean geometry, 252, 278, 285in inversive geometry, 141in projective geometry, 128, 130–131See also under order

DA COLLINA, Mareno (1960–), xDA COLLINA, Mario (1921–), xDAGNINO (colonello cav., 18??–19??), 48DAL POGGETTO, Maria. See Maria PIERID’AMICO, Francesco (1880–19??), 40, 75,

396–97, 410, 413D’ANGIÒ, Giuseppe Gioeni. See GIOENI

DANIELE, Pietro Ermenegildo (1875–1949),96, 413

DANTE. See ALIGHIERIDARBOUX, Gaston (1842–1917),

121, 137, 139–40, 413, 425DARMSTAEDTER, Ludwig (1846–1927), 2, 398D’ASBURGO. See HABSBURGDAUBEN, Joseph Warren (1944– ),

296, 406, 413DAVINI, Marianna. See Marianna LUPORINI

DAVIS, Philip J. (1923–), xdecidability

of elementary geometry, 351–52See also under theory

DEDEKIND, Caroline Marie HenrietteEmperius (1799–1882), 76, 84

DEDEKIND, Julie Marie Sophie(1825–1914), 76

DEDEKIND, Julius Levin Ulrich(1795–1872), 76

DEDEKIND, Julius Wilhelm Richard(1831–1916), 76–77, 295, 306,404, 409, 413, 415, 419

continuity and real numbers, 76, 263,273, 289, 293–94, 305

finiteness definition, 290, 296, 300, 314logic, 299, 301, 305, 307, 315–16, 319natural numbers, 76, 296, 300–302, 304,

308, 313–15, 317–20, 322, 324, 326See also under continuity; cut;

real-number arithmeticDEDEKIND, Karl Adolf (1829–1909), 76DEDEKIND, Richard

See Julius Wilhelm Richard DEDEKINDdeduction. See consequencedefinability, 331, 358, 360definiendum, definiens, 265definition, 265, 302, 341, 358

by abstraction, 392equality by (see under equality)explicit, 265hypothesis of, 265implicit, 127, 265, 267–68, 307–8nominal, 265, 268, 306–7, 392real, 265recursive, 298–300, 314–15, 319, 324

DE FRANCHIS, Michele (1875–1946),36, 42, 44, 77, 89–90, 450

DEHN, Max (1878–1952), 96, 151, 380,414–15, 421, 434, 441, 454, 456

470 Index

DELOGU, Pietro (1857–1932),38, 387, 414, 441

DEL RE, Alfonso (1859–1921), 36, 42, 77, 430rivalry with Pieri, 22, 23, 25, 27–29, 31

Denmark, mathematics in, 121–22DENTON, William Wells (1882–1961), 456DE PAOLIS, Riccardo (1854–1892),

12, 20, 65, 78, 129, 147, 375–76,395, 398, 403, 414, 437, 449

1892 correspondences paperSee under Mario PIERI

depression. See economic depressionDE SANTIS, Maria Gabriela (19??–), 11, 16, 414DESARGUES, Girard (1591–1661)

See under theoremdescending. See progressiondescriptions, theory of, 341descriptive geometry. See under geometrydesign courses, 11, 34, 46, 332DE STEFANI, Carlo (1851–1924), 70, 414determinacy. See under axiomdiagrams. See under geometrydiametrically opposite points, 194DI BORBONE. See BOURBON

DICKSTEIN, Samuel (1851–1939), 124, 398DIEUDONNÉ, Jean Alexandre Eugène

(1906–1992), 316, 414–415DI FARNESE. See FARNESE

differential and integral calculus, 333, 346foundation for, 76, 291–93, 300, 335multidimensional, 53, 397See also under Giuseppe PEANO

differential equations, 301, 334differential geometry. See under geometryDI LEONARDO, Maria Vittoria (????– ),

323, 325, 328, 414dimension, 186, 277, 283, 287–88,

334, 358, 360, 362, 394DINI, Ulisse (1845–1918), 12, 13,

67, 397–98, 450, 461direct. See under isometry; motionDIRICHLET, Johann Peter Gustav Lejeune

(1805–1859), 76DI SAVOIA, DI SAVOJA. See SAVOY

DI SIENO, Simonetta (19??–), 414, 431disjunction. See under symboldissection, 348, 394distance, 155, 160, 258, 263, 284–85distributivity, 294

doctorate, Italian, 7, 11See also under Mario PIERI

DONATI, Luigi (1846–1932), 10, 98, 417D’OVIDIO, Enrico (1843–1933) 20, 21, 49, 78–

79, 97, 332–33, 405, 414, 418, 450, 460referee for chair competitions, 22–23, 26reviewer, 57–58, 131, 145–46, 148, 151,

375, 377, 380–81, 437–38, 440D’OVIDIO, Francesco (1849–1926), 79, 97DRUCKER, Thomas (19??–), 415DUFLOT, Jeanne (19??–), 406DUGAC, Pierre (1926–2000), 77, 314, 415DUPORCQ, Ernest (1873–1903), 422DUREN, Peter (1935–), 415

École Normale Supérieure, 11economic depression, ix, 366, 368–70economic unrest, 17, 44edge of angle or triangle, 163, 204–5EDWARDS, Harold Mortimer, Junior

(1936–), 415EHRENFEUCHT, Andrzej (1932–), 113EHRENFEUCHT, Krystina Tarski (1938–),

113, 352EISELE, Carolyn (1902–2000), 74, 80, 415elementary function. See under functionelementary geometry

Brodén, 68, 153–154, 407, 412, 436Hilbert, 58–59, 148–51,

274–77, 288, 350, 421 Pieri, vii, 54, 58, 153–54, 156–270,

278–87, 349, 355–58, 381, 442Tarski, viii, 58, 288, 350–56, 368, 394, 452

elicomozione. See transformation: screwelliptic geometry. See under geometryelliptic polar system. See polar systemEMCH, Arnold (1871–1959), 408, 415Emilia–Romagna, xvii, xix, 7, 11, 16, 44empty set, 316Encyclopedia of Elementary Mathematics

(Italian), 65, 327–29, 404See also under BERZOLARI

Encyclopédie des mathématiques pures etappliquées, 46, 87, 430

Encyklopädie der mathematischenWissenschaften, 2, 46, 87, 325, 429

endof arc or semicircle, 252of segment, 195

Index 471

ENGEL, Friedrich (1861–1941), 96, 105–6,112, 415, 419, 421–22, 427, 447

ENRIQUES, Federigo (1871–1946), 24, 42,72–73, 79–80, 107, 398–99, 410, 460

correspondence with Castelnuovo,23, 72, 80, 406

on elementary geometry, 123, 128–29, 133,153, 160, 278, 285, 287, 368, 415–16

rivalry with Pieri, 23, 25–29, 31–32,36, 75, 94, 98, 119, 411

See also under BOTTAZZINI

enumerable. See under classenumerative geometry. See under geometryequality

by definition, 163, 265in logic, 316–17, 336, 350–52of transformations, 270See also under symbol

equidistance, 153quaternary, 153, 281, 351, 353–54, 357–58,

360–62 (see also under congruence: ofpoint pairs; congruence: of segments)

ternary, vii, 152–54, 157, 160–61, 165, 272,278, 280, 282, 286, 341–42, 349–50,357–58, 360–62, 366, 368

equilaterality, 362equinumerous sets, 296equivalence, logical, 163, 348

See also under symbolERBA, Giuseppe Bartolomeo (1819–1895), 332Erlanger program. See under KLEIN

ERDÉLYI, Arthur (1908–1977), 114EUCLID (325?–265? B.C.), 95, 102, 125–26,

147, 149–51, 288, 291, 338, 369, 416in Point and Sphere, 167, 175, 180–83,

185, 187, 192, 199, 206, 209–210,222, 224–28, 230–233, 262, 263

postulate (see parallel postulate: Bolyai’s)Euclidean. See under field; geometryEULER, Leonhard (1707–1783),

18, 76, 248, 284, 416EWALD, Günter (1929–), 143EWALD, William Bragg (1954–), 291, 409,

413, 416, 422, 445excess-intersection theory, vii, 52exponentiation, 314, 318–19existential. See under sentenceextension

of predicate, 298–99, 302, 304of segment, 200

external pointto circle, 195to sphere, 194to triangle, 205

FAÀ DI BRUNO, Francesco (1825–1888),107, 332

FABBI, Angelo (19??–), 44, 46, 110, 117, 416faculty ranks, Italian, 11FAIFOFER, Aureliano (1843–1909), 71–72FANO, Gino (1871–1952), 36, 72, 80–81,

87, 89, 94, 132, 337, 411, 416, 460rivalry with Pieri, 29, 31–32,on foundations of geometry, 53, 65, 107–8,

129, 131, 133, 405, 416, 425FÄRBER, Karl (1863–1912), 377, 438, 448, 455FARNESE, Ranuccio I of (1569–1622), 41Fascism, Italian. See under ItalyFAVA, Narciso (18??–19??), 48–49, 399, 416FÀVARO, Antonio (1847–1922), 36, 398, 411FAZZI, Carlo (18??–????), 398FEFERMAN, Anita Burdman (19??– ),

113, 347–48, 416FEFERMAN, Solomon (1928– ),

113, 347, 348, 416 FEHR, Henri (1870–1954),

290, 382, 412, 442, 445FEIGL, Georg (1890–1945),

404, 415, 417, 422, 426FEIGL, Herbert (1902–1988), 127, 405FEINBERG, Barry (1938–), 386, 416, 441female mathematicians, 40, 365FERDINANDO II. See Ferdinando II

of BOURBON

FERGOLA, Emanuele (1830–1915), 79FERMAT, Pierre de (1601–1665), 454FERMI, Enrico (1901–1954), 114FERRARIS, Galileo (1847–1897), 64, 416FERREIRÓS DOMÍNGUEZ, José (19??– ),

292, 296, 299, 314–15, 323, 416FICHERA, Filadelfo (1850–1909), 33–34FICHERA, Gaetano (1922–1996), 115, 417FIEDLER, Otto Wilhelm (1832–1912), 117field

Euclidean, 263, 351, 355formally real, 354ordered, 351, 354, 356Pythagorean, 354real-closed, 351–52theory, 351

472 Index

figure, 164, 249finitary methods, 60–61finiteness. See under Julius DEDEKIND

FINZI, Bruno (1899–1974), 110FINZI, Cesare (1836–1908), 12, 417FIORINI, Matteo (1827–1901),

8, 10, 28, 81, 398, 423FISCHER, Michael J. (1942–), 352, 417FISCHER-BENZON, Rudolph Jacob

Diedrich von (1839–1911), 456fixed point, 52, 175–76, 270FOÀ, Emanuele (1892–1949), 417FOLKERTS, Menso (1943–), 412, 417, 453following, 163, 249, 252

See also under symbolfoot, 187footnotes, ix, 4, 62, 158FORDER, Henry George (1889–1981), 286, 417formal sum, 292, 294formalism, 307formally real. See under fieldFormolario, Formulaire, formulary

See under Giuseppe PEANO

foundation. See under axiomfoundations of geometry. See under geometryfoundations of mathematics,

85, 116, 128, 298, 327, 343See also under Mario PIERI; RUSSELL;

and school: PeanoFOURET, Georges François (1845–1921),

101, 398, 417France, xvii–xix, 18FRAENKEL, Abraham Adolf (1891–1965),

290, 304, 409, 417, 445FRANCESCONI, Stefano (19??–), 10, 63,

68, 70, 81, 101, 118, 417FRANZ I. See Franz I of HABSBURG

FRÉCHET, Maurice (1878–1973), 124, 398FREGE, Friedrich Ludwig Gottlob (1848–

1925), 298–307, 316, 347, 405, 417,419, 421, 431–32, 434, 447

[1879] 1970 Begriffsschrift, 298–99, 304[1884] 1953 Foundations, 298, 300, 304–51893–1903 Basic Laws, 301–2logic, view of, 298–99, 301, 305, 340See also under Russell’s antinomy;

successorFREGUGLIA, Paolo (19??–), 156, 402, 406FREUDENTHAL, Hans (1905–1990), 85, 127,

313, 341, 407, 415, 418

FRICKE, Robert (1861–1930), 414, 421, 425FRIEDMANN, Sigismondo (1852–1917), 12FRITSCH, Rudolf (1939–), 96, 418FUBINI, Guido (1879–1943), 34, 67,

78, 81, 108, 114, 398, 450FUJIWARA, Matsusaburô (1881–1946), 423FULTON, William (1939–), 52–53, 418function, 269, 314–15

application, 269, 273bijection, 269composition, 273elementary, 292, 299holomorphism, 140injection, 269, 314reciprocal (see function: bijection)See also under transformation

functional analysis, 123fundamental sequence or series, 254, 292

coherent, 293fundamental theorem of projective geometry,

121, 137, 207, 214, 336, 338, 380, 388FURCH, Robert Otto (1894–1967), 143FURTH, Montgomery (1933–), 417fusionism, 147–48, 150, 154–55, 157, 160

GABELLI, Aristide (1830–1891), 17, 418GALOIS, Evariste (1811–1832)

theory, 66, 76GANDINO, Giovanni Battista (1827–1905), 96GARIBALDI, Giuseppe (1807–1882), 41, 65, 80GARIO, Paola (19??–). See BOTTAZZINI

GAROSI, Gino (????–????), 444GAUSS, Carl Friedrich (1777–1855),

76, 93, 111, 125, 293–94, 418, 421GAZZANIGA, Paolo (1853–1930), 118, 455GEISER, Carl Friedrich (1843–1934), 100, 418GEMINIANI, Francesco (1687–1782), 4general topology. See under topologyGENOCCHI, Angelo (1817–1889),

20, 82, 91, 107, 332–36, 418GENTILE, Giovanni (1875–1944), 366, 453

See also under lawsgeometrography, 89geometry

absolute, 145, 151, 273, 287, 340algebraic, vii, ix, 51–53, 75,

84, 103, 107, 123, 406birational, vii, 52Cartesian, 351, 354

Index 473

geometry (continued)complex projective (see under

projective geometry)descriptive, 11, 29, 31, 120diagrams in, 58differential, 53, 116, 120, 383elementary (see elementary geometry)elliptic, 356, 358, 360–61enumerative, vii, 2, 46–47, 51–52, 73,

104–05, 112, 121, 337, 374, 376, 396Euclidean, 145, 149, 151, 157, 161, 274,

349, 356–57, 361, 368, 370 (see alsoelementary geometry)

foundations, 53–54, 58, 84, 95, 106, 113,117, 123–26, 142, 147, 156, 161, 284, 337,342, 350, 352, 354, 366, 368, 371, 379, 394

fundamentals (complementi), 22hyperbolic, 145, 356, 360–61inversive, 57, 93, 99, 137–43line, 134multidimensional, 53, 108, 117, 129, 133,

140, 287, 351, 356, 358, 361, 395non-Archimedean, 81, 117–18non-Euclidean, 41, 81, 84, 86,

112, 284, 287, 358, 394projective (see projective geometry)real enumerative, viitransformational, 284See also compass

GÉRARD, Louis (18??–19??), 263, 418GERGONNE, Joseph Diaz (1771–1859),

127, 418GERHARDT, Karl Immanuel (1816–1899), 427GERICKE, Helmut (1910–), 417GERLA, Giangiacomo (????– ), 350, 418GIACARDI, Livia (1952–), x, 18, 22, 69, 72, 79,

81–82, 92, 108, 123, 129, 333, 411, 418GIAMBELLI, Giovanni Zeno (1879–1953),

42, 49, 286, 399, 418GIAMPAGLIA, Niccolò (18??–19??), 34, 40,

82, 393, 395–96, 410, 419, 443GIANNATTASIO, Antonio (19??–), 95, 419GIBSON, George Alexander (1858–1930), 447GIBBS, Josiah Willard (1839–1903), 53, 84GIGLI, Duilio (1878–1933), 65, 419

See also BERZOLARI: EncyclopediaGILLIES, Douglas Angus (????– ),

294, 302, 304, 315, 419GILLISPIE, Charles Coulston (1918–), 419ginnasio, 7

Gioenia, Accademia, 1, 34, 41See also under journals

GIOENI D’ANGIÒ, Giuseppe (1747–1822), 41GIOLITTI, Giovanni (1842–1928), 26, 43GIORGIO, Gino (avv. 18??–19??), 48Giornale di matematiche. See under journalsGIUDICE, Francesco (1855–1936), 18, 337GIUSFREDI, Beatrice. See Beatrice CAMPETTI

GIVANT, Steven Roger (1945?– ), x, 113, 156,347, 350–55, 357, 360, 369, 419, 452, 460

glide reflection. See under reflectionGOBLOT, Edmond (1858–1935), 147, 419GÖDEL, Kurt (1906–1978), 61, 85,

120, 305, 319, 347, 419–21GOHEEN, Harry (19??–????), 149GOLDFARB, Warren (1949– ), 351, 419GORDAN, Paul Albert (1837–1912), 112Göttingen, 84, 87, 107

Library, 460GOW, Roderick Ian (1948–), 103, 419GRANDJOT REINS, Karl (1900–1979), 319, 409graphical statics, 75, 99, 106GRASSI-CRISTALDI, Giuseppe (18??–19??),

32, 37, 388, 441GRASSMANN, Hermann Ernst (1857–1922), 83GRASSMANN, Hermann Günter (1809–1877),

82–84, 295, 300, 408, 419, 421, 448on arithmetic, 294, 298, 305, 318, on multidimensional methods,

53, 334, 406, 433See also under natural-number arithmetic

GRASSMANN, Justus Günther (1779–1852), 82GRASSMANN, Robert (1815–1901), 83GRATTAN-GUINNESS, Ivor (1941–), v, x, 291,

294, 301–2, 341, 364, 424GRAY, Jeremy (1947–), x, 59, 420, 430greatest-lower-bound principle, 291–93GREITZER, Samuel L. (1905–1988), 75, 420GRIFFIN, Frank Loxley (1881–1969),

369–70, 420GRIM, 414GRIMALDI, Giovan Pietro (1860–1918),

32, 37, 387, 389, 420group. See transformation: groupGRUGNETTI, Lucia (19??–), 406, 430GRUNSKY, Helmut (1904–1986),

409, 415, 417, 426GUCCIA, Giovan Battista (1854–1936), 398GUERRAGGIO, Angelo (1948–),

119, 414, 420, 460

474 Index

GUMBEL, Emil Julius (1891–1966), 447GUNDEL, Hans Georg (1912–), 420, 436GÜNTHER, Siegmund (1848–1923), 398GUPTA, Haragauri Narayan (19–),

x, 313, 354–57, 420Gymnasium. See ginnasio

Habsburg dynasty, xvii, 41HABSBURG, Franz I of (1768–1835), 41HABSBURG, Leopoldo I of (1747–1792), xvii, 41HABSBURG, Leopoldo II of (1797–1870),

xvii, 41HABSBURG, Maria Luigia of (1791–1847),

xvii, 41half-line, 146, 192, 200, 272

See also under symbolhalf-plane, 192, 202, 272half-space, 204half-turn. See reflection: across a lineHALSTED, George Bruce (1853–1922),

64, 162, 198, 263, 420, 422, 445HAMBURGER, Meyer (1838–1903), 408, 433HAMEL, Georg (1877–1954), 360, 420HAMILTON, William Rowan (1805–1865), 53harmonic conjugation, 134harmonic points, 131, 142, 143HASKELL, Mellen Woodman (1863–1948), 425HAUSSNER, Robert Karl Hermann

(1863–1948), 378, 438HAWKINS, Thomas (1938–), 3, 133, 134, 421HEATH, Archie Edward (1887–1961), 84, 421HEATH, Sir Thomas Little (1861–1940),

403, 416HEAVISIDE, Oliver (1850–1925), 53, 84HEDRICK, Earle Raymond (1876–1943),

134, 421, 425HEEGAARD, Poul (1871–1948), 407HEIBERG, Johan Ludvig (1854–1928),

121, 416, 421HEIJENOORT, Jean van (1912–1986),

307, 316, 407, 419, 421, 431HEINE, Heinrich Eduard (1821–1881), 104

See also under theoremhelicoid. See under surfaceHELLINGER, Ernst David (1883–1950), 425HELMHOLTZ, Hermann Ludwig

Ferdinand von (1821–1894), 125HEMPEL, Carl Gustav (1905–1997), 452HENKIN, Leon Albert(1921–2006), 318–19,

324, 421

HERMANN, Robert (1931–), 426HERRE, Paul (1876–1962), 47, 398HERSH, Reuben (1927–), xHessen, Historische Kommission, 460HEYTING, Arend (1898–1980), 362hierarchy, Kleene–Mostowski, 271HILBERT, David (1862–1943), v, 2, 38, 41,

59–61, 84–85, 87, 104–5, 118, 127, 271,275, 300, 369, 378, 405–6, 416, 418,420–22, 424, 433, 439, 460

[1899] 1971 Foundations of Geometry,58–59, 84, 106, 127, 143, 148–51, 155– 56,162, 198, 263, 274, 275, 276–78, 280,282–83, 285, 288, 342, 348, 350, 354, 357,360, 366–67, 369, 414, 436, 445, 453

See also under school; styleHIRST, Thomas Archer Hirst (1830–1892), 378history of mathematics, v, vi, viii, 3, 63,

80–81, 91, 121Italian, 80, 91, 114, 364, 406, 445

HOBSON, Ernest William (1856–1933), 423HOFFMAN, Alan Jerome (19??–), 143, 422HOLGATE, Thomas Franklin (1859–1945), 445Holocaust. See World War II: effects on Jewsholomorphic function. See under functionhomogeneous coordinates

See under coordinateshomography. See under transformationhomothety. See under transformationHOPPE, Reinhold Eduard (1816–1900),

425, 432HUBER, Birkett (19??–), 52, 422HUDSON, Hilda Phoebe (1881–1965), 139, 422HUNTINGTON, Edward Vermilye (1874–1952),

327, 340, 350, 422, 445HURWITZ, Adolf (1859–1919),

84, 86, 105, 292, 455hyperbolic geometry. See under geometryhypermedial point, 212hyperplane reflection. See under reflectionhypersurface, 374, 396

classification, 52quadric, 57

hypersymmetric points, 212hypothesis of definition. See under definitionhypothetical-deductive system, v, vii, 54, 56,

58, 59, 126–28, 141, 147, 151, 154, 157,161–62, 286, 290, 320, 322, 363, 369, 381

See also axiomatic method

Index 475

identity. See under transformationideography. See Giuseppe PEANO:

symbolic notationIdo, 74illumination, constant. See under curveimaginary point, 56, 135, 393implication. See under symbolincaricato. See professoreincidence, 149, 274, 334

relation between lines (see underintersection)

See also under postulateinclusion, 165, 314, 349

See also under symbolinconsistent collection, 304independence

of arithmetic axioms, 312, 318, 325–26, 328of axioms in general, 54, 59, 61, 131, 148,

277, 287–88, 320, 342of primitive notions, 95, 321of Tarski’s geometric axioms, 356See also under parallel postulate

indirect. See under isometry; motioninduction (mathematical), 60, 294,

298–99, 313, 322, 343in Dedekind, 314–15Peano’s postulate, 58, 308, 311,

315, 317, 323, 325–26Pieri’s postulate, 310, 322–23, 325–26definition by. See definition: recursiveSee also minimum principle

inequality. See under symbolinference rule, 316infinite descent, 313infinite set. See Julius DEDEKIND: finitenessinfinity

point at, 138–40, 277, 393See also under axiom

injection. See under functionINGOLD, Louis (1872–19??), 134, 421INGRAMI, Giuseppe (18??–19??),

394–95, 422, 440Institute for Advanced Study

See under Princetoninstitute. See technical instituteinstruction, minister of

See minister of instructioninteger modulo n. See under arithmeticintegral calculus. See differential

and integral calculus

Interlingua. See Giuseppe PEANO: AcademiaIntermédiaire des mathématiciens

See under journalsintermediate value. See under theoreminternal point

to angle, 204to arc or semicircle, 252to circle, 195to segment, 195to solid, 349to sphere, 194to triangle, 205

International Congress of Mathematicians1897 Zürich, 891900 Paris, 60, 85, 104, 148,

281, 341–43, 422, 4491904 Heidelberg, 60, 4221908 Rome, 53, 328, 4221912 Cambridge, 2, 47, 328, 4231928 Bologna, 98

International Congress of Philosophy1900 Paris, 1, 36, 74, 95, 102, 148, 153,

281, 302, 320, 341–43, 381, 406, 412,423 (see also under RUSSELL)

1904 Rome, 1241911 Bologna, 80

Internet repositories, academic, 401intersection, 165, 314

multiplicity, 51, 53, 378relation between lines, 56, 134See also excess-intersection theory

and under symbolinterval. See segmentintuition, geometric, 54, 95, 108, 126–27, 131,

135, 161, 266–67, 283, 286–87, 354inverse. See under transformationinversion. See under transformationinversive

group, 139, 140property, 57See under geometry

involution. See under transformationirrational number, 289, 293, 382isomer, 218isometric figures, 218

See also under congruenceisometry, 154–55, 218, 271

classification, 241, 247–48See also motion

476 Index

isomorphicsimply infinite systems, 300well-ordered sets, 297

ISRAEL, Giorgio (1945– ), 291, 423Italian Mathematical Union, 98Italian universities, 11

hiring policies, 16, 25private, 16public, xviii, 16

Italycities, xviiiChamber of Deputies, 18colonialism, 43corruption, 26, 43economy, 15, 23Fascism, 72, 80–81, 90–92, 119, 366government, 75, 366history, xviiregions, xvii, xixschool system, 7, 15, 366Senate, 18, 27Unification, xvii, 4, 11, 18, 41,

63, 65–66, 70, 78, 85, 109See also Italian universities; literacy

IZUMI, Shin-ichi (1904–1990), 143, 423

Jacobian. See under varietyJADANZA, Nicodemo (1847–1920),

20, 81, 85, 423, 432Jahrbuch über die Fortschritte der

Mathematik. See under journalsJAMES, Glenn (1882–1961), 412, 423JANELLI, Angiolina. See Angiolina PIERI

JESPERSEN, Julie. See Julie ZEUTHEN

JESPERSEN, Louise. See Louise ZEUTHEN

JEVONS, William Stanley (1835–1882), 316JOHNSON, Jeremy R. (196?–), 352, 411, 417join

2 points, 54, 56, 167 (see also symbol: line)3 points, 167, 170 (see also symbol: plane)

JOLLES, Stanislaus (1857–19??), 456JORDAN, Marie Ennemond Camille

(1838–1922), 74, 86JORDAN, Zbigniew (1911–1977), 351, 423JOURDAIN, Philip Edward Bertrand

(1879–1919), 289–90, 292–94, 304–5,316, 327–28, 409, 423–24, 431, 456

Journal für die reine und angewandteMathematik. See under journals

Journal of Symbolic Logic. See under journals

journalsAcademia pro Interlingua, 345Annuario scientifico ed industriale, 90Atti della Accademia Gioenia di Scienze

Naturali in Catania, 34, 35Battaglini (see journals: Giornale di

matematiche)Bollettino di bibliografia e storia della

scienza, 92Bulletin des sciences mathématiques, 402Crelle’s (see journals: Journal für die reine

und angewandte Mathematik)Geisteswissenschaften, 47Giornale di matematiche, 402Intermédiaire des mathématiciens, 89Jahrbuch über die Fortschritte der

Mathematik, 299, 329, 333, 368, 402Journal für die reine und angewandte

Mathematik, 93Journal of Symbolic Logic, 402Mathematicae Notae, 91Mathematical Reviews, 402Mathematische Annalen, 86Revista de mathematica, 336, 343Revue de métaphysique et de morale, 341Revue des mathématiques, 336Rivista di matematica, 71, 301, 336–37,

342–43Rivista populare di politica, lettere

e scienze sociali, 43Transactions of the American

Mathematical Society, 388

KAGAN, Benjamin Fedorovich (1869–1953),124, 385, 398, 441

KALMÁR, László (1905–1976), 319KANNENBERG, Lloyd Chambers (1939–),

419, 433KANT, Immanuel (1724–1804), 266, 305,

412, 416philosophy of arithmetic, 84philosophy of geometry, 102

KANTOR, Seligmann (1857–19??), 398KARZEL, Helmut (1928–), 143, 145, 424KASNER, Edward (1878–1955), 142–43, 424KELVIN, Lord. See William THOMSONKENNEDY, Hubert Collings (1931–),

27, 41, 69, 73, 82, 95–96, 301–2, 316,320, 332–338, 341–42, 345–47,364–65, 399, 424, 433, 435, 460

Index 477

KEYSER, Cassius Jackson (1862–1947),127–28, 424

KILLING, Wilhelm (1847–1923),41, 133, 147, 424

KIRKOR, Andrzej (19??–), 426KIST, Günter (1943–), 145, 424KLEENE, Stephen Cole (1909–1994)

See hierarchyKLEIMAN, Steven Lawrence (1942–),

x, 51–52, 105, 122, 424, 448KLEIN, Christian Felix (1849–1925), 55, 56,

67, 80, 84, 85–87, 105, 107, 117, 121,125, 133, 135, 137, 143, 146, 340, 365,383, 398, 408, 425–26, 430–31,440, 444, 446, 449, 459

1872 Erlanger program, 56–58, 80, 86–87,107–8, 116, 133–34, 139–42, 145,154, 421, 425

KLINE, Morris (1908–1992), 291, 426, 456KNEEBONE, Geoffrey Thomas (1918–2003),

294, 301, 305, 426KNOBLOCH, Eberhard (1943–), 446KOHN, Gustav (1859–1921), 148, 426KOLLÁR, János (19??–), 426, 450KOPPELMAN, Elaine (1937–), 73, 426KOSTANT, Ann (1937–), xKOTARBINSKI, Tadeusz Marian

(1886–1981), 429KÖTTER, Fritz Wilhelm Ferdinand

(1857–1912), 375–77, 379, 415,426, 434, 437–39, 445, 455

KOWALEWSKI, Gerhard WaldemarHermann (1876–1950), 454

KRATZER, Adolf (1858–), 422Kreisverwandtschaft

See transformation: circularKROLL, Hans-Joachim (1942–),

143, 145, 424KRONECKER, Leopold (1823–1891), 95KRYNICKI, Micha (19??–), 271, 426KUMMER, Ernst Eduard (1810–1893), 104–5KURATOWSKI, Kazimierz (1896–1980),

350, 426KUZAWA, Mary Grace (1918–), 367, 426KWIETNIEWSKI, Stefan (1874–1940),

381, 415, 443

LAGRANGE, Joseph Louis (1736–1813), 18LAISANT, Charles-Ange (1841–1920), 89, 426

LAMPE, Emil (1840–1918), 377, 379–80, 408,414–16, 418–19, 422–27, 430–32, 439,441–42, 445, 448–50, 454–57

LANDAU, Edmund Georg Hermann(1877–1938), 290, 294, 319, 426

LANDI, Claude P. (18??–19??), 446LANTELME, Giuseppe (????–????), 332larger-than relation

for angles, 223for cardinals, 296for natural numbers, 311, 313–14, 319for ordinals, 297for real numbers, 293for segments, 220, 272

Latin, 316, 343Uninflected, 42, 49, 74, 301,

336–37, 343, 345, 365lattice theory, 143, 350LAUGEL, Leonce (18??–19??), 421laureate. See doctorateLAURICELLA, Giuseppe (1867–1913),

34, 37, 40, 63, 88, 393, 395–96,413, 419, 426, 443, 450, 459

LAVAGGI, Luigi (18??–19??), 46, 88LAVRENKO, N. I. ( , . ., 19??–),

41, 406LAWAETZ, Sophie. See Sophie ZEUTHEN

lawsCasati and Mamiani, 16, 25Gentile, 90

LAX, Anneli (1922–1999), xLAX, Peter David (1926–), xLAZARSFELD, Robert (19??–), 426, 450LAZZERI, Giulio (1861–1935), 99, 239, 426LEBESGUE, Henri Léon (1875–1941), 89LEGENDRE, Adrien-Marie (1752–1833),

185, 227, 426LEIBNIZ, Gottfried Wilhelm von (1646–1716),

74, 83, 154, 167, 280–81, 284, 307,340–43, 361, 427, 454

LEJA, Franciszek (1885–1979), 427, 445LEMMERMEYER, Franz (1962–), 421LEMOINE, Émile Michel Hyacinthe

(1840–1912), 41, 88–89, 426, 450length of segment, 259LENNES, Nels Johann (1874–1951), 116, 454LEOPOLDO I. See Leopoldo I of HABSBURG

LEOPOLDO II. See Leopoldo II of HABSBURGLESNIEWSKI, Stanis aw (1886–1939), 347, 349less-numerous set, 296

478 Index

LESSONA, Michele (1823–1894), 332letters. See under Mario PIERI

LEVI, Albina Bachi (1886–1951), 90LEVI, Beppo (1875–1961), 2, 10, 12, 15, 20, 22,

34, 36, 41–42, 45, 46–49, 61, 65, 79,89–91, 98, 107, 119, 124, 161, 254,373, 398–99, 411, 427, 460

LEVI, Decio (1885–1917), 90LEVI, Donato (1834–1885), 332LEVI, Emilia. See Emilia RESTA

LEVI, Eugenio Elia (1883–1917), 90LEVI, Giulio (19??–????), 90–91LEVI, Laura (1915–2003), 42, 90–91, 427LEVI-CIVITA, Tullio (1873–1941), 114, 342libera docenza, 11liberal-arts education, 369Library of Congress, 460licenza, 12liceo, 7LIE, Sophus (1842–1899), 41, 56,

86, 105, 121, 146, 427, 446limit, inferior or superior, 255, 273, 285Lincei, Reale Accademia dei, 5, 11, 342LINDEMANN, Carl Louis Ferdinand von

(1852–1939), 84LINDENBAUM, Adolf (1904–1941?),

358, 359, 360, 362, 427, 452, 460LINDGREN, Uta (19??–), 417, 453line

defined, 133–34, 138, 146, 154,170, 272, 281, 340–41

geometry (see under geometry)at infinity, 277reflection (see under reflection)undefined, 128–29, 149, 274, 338See also under symbol

linguistics, 83, 97, 123See also under Giuseppe PEANO

LIOUVILLE, Joseph (1809–1882), 140LIPSCHITZ, Rudolf Otto Sigismund

(1832–1903), 86literacy, 7, 15LOBACHEVSKY, Nikolai Ivanovich

( , ,1792–1856), 41, 125

Prize, v, 2, 38, 41, 64, 84, 89,106, 112, 342, 406, 445

logic, 22, 59, 95, 123, 126, 267, 269, 271,316, 320, 325, 335–37, 341–42, 348,352, 368, 381–82, 387, 402, 411

logic (continued)elementary or first-order, viii, 156, 288, 316,

350–51, 353, 355–356, 368, 370–71See also under Georg CANTOR; Julius

DEDEKIND; FREGE; Giuseppe PEANO;Mario PIERI; RUSSELL

logical constants and terms, 265, 386See also under symbol

logical positivism, 127logicism, 102, 120, 298, 307logistic, 348LORIA, Achille (1857–1943), 92LORIA, Gino (1862–1954), 31, 91–92, 105,

107–108, 152, 333, 398, 409, 418, 447, 460as reviewer, 92, 123, 150–51, 335, 374–78,

380, 393, 397, 407–8, 410, 414, 416, 427,433, 436–41, 443, 447, 449–50, 455

LOVE, Augustus Edward Hough(1863–1940), 423

LOVETT, Edgar Odell (1871–1957),341, 381, 423, 427, 440

LOWE, Victor (1907–1988), 120, 428LÖWENHEIM, Leopold (1878–1957), 423, 428LRM3D2, 405Lucca, xvii, xviii, 1, 4, 6, 44, 49

Royal Academy of Science, Letters,and Arts, 1, 3, 4, 11, 49

LUCCHESI, Camillo (18??–????), 398LUCIANO, Erika (19??–), vi, 74, 341, 428LUGLI, Aurelio (1853–1896), 18LUITPOLD, Prinzregent von Bayern.

See Luitpold of WITTELSBACH)UKASIEWICZ, Jan (1878–1956),

294, 313, 420, 428LUPORINI, Erminia. See Erminia PIERI

LUPORINI, Gaetano (18??–????), 7LUPORINI, Luigi (????–18??), 4LUPORINI, Marianna Davini (????–18??), 4LÜROTH, Jakob (1844–1910), 425

MACCAFERRI, Eugenio (1870–1953),2, 392, 398, 407, 428, 443

MACCOLL, Hugh (1836–1909), 316MACHOVER, Moshé (1936–), 313, 428MACK SMITH, Denis (1920–), 26, 44, 428MAC LANE, Saunders (1909–2005), 116, 428Magnitude, continuous geometric, 291–92MAGNUS, Heinrich Gustav (1802–1870), 104MAKOWIECKA, Helena (19??–), 362, 428

Index 479

MAMIANI, Terenzio, Conte della Rovere(1799–1885). See under laws

MANARA, Carlo Felice (1916–), 129, 428MANFREDI, Bianca (1920?–1999),

47, 71, 88, 106, 428MANN, Thomas (1948–), 402, 428MANSION, Paul (1844–1919), 167MARACCHIA, Silvio (19??–), 366, 428MARCHISOTTO, Elena Anne Corie (1945–),

iv–v, x, 146, 328–29, 336,357, 411, 416, 429

MARCHISOTTO, Joseph Anthony, Jr. (1942–), xMARCOLONGO, Roberto (1862–1943),

46, 53, 69, 378, 392, 407, 429, 443MARCONI, Guglielmo (1874–1937), 100

Comitato International, 461MARIA LUIGIA. See Maria Luigia of HABSBURG

MARIA LUISA. See Maria Luisa of BOURBONMARIE-LOUISE. See Maria Luigia of

HABSBURG

MARINO, Teresa (????– ), 323, 325, 328, 414MARLETTA, Giuseppe (1878–1944), 34, 40,

92, 388, 398, 402, 408, 410, 429Mars. See under Mario PIERI

MARSILI, Luigi (????–18??), 70MARTIN, George Edward (1932–),

145, 147, 151, 429MARTINETTI, Vittorio (1859–1936), 22, 36, 411MARTINI, Eligio (1828–1896), 332MARTINI, Ferdinando (1841–1928), 26MARTOLINI, Guglielmo (18??–????), 12MASOTTO, Guido (19??–). See under

BRIGAGLIA

Mathematicae Notae. See under journalsmathematical induction. See inductionMathematical Reviews. See under journalsmathematics

education (see pedagogy)history (see history of mathematics)See also under RUSSELL

Mathematische Annalen. See under journalsMathesis, 18, 20, 49, 66, 71, 326–27MAXWELL, James Clerk (1831–1879), 75MAY, Kenneth Ownsworth (1915–1977), 429MAYER, Christian Gustav Adolph

(1839–1907), 418, 445MCCALL, Ronald Storrs (1930–), 423, 429MCCLEARY, John (1952–), 446MCCONNELL, Albert Joseph (1903–1993),

103, 429

MCKENZIE, Ralph Nelson Whitfield(1941–), 452

MCKINSEY, John Charles Chenoweth(1908–1953), 352, 357, 452

mean point. See midpointmechanics, 12, 15, 383median of triangle, 233medio-symmetric point, 212, 285MELDRUM, John (19??–), 406membership. See under symbolMENEGHINI, Giuseppe (1811–1889), 70MENGER, Karl (1902–1985),

350, 361, 366–67, 429MÉRAY, Hugues Charles Robert (1835–1911),

218, 429mereology, 349, 350MERIANO (18??–1935?), 5 MERIANO, Gemma Pieri (????–1961?) 5, 9, 97MERZBACH, Uta Caecilia (1933–), 415MEYER, Friedrich Wilhelm Franz

(1856–1934), 87, 105, Encyklopädie, 2, 87, 429,Encyclopédie, 87, 325, 393, 430 reviewer, 413, 424, 427, 455

MIANOWSKI, Józef (1804–1879), 367, 381, 443Foundation, 367–68

MICHAELIS, G. J. (18??–19??),412, 431, 434, 436

middle school, 7, 20midpoint, 146, 155, 164, 174,

272, 283, 357, 362See also under symbol

Milan, Istituto Tecnico Superiore, 66minimum principle, 313, 323, 327–28

weak form, 58, 310, 313, 322–23,325–26, 328

See also inductionminister of instruction, 16, 26–27, 36, 38, 43,

63, 66, 118MINKOWSKI, Hermann (1864–1909), 84MIQUEL, Auguste (????–18??)

See under theoremMITCHELL, Ulysses Grant (1872–19??), 456MITTELSTRASS, Jürgen (1936–), 430, 453MÖBIUS, August Ferdinand (1790–1868)

57, 93, 139, 140, 144, 403, 430, 436See also under transformation

modus ponens, 316MOESE, Henryk (????–19??), 451–52

480 Index

MOHRMANN, Hans (1881–1941), 2, 410,415–16, 429–30, 449, 457

MOIGNO, Abbé François Napoléon Marie(1804–1884), 410

MOLK, Jules (1857–1914), 87, 393, 429–30 MOLLAME, Vincenzo (1848–1911)

32, 34, 75, 93–94, 411, 430MOLLWEIDE, Karl Brandan (1774–1825), 93MONGE, Gaspard (1746–1818), 33monoid, 377monotonic sequence, 254MONTALDO, Oscar (19??–), 406, 430MONTANARI, Enrica. See Enrica PIERI

MONTESANO, Domenico (1863–1930),36, 40, 42, 94, 430, 448

and Pieri’s chair quest, 22, 25–27, 29, 398MONTGOMERY, Deane (1909–1992), 116, 430MOORE, Eliakim Hastings (1862–1932),

116, 342MOORE, Gregory H. (1944–),

59, 351, 420, 430MOORE, Robert Lee (1882–1974),

116, 342, 351, 431MORAW, Peter (1935–), 420, 436MORRICE, George Gavin (1859–1936), 425MORRISON, David Robert (1955–), 426, 450MOSSOTTI, Ottaviano Fabrizio (1791–1863),

66, 78MOSTOWSKI, Andrzej (1913–1975)

See hierarchymotion, 139, 161, 248, 271, 394

direct, 57, 145–47, 150, 152,271, 278, 280–81, 338, 340

indirect, 145–46, 271MÜLLER, Felix (1843–1928), 416, 423, 455multidimensional. See under differential

and integral calculus; geometry;projective geometry

multiplication. See productmultiplicity. See under intersectionmusic, 4, 7, 70, 76, 83, 88, 92–93, 101, 103MUSSOLINI, Benito (1883–1945), 119, 366

Nachlass. See under Mario PIERI

NAGEL, Ernest (1901–1985), 128, 418, 431NANNEI, Enrico (1864–1961), 398Naples, xvii, xviii, 41

university, 5, 16, 22, 25NAPOLEON. See BONAPARTE

Napoleonic wars, xvii, 4, 11, 18, 41

NAPOLITANI, Pier Daniele (1951–), xNARDI DEI, Angiolo (1833–1913), 12, 74, 94NARDINI, Rodolfo (18??–19??), 385NASTASI, Pietro (1941–), 63, 65, 73,

88, 101, 119, 414, 420, 431, 460NATORP, Paul Gerhard (1854–1924),

328, 423, 431natural-number arithmetic, 291, 294

Cantor’s theories, 297consistency, 60, 85, 290, 306 (see also

under Mario PIERI)Dedekind’s theory (see under

Julius DEDEKIND)foundation, 290, 302, 313Frege’s theory, 298, 305Grassmann’s work, 84, 294Padoa’s reduction, 321, 322, 325Peano’s system, 84, 301, 307,

316–22, 370, 405, 409, 449Pieri’s axioms (see under Mario PIERI)

natural-number definition 290, 298, 306–7negation. See under symbolNERVO, Natalia (19??–), vi, 446NEUGEBAUER, Otto (1899–1990), 426NEUMANN, Hanna von Caemmerer

(1914–1971), 455NEWSON, Mary Francis Winston

(1869–1959), 421NEWTON, Sir Isaac (1643–1727), 316NIDDITCH, Peter Harold (1928–1983),

299, 302, 431, 434NIELSEN, Niels (1865–1931), 448NIERI, Idelfonso (1853–1920), 101NOBLE, Charles Albert (1867–1962), 425NOETHER, Emmy (1882–1935), 414NOETHER, Max (1844–1921), 65, 111, 122, 431non-Archimedean. See under geometrynon-Euclidean. See under geometrynormal

to a variety, 374–75See also orthogonality

normal school, 7NORTH, John David (1934–), 146, 431notation

in Point and Sphere, 158, 162–64, 270Pieri’s usage in general, ix, 59, 87,

126, 306, 323, 340, 366–67, 383See also under Giuseppe PEANO; symbol

number theory, 313

Index 481

obituaries. See under Mario PIERI

O’CONNOR, John J. (1945–), 87, 431OETTINGEN, Arthur Joachim von

(1836–1920), 444OHM, Martin (1792–1872), 291–92, 300,

404, 431OHRTMANN, Carl (18??–????), 409, 450one, 258operation on points. See transformationopposite. See diametrically opposite

and under orderorder

on a circle, 252in elliptic geometry, 356, 360–61on a line, 251opposite, 252, 254See also betweenness; cyclic order; larger-

than; and under field; postulateordinal number, 297, 313, 323

See also under arithmeticordinario (see professore)ORE, Øystein (1899–1968), 414orientation, 145, 247, 271origin, 257

of half-plane, 202of ray, 200

orthogonality, 154–55, 280of line and plane, 147, 183, 188, 272of lines, 147, 155, 179, 272, 280, 362of planes, 192See also under symbol

OSTROWSKI, Alexander Markowich(1873–1986), 374, 410, 438, 457

OTTE, Michael (1938–), 404

pacifism, 74, 102PADOA, Alessandro (1868–1937) vi, 94–95,

167, 281, 316, 318, 320–27, 337, 341–42,392, 398, 419, 424, 431, 446

[1900] 1901 Essay on an AlgebraicTheory of Integers, 320–22, 360

[1900] 1902 New System of Definitions forGeometry, 148, 161, 278, 281, 341–42

1902 Theory of Natural Numbers,308, 310, 314, 321–23

method, 321, 360–62Padua, xviii

university, 16PAGLI, Paolo (19??–), 424PAGLIERO, Giuliano (18??–19??), 398

PALLADINO, Franco (1945– ), x, 20, 22, 27,34, 62, 73, 147, 156, 383, 432, 438

See also under BORGA

PAMBUCCIAN, Victor (1959– ), 362, 432PANETTI, Modesto (1875–1957), 85, 432PANNELLI, Marino (1855–1934), 36PANNWITZ, Erika (1904–1975), 409Papal States, xvii, 11PAPPUS (290–350). See under theoremparallel postulate

Bolyai’s form, 155, 206, 209, 278, 285independence, 125, 287Playfair’s form, 149, 155, 274, 277

parallelism, 229of line and plane, 232of lines, 155, 229of planes, 231

PARDUCCI, Amos (1877–1949), 5, 79, 97, 432Parma, xvii, xviii, 11, 41, 44

university, 16, 41, 44, 90, 460PARSONS, Charles Dacre (1933– ),

300, 305, 432PASCAL, Blaise (1623–1662), 117

See also under theoremPASCAL, Ernesto (1865–1940), 110, 398PASCH, Moritz (1843–1930), 54, 95–96,

132, 161, 197, 398, 415, 418,424, 432–33, 436, 449, 460

1882b Lectures on Geometry, 58, 126–28,146–48, 338, 351, 394

on real numbers, 289–90, 293–94, 305See also under postulate; real-

number arithmetic)PÀSCOLI, Giovanni (1855–1912), 96PEANO, Bartolomeo (????–1888), 332, 333PEANO, Bartolomeo (????–????), 332, 333PEANO, Carola Crosio (????–1940), 333 PEANO, Francesco (????–????), 332, 333PEANO, Giuseppe (1858–1932), v–ix, 20,

53–54, 59–60, 69, 71, 73–74, 82, 84, 87,95, 96, 102, 107–8, 114–15, 127, 129,156, 167, 294, 299–305, 303, 315–22,327–28, 331–47, 339, 363–68, 380, 392,395–96, 398, 409, 412, 417, 419, 424,428, 431–36, 446, 450, 453, 460

[1888] 2000 Geometric Calculus,viii, 53, 84, 334, 433

1889 Principles of Geometry PresentedLogically, 128, 334–35, 338, 340, 433

482 Index

PEANO, Giuseppe (continued)[1889] 1973 Principles of Arithmetic,

58, 294, 301, 303, 308, 313–14, 316,318–20, 324, 334–35, 343, 433

1891d Concept of Number,317–18, 320–21, 336–37, 434

1894 Fundamentals of Geometry,127, 147, 338, 340, 351, 434

1895–1908 Mathematical Formulary, 66,69, 71, 269, 301, 306, 317–18, 321, 336–37, 342–43, 344, 345, 364, 409, 436

1903 Geometry Based on Ideas of Point andDistance, 161, 280, 341, 351, 394, 434

Academia pro Interlingua, 42, 49, 345birth, 332calculus text, 82, 333, 336, 418death, 346differential equations research, 334early career, 333family, 332–33Lingua (see Latin: Uninflected)linguistics research, 345, 364–65logic, view of, 59, 301, 305–6, 308,

315, 318–19, 327, 340, 342, 386Pieri, relationship with

(see under Mario PIERI)politics, 333, 346, 365postulates (see under natural-

number arithmetic)praising or supporting Pieri, viii, 27, 38,

43, 56, 124, 131, 135, 153, 325, 329,378, 399, 414–15, 439–41, 443

professorship at Turin, 335, 346Segre polemic, 108, 337, 449schooling, 332students, 71, 80, 89, 94, 365, 396 symbolic notation, ix, 59, 87, 126, 301,

306, 316, 319, 325, 334, 336, 338, 340,342–45, 364–367, 383, 395–96

space-filling curve (see under curve)teaching, 108, 333, 345–46, 364, 395university studies, 332Veronese polemic, 117See also school: Peano and under

real-number arithmeticPEANO, Michele (????–????), 332–33PEANO, Rosa (????–????), 332PEANO, Rosa Cavallo (????–1910), 332PEARSON, Patricia M. Cowan (1928?–),

x, 145, 151, 369–70, 436

pedagogy, mathematical, 40, 98, 108,327, 336, 346, 351, 365, 367

for arithmetic, 322–23, 326, 395–96for geometry, 29, 117, 127, 147–48,

150, 153–54, 157, 160, 162,266, 286–88, 348, 393–94

See also MathesisPEDRAZZI, Maino (19??–), 156, 436PEIRCE, Charles Sanders (1839–1914),

296, 316, 334, 413, 436PEJLARE, Johanna (19??–),

68, 153–54, 407, 436PELLEGRINI, Aldo (????–1940), 6PELLEGRINI, Geminiano (18??–????), 5–6PELLEGRINI, Paolina Livia Pieri

(1865–1959?), 5–6PENNACCHIETTI, Giovanni (1850–1916),

32, 34, 37, 96, 413PENSA, Angelo (18??–19??), 286, 436pentaspherical. See under coordinatesperiphery. See boundarypermutable transformations, 270perpendicularity. See orthogonalityPETERSEN, Julius Peter Christian

(1839–1910), 120–22PETKANTSCHIN, Boyan Lazarov ( ,

, 1907–1987), 143, 436PETRI, Cesare (18??–????), 398PFAFF, Johann Friedrich (1765–1825), 93PFANNER, Pietro (1864–1935), 48–49PHILLIPS, Esther Rodlitz (1933–), 430, 436philosophy of mathematics and science,

80, 125–28, 147, 160, 162, 266–67,286, 341, 343, 381–82, 394

physics, mathematical, 123, 394PICARD, Charles Émile (1856–1941), 64, 74PICKERT, Günter (1917–), 96, 436, 460Piedmont, xvii, xix, 11, 16, 18PIERI, Alfonso (nephew, 1890–1890), 5PIERI, Angiolina Anastasio Janelli (wife,

18??–19??), 5–6, 36, 45, 385, 391, 459PIERI, Beppina Bastian (sister-in-law,

????–????), 5PIERI, Enrica Montanari (sister-in-law,

18??–????), 5, 27, 97PIERI, Erminia Luporini (mother, 1826–1914),

4–5, 39PIERI, Felice Ettore Pacifico Giovanni

(brother, 1866–1920?), 5–6

Index 483

PIERI, Ferruccio Fabio (brother,1864–1933?), 5–6, 39, 390, 436

PIERI, Gemma (niece). See Gemma MERIANO

PIERI, Gemma (sister). See Gemma CAMPETTI

PIERI, Maria Dal Poggetto (sister-in-law,18??–19??), 5–6, 39

PIERI, Mario (1860–1913), i, 19, 24, 1301884b,c dissertations,

12, 14, 15, 50, 374–75, 4591889a translation of Staudt 1847, 20, 107,

111, 123, 137, 335–336, 338, 3931889b Triple Tangents, 20, 21, 3751895a, 1896a,b Principles Governing

Geometry of Position, 124, 128,137, 305–6, 338, 379

1896c Postulates for Projective Geometryof hyperspaces, 128, 379

1897c Primitive Entities of ProjectiveGeometry, 128–29, 380

1898b New Method for DevelopingProjective Geometry, 133, 380

1898c Principles of Geometry of Position,38, 57, 59, 102, 120, 129, 131, 134, 137,306–307, 327, 338, 340, 343, 367,371, 380

1900a Point and Motion, 57–58, 84, 124,126, 134, 145, 148–57, 271–74,276–78, 280–82, 285–87, 327,340–343, 367, 369–71, 381

[1900] 1901 Geometry as a Purely LogicalSystem, 1, 36, 74, 126–27, 148, 153, 278,306, 320–22, 327, 341–43, 379, 381

1901b Principles of Line Geometry,124, 134, 380

1904a Staudt’s Fundamental Theorem,137, 306, 336, 343, 380

1905c, 1906a Complex ProjectiveGeometry, 134–35, 380

1906d New Logico-Mathematical Direction,289, 304, 307, 343, 382

1906e Definition of the Irrationals,289, 328, 343, 382

1906f Staudtian HomographyDefinition, 137, 380

1906g Consistency of Arithmetic,60, 74, 289–90, 306–308, 326–27,340, 343, 363, 382

1907a Axioms of Arithmetic, viii, 3, 58, 289,300–301, 308–13, 321–28, 343, 370, 382

PIERI, Mario (continued)1908a Point and Sphere, viii, 3, 58, 84, 90,

145, 153–271, 278, 280–87, 340, 349–51,355–58, 361, 366–68, 371, 381

1911d, 1912c Geometry of Inversions,137, 141, 145, 381

on algebraic geometry, viii, 1, 12, 38, 50– 53,73, 75, 99, 103, 343, 363–64, 370, 374–78

archive (see Mario PIERI: Nachlass)on arithmetic and logic, vii–viii, 1, 3, 38,

61, 77, 290, 305–13, 320, 322–29,342, 363, 370, 381–382

birth, 4burial, 48–50, 90, 459chair quest, 22–32, 384character, 2, 23, 42–43, 47, 50,

124, 131, 363, 383, 390–92death, 2, 6, 47on differential geometry,

viii, 1, 15, 50–51, 53, 374early career, 5, 17, 20, 22, 27, 31, 383editing De Paolis 1892, 78, 123–24, 393, 414election to academies, 34family, 3–8, 9, 17, 27, 36, 37, 39, 42–44, 45,

46–49, 96, 382–83, 385, 388, 390–91, 459finances, 8, 34, 43–44, 382, 388, 390formulas, ix, 1, 52on foundations of geometry, vii, ix–x, 1, 32,

38, 54–58, 61, 86, 118, 123–25, 145–56,305–6, 327–28, 331, 340–42, 347–51, 363,366–70, 374, 381, 385, 394–95

on foundations of projective geometry,vii, 99, 111, 121, 128–37, 287,328, 336, 367, 370, 379–80

health, 8, 43, 46–47, 117, 388–89, 391–92interest in pedagogy, 20, 49, 147, 154,

160, 162, 265–67, 286–87, 322,326–27, 368, 392–96

on inversive geometry, vii, 46, 137, 139,141–42, 145, 363, 370, 381, 443

knighthood, 2, 42, 389lecture notes, published,

ix, 20, 22, 46, 397–98lecture notes taken by, 12, 392, 397–99legacy, 2, 61, 115, 363–71letters from, ix, 382–92, 399letters to, vi, ix, 382, 398, 403libera docenza, 20, 22licenza, 12

484 Index

PIERI, Mario (continued)logic, view of, 59, 102, 272–73, 289–90,

306–8, 327, 340, 342–43, 363, 386marriage, 5–6, 36, 385Mars drawing, 8, 9, 397, 459music, 7, 42Nachlass, ix, 6, 398–99obituaries of, 49, 61, 90, 399Peano, relationship with, ix, 22, 27, 42, 49,

59, 338, 340, 342–43, 345, 363, 386philosophy of mathematics, viii, x,

102, 147, 160–62, 266–67, 289,306–8, 343, 363, 370, 386

physical stature, 8, 43Poincaré dispute, 58, 74, 290, 326, 343professorship at Catania, 4, 32,

34, 38, 384–85, 387, 390professorship at Parma, 4, 44, 46–47, 390promotion, 36, 37, 38, 386, 459as referee, 34, 42, 388–89relation (see equidistance: ternary)reviews by, ix, 38, 393–97schooling, 5, 7–8, 100, 382, 397student records, 8, 10, 12, 382–83, 459students, 34, 38, 40, 70, 75, 82, 92,

104, 388–89, 393, 395–97university studies, 5, 8, 10,

12–15, 382–83, 397on vector analysis, viii, 1,

46, 53, 343, 364, 378workload, 33–34, 44, 384, 387–90works (collected or listed), vi, ix, 3, 50,

329, 373–74, 379, 392, 398, 401See also under notation; style; successor

PIERI, Nisedi (mother of sister-in-law, 18??–????), 6

PIERI, Paolina (sister). See PaolinaPELLEGRINI

PIERI, Pellegrino (father, 1826–1882),4, 5, 17, 96, 443

PIERI, Pellegrino [Rinuccio] (nephew,18??–19??), 5, 8, 97, 398

PIERI, Piero (nephew, 1893–1979), 5, 97PIERI, Rinuccio

See Pellegrino [Rinuccio] PIERI

PIERI, Romano (grandfather, ????–18??), 4PIERI, Silvio Dante (brother, 1856–1936),

5, 7–8, 9, 17, 34, 39, 49, 79, 96–97,398, 405, 432, 444, 459

PIERI, Teresa Ricci (grandmother,????–18??), 4

PIERI, Teresa (sister). See Teresa BRANCOLI

PIERI, Virginia (sister)See Virginia ANASTASIO

PIERPONT, James (1866–1932), 342, 444PIESYK, Zbigniew (????– ), 355, 420PINCHERLE, Salvatore (1853–1936),10, 25–29,

30, 46, 91, 97–98, 101, 108, 292, 398,402, 444, 449, 453, 455, 459

PINL, Maximilian (1897–1981), 454PIRONDINI, Geminiano (1857–1914),

10, 98–99, 426Pisa, xvii, xviii, 11

Scuola Reale Normale Superiore, 4, 8,10–12, 13, 17, 23, 50, 66–67, 78, 459

university, 10–12, 15–16, 50, 459, 461PITTARELLI, Giulio (1852–1934), 36, 411PLANA, Giovanni (1781–1864), 82plane

antipolar, 238defined, 129, 146, 154, 167,

170, 272, 281, 341at infinity, 277of symmetry, 187undefined, 149, 274reflection (see under reflection)See also under polar plane; symbol

PLATO (427?–347? B.C.), 436PLAUMANN, Peter Klaus (19??–), 424, 444PLAYFAIR, John (1748–1819). See under

parallel postulatePLÜCKER, Julius (1801–1868),

53, 86, 134, 139, 444poetry, 5, 6, 8, 97, 101, 337POGGENDORFF, Johann Christian

(1796–1877), 399, 444POGODA, Zdzis aw (????– ), 350, 411POINCARÉ, Jules Henri (1854–1912),

38, 84, 86, 116, 133, 421, 445See also under Mario PIERI

Point and Motion. See under Mario PIERI

Point and Sphere. See under Mario PIERI

point (defined), 349point reflection. See under reflectionPoland under Russia, 347, 367polar plane

for sphere, 238for two points, 188

polar system, elliptic, 278

Index 485

pole-placement problem, 52poles of a sphere, 194polyhedra, 17PONCELET, Jean-Victor (1788–1867), 104PONZI, L. (18??–19??), 397, 443positivism. See logical positivismpostulate, 54, 95, 162, 266–68

American theorists, 116Archimedes, 117, 137, 141, 213, 273,

276, 278, 284–85, 287, 336, 355circle, 355, 358, 361–62completeness, 149, 151, 276, 350congruence, 146, 149, 274continuity, 141, 146–47, 149, 151, 155, 263–

64, 273, 277–78, 285, 287, 353, 355, 362Euclid (see parallel postulate: Bolyai)incidence, 11, 95, 146, 149, 274order, 95, 149, 274Pasch, 95, 196, 198, 274, 283, 353, 356, 451parallel (see parallel postulate)SAS congruence, 149, 150, 274See also axiom

postulates, choice of, 126, 267, 282, 286, 394postulates, Peano. See under

natural-number arithmeticPOTTS, Donald (1921–2001), xpreceding, 251–52predicate, 298–99prefix. See under quantifierPRESBURGER, Mojzesz (1904–1943?), 445, 351PRESS, Volker (1939–1993), 420, 436PRIHONSKY, Frantisek (1788–1859), 405primitive notions, v, vii, 54, 126–27, 147, 149,

266–68, 320–21, 350, 358, 360–63, 394binary relations as, 360–363

PrincetonInstitute for Advanced Study, 116university, 116, 461

PRISTEM, 403, 445, 460Privatdozent, 11product

of cardinals, 297of natural numbers,

299–300, 314, 318–19, 344of ordinals, 297of scalars, 354of transformations, 270, 273

(see also under symbol)professore (all ranks), 11, 16progression, ascending or descending, 254–55

projective geometry, vii, 8, 11, 20, 22, 51,56, 71–73, 99, 116, 121, 125, 143,277–78, 283, 379, 393, 397

analytic, 60, 93, 140complex, vii, 52, 56, 111,

134–35, 145, 370, 380consistency, 59–60finite, 116foundations, vii, 1, 53–54, 62, 78, 80,

95, 106, 111, 116, 120, 128–33,284, 338, 371, 379–80

multidimensional, vii, 51–54, 57, 65,80, 107, 117, 128–29, 135, 145,338, 374, 376, 379, 396

protothetic, 348, 350provinces, xvii

See also Emilia–Romagna; Piedmont;Tuscany; Sicily

sentence. See under sentencePUCCINI, Giacomo (1858–1924), 4PYTHAGORAS (569? B.C.–475? B.C.), 41, 263Pythagorean. See under field

quadric. See under surface.quadrilateral. See complete quadrilateralquantifier, 271, 348

elimination, 351, 361prefix, 271over sets, 315

quasi-interpretable. See under theoryQueen’s University, 460

RABIN, Michael Oser (1931–), 352, 417RANUCCIO I. See Ranuccio I of FARNESE

RASCHI, Leonida (1831–1917), 46, 99rational number, 392

See also under arithmeticRAUSENBERGER, Otto (1852–19??), 263, 445ray. See half-lineRAZZABONI, Amilcare (1855–1920), 26real-closed. See under fieldreal enumerative. See under geometryreal-number arithmetic, 257–58, 263–64,

276, 278, 285, 288, 289–92Cantor, 293Dedekind, 76, 293–94, 299Pasch, 293Peano, 293, 334Russell, 293Weierstrass, 292, 305

486 Index

Realschule, 7reciprocal

radii (see transformation: inversion)See also function: bijection

recursive definition. See under definitionrecursive function theory, 315recursive proof. See inductionReed College, 369–70reflection, 154

hyperplane, 140line, 139, 147, 155, 164, 176 (see also

transformation: ribaltamento)plane, 140, 145, 147, 155, 178, 186–87, 248point, 145, 155, 164, 174–75, 280glide, 245, 248rotary, 243, 248See also under symbol

reflexive. See under relationregion. See under Italyregular open set, 349REID, Constance (1917–), 85, 100, 445, 460REINHARDT, Curt (1855–1938?), 430relation, 249, 269

Pieri’s (see equidistance: ternary)reflexive, 165symmetric, 165transitive, 165univocal, 268 (see also function)See also betweenness; cyclic order;

equality; equidistance; equilaterality;larger-than; order

RENSCHUCH, Bodo (19??–), 105, 408representation, 266

See also transformationRESTA, Emilia Levi (1921–), 90–91, 460Revista de mathematica. See under journalsreviews, sources of, 402revolutions. See 1848.Revue de métaphysique et de morale.

See under journalsRevue des mathématiques.

See under journalsREYE, Carl Theodor Reye (1838–1919),

57, 71, 72, 99–100, 106, 136,137, 139, 418, 445, 448, 453

REYES Y PRÓSPER, Ventura (1863–1922), 398ribaltamento. See under transformationRICCÒ, Annibale (1844–1919),

17, 32, 37, 398, 445RICHARD, Jules (1862–1956), 326

RIEMANN, Georg Friedrich Bernhard(1826–1866), 66, 76, 86, 98–99, 125

sphere, 140See also under theorem

RIGHI, Augusto (1850–1920), 8, 9, 25,100, 397, 409, 445, 451, 461

Right angle (see under angle)RINDI, Scipione (1859–1952), 3, 4, 6–8, 20,

22, 34, 36, 47, 49, 61, 101, 123–24,375, 398–99, 408, 438, 446

RINOW, Willi (1907–1979), 425Rivista di matematica. See under journalsRivista populare di politica, lettere e scienze

sociali. See under journalsRIZZI, Fortunato (1880–1965), 99, 446ROBERTSON, Edmund Frederick (1943–),

87, 431ROBINSON, Raphael Mitchel (1911–1995),

361–62, 446RODRIGUEZ-CONSUEGRA, Francisco A. (1951–),

x, 294, 307, 328, 341, 386, 429, 446ROERO, Clara Silvia (1952–), vi, 11, 18,

32, 69, 74, 95, 341, 418, 428, 446Peano CDs, vi, 336, 345, 347,

377–79, 393–94, 446ROGERS, Reginald Arthur Percy

(1874–19??), 447ROHN, Karl (1855–1921), 451ROLLAND, Romain (1866–1944), 88–89, 446ROMAGNOLI, Raffaello (19??–), xRoman Catholic Church, xvii, 4Rome, xvii, xviii, 11

archives, 459university, 16, 22–23, 75

Rosario, university, 91ROSENTHAL, Arthur (1887–1959), 451ROSS, Robert Hemphill (1949–), 403ROSSI, Luigi (18??–????), 398ROTA, Gian-Carlo (1932–1999), 148, 446rotary reflection. See under reflectionrotation. See under transformationROTHE, Rudolf Ernst (1873–1942), 378, 443ROWE, Charles Henry (1893–1943), 447ROWE, David E. (1950–), x, 56, 446ROYDEN, Halsey Lawrence (1928–1993),

361, 447RUFFINI, Ferdinando Paolo (1823–1908),

8, 28, 101, 444RUFFINI, Paolo (1765–1822), 101ruled. See under surface

Index 487

RUSS, Steven (19??–), 405RUSSELL, Bertrand Arthur William (1872–

1970), viii, x, 47, 74, 102–3, 120, 125,300, 304–5, 327, 339, 341–42, 392, 402,406, 417, 419–20, 428, 446–48, 456

1900 Paris epiphany, 74, 102, 302, 3411903 Principles of Mathematics,

38, 102, 129, 249, 290, 294, 302,304, 328, 338, 386, 392

Archives, 416, 459letter from Pieri, 307–8, 386, 398, 441, 459on real numbers, 289–90, 293–94, regard for Pieri, v, 2, 38, 129, 328, 338, 343 view of logic, 386, 307, 327, 386view of mathematics, 307See also under real-number arithmetic;

Russell’s antinomy; A. N. WHITEHEAD

Russell’s antinomy, 102, 302, 304–5, 447Frege’s response, 302, 304, 417

RUSSO, Achille (1866–1955), 37RUZIEWICZ, Stanis aw (1888–1941), 450

SAINT-SAËNS, Camille (1835–1921), 88Salesians of Don Bosco, 460SALKOWSKI, Erich (1881–1943),

405, 412, 431–32, 448, 456Salmon, George (1819–1904),

52, 103, 117–18, 419, 429, 447SALUTA, Andrea (18??–19??),

40, 104, 410, 447SALUZZO, Giuseppe Angelo, Count of

Monesiglio (1734–1810), 18SANNIA, Achille (1823–1892), 79, 85, 93SANTAGATA, Domenico (1812–1901), 8, 10Sant’Andrea di Compito, 6, 44, 47, 49, 459Sardinia

kingdom, xvii, 18region, xvii, xix, 16

SAS. See under postulateSAVI, Paolo (1798–1871), 70SAVOIA, SAVOJA. See SAVOY

SAVOY, Carlo Emanuele II of (1634–1675), 18Savoy dynasty, xvii, 18SAVOY, Umberto I of (1844–1900), 70SAVOY, Vittorio Amedeo III of (1725–1796), 18SAVOY, Vittorio Emmanuele II of

(1820–1878), xvii, 18SAVOY, Vittorio Emmanuele III of

(1869–1947), 423

SCACCIANOCE, Rosario (1880–19??),40, 43, 104, 389, 410, 447

scale determined by two points, 155, 212, 354Scandinavian Congress of Mathematicians,

1911, 153, 448SCANLAN, Michael (1957?–), 116, 278, 360, 448SCHAPPACHER, Norbert (19??–), 421SCHAYER, W. (????–????), 348, 452SCHEIBNER, Wilhelm (1826–1908), 430SCHEPP, Adolf (1837–????), 418SCHIFF, Hugo Josef (1834–1915), 332 SCHILLING, Friedrich Georg (1868–1950), 425SCHLEGEL, Viktor (1843–1905),

380, 408–9, 433–34, 440, 448SCHLESINGER, Ludwig (1864–1933), 41SCHMID, Anne-Françoise (1949– ), 305, 448SCHNEEWIND, Jerome Borges (1930–), 428SCHOENEBERG, Bruno (1906–1995), 87, 408SCHOLZ, Heinrich (1884–1956), 454, 456SCHÖNFLIES, Arthur Moritz (1853–1928),

377, 398, 438–39school

Hilbert, vii, ix, 84–85, 143, 284Peano, vii, 1, 2, 22, 27, 32, 40, 54, 58–

59, 66, 69, 71–74, 80, 84, 95, 102,108, 110, 114, 80, 123, 126–27, 131,153, 300–301, 305, 308, 320, 326, 331,335–38, 341–42, 346, 363–66, 369, 405

Segre, vii, ix, 1, 22, 32, 51, 72, 107,123, 131, 333, 337–38, 346

system (see under Italy)Tarski, viii, 58, 113, 352–53,

356, 361, 367–68, 370Volterra, 123See also logical positivism; middle school;

normal school; postulate: Americantheorists; technical school; Vienna Circle

SchraubenbewegungSee transformation: screw

SCHRÖDER, Friedrich Wilhelm Karl Ernst(1841–1902), 104, 316, 334, 341

SCHRÖTER, Heinrich Eduard (1829–1892),95, 105

SCHUBERT, Hermann Cäsar Hannibal(1848–1911), vii, 52, 104–5, 112, 325,396, 398, 408, 417, 424, 448, 453, 456

calculus, 52, 104, 112SCHUBRING, Gert (19??–), 406, 448SCHUMANN, Richard (1864–1945), 456

488 Index

SCHUR, Friedrich (1856–1932),41, 86, 99–100, 105–6, 124,148, 263, 398, 448, 453

SCHUR, Issai (1875–1941), 148SCHWABHÄUSER, Wolfram (1931–1985),

350, 354–57, 360–62, 448SCHWARTZE, Heinz (1924–2006), 84, 448SCORZA, Bernardino Gaetano (1876–1939),

46–47, 94, 106, 448SCOTT, Charlotte Angas (1858–1931),

343, 422, 449screw. See under transformationSCRIBA, Cristoph J. (1929–), 84, 406, 408, 413Scuola Reale Normale Superiore.

See under Pisasecant space, 377, 395–96segment, 128, 146, 195, 272, 335, 394

open, 277projective, 54, 128, 130, 131, 134–5, 338See also length and under symbol

SEGRE, Beniamino (1903–1977), 98, 449SEGRE, Corrado (1863–1924), 20, 36, 42,

55, 69, 107–8, 118, 383, 393, 398–99,410, 428, 440, 449–50, 460

algebraic geometry, v, vii, ix, 51, 53,72, 78–79, 87, 89, 91, 105, 123–24,133, 333, 405, 414, 418, 449

and Pieri’s chair quest, 23, 26, 28–29projective geometry, 53, 80, 123, 335, 393rift with Peano, 108, 114, 337–38,

346, 428, 434, 449supporting Pieri, 56–58, 131, 135, 145–46,

157, 160, 376, 380–81, 436–37, 441–42See also under school

SEGRE, Michael (1950–), 291–92, 301,320, 332, 335–337, 342, 346, 449

SEIDENBERG, Abraham (1916–1988), 96, 449semicircle, 252SEMPRONIO, 268Senate. See under Italysense, 247, 251–52

See also cyclic order; ordersentence

classification by quantifier prefix,271, 276–77, 282–83

existential, 271, 271, 373

universal, 271separation. See cyclic order

sequence, fundamentalSee fundamental sequence

sequent of natural number, 311, 324series, fundamental

See fundamental sequenceSERNESI, Edoardo (19??–)

See under BRIGAGLIA

SERRET, Joseph Alfred (1819–1885), 78set theory,

antinomies, 69, 304axiomatic, 290, 300, 304Cantor’s development, 296–97, 299, 305in Dedekind, 77, 293, 299, 305, 314–15 general usage, 313, 316, 324, 360 in Point and Sphere, 157, 294,

305, 350, 352, 360Pieri’s natural-number model, 61, 290Tarski’s work, 113, 348See also class

SETON-WATSON, Christopher (1918–),43–44, 449

SEVERI, Francesco (1879–1961),42, 77, 104, 114, 290, 398, 450

SEVERINI, Carlo (1872–1951), 34, 98, 108, 450SFORZA, Giuseppe (18??–1927), 306, 398shadow

of line, 202of plane, 204of point, 200

SIACCI, Francesco (1839–1907),20, 27, 91, 109, 332, 403

SIBIRANI, Filippo (1880–1957),46, 63, 98, 109, 454

Sicilies, Kingdom of Two, xvii, 41Sicily, xvii, xix, 16, 41side

of line in plane, 203of plane, 204of point on line, 202

signed integer. See under arithmeticSIGNORINI, Antonio (1888–1963), 110, 450SILLA, Lucio (1872–1959), 88, 450similar figures, 207, 160similarity, 57, 139, 154, 161, 206–7, 281

classification, 278, 284in hyperbolic geometry, 361

simply infinite system, 300, 314existence, 300, 307, 314

SINACEUR, Hourya (1940–), 351, 450singleton. See under symbol

Index 489

singular point, vii, 12, 50–51, 375SINIGALLIA, Luigi (1864–1944), 46, 110SINSHEIMER, Thomas (1943–), xskew lines, 246SKOF, Fulvia (1932–), 71, 399, 450SKOLEM, Thoralf Albert (1887–1963), 452smaller-than. See larger-thanSMID, Lucas Johannes (????–????), 143, 455SMITH, David Eugene (1860–1944), 89, 450SMITH, Helen Marie Patteson (1940–), xSMITH, James Thomas (1939– ), iv, x, 357, 450solid, 349SOMIGLIANA, Carlo (1860–1955),

10, 79, 110, 403, 417, 450SOMMER, Julius (1871–1943), 380, 440, 451SONCINI, A. (18??–19??), 397, 443SOTTILE, Frank (1963–), x, 52, 422, 450SPERANZA, Francesco (1932–1998), xSPEZIA, Giorgio (1842–1911), 332SPEZIALI, Pierre (1913–1995), 78, 81, 108, 450sphere, 160, 349

concentric, 349fundamental, of anti-inversion, 238in inversive geometry, 137, 140–42 in Point and Sphere, 58, 157, 165, 281 in Point and Motion, 146, 272polar, 194, 272as tool, 154, 157, 283tangent, 167See also under Riemann; symbol

SPOGLIANTI, Maria (19??–), 129, 428Springer-Verlag, 460SSS triangle-congruence. See under theoremSTÄCKEL, Paul (1862–1919), 451, 455STAHL, Wilhelm (1846–1894), 432, 449, 455STAMM, Edward (????–1940), 345, 450statics. See graphical staticsSTAUDT, Georg Karl Christian von

(1798–1867), 54, 110–11,136, 408, 429, 431, 444

1847 Geometrie der Lage, vii, 57,95, 99, 129, 160–61, 269, 450

collineation or homography, 207, 380complex projective geometry, 56, 134–36Pieri’s 1889a translation of Staudt 1847,

20, 107–8, 123, 335, 338, 379, 393, 437See also fundamental theorem

of projective geometrySTEELE, Donald A. (19??–????), 405STEFANINI, Annibale (18??–????), 398

STEINER, Jakob (1796–1863),83, 101, 105, 139, 450

STEINHARDT, Fritz (1922–1993), 426STEINITZ, Ernst (1871–1928),

404, 431, 448, 454STÉPHANOS, Cyparissos (1857–1918), 398stereographic projection, 138STOLZ, Otto (1842–1905), 291, 408, 425, 450STRAMBACH, Karl (1939–), 424, 444STRANEO, Paulo (1874–1968), 108, 450straordinario. See under professoreSTÜCKRAD, Jürgen (1948–), 52, 450STUDY, Eduard (1862–1930), 41, 105,

111–12, 133, 405, 408, 415, 436, 451STURM, Rudolf (1841–1919), 445

sequence, 351STURMFELS, Berndt (1962–), 52, 422style,

Hilbert, 148–51, 276, 283, 354Pieri, 58, 126, 141, 147, 150–51, 156,

158, 162, 171, 272–73, 278, 283–86,323, 340, 350, 356, 368, 370

Tarski, 350, 354, 356successor, 58, 299, 308, 315–6, 321–22

Frege’s definition, 298Pieri’s definition, 290, 307Zermelo’s construction, 300See also under symbol

sumof angles, 220, 228, 395of cardinals, 297in mereology, 349of natural numbers, 299–300,

311, 314–15, 318–19, 324of ordinals, 297of scalars, 354of segments, 210, 220, 226, 272, 392

sundial, 52, 376superposition, 218

argument, 150, 338SUPPES, Patrick (1922– ), 297, 421, 431, 451surface

algebraic, vii, 12, 15, 20–21,51, 72, 75, 375–76, 383

birationally equivalent, 79helicoidal, 51, 374quadric, 51, 57, 140, 375ruled, vii, 51–52system, 378

Sweden, mathematics in, 68, 153–54

490 Index

symbolabstraction, 316angle notation, 163, 204 . . . . . Â.BC, A.BCarc, 252 . . . . . . . . . . . . . . . . . . . . . . . . O(AB)between, 272, 351 . . . . . . . . . . . q-p-r, Bqprcollinear, 357 . . . . . . . . . . . . . . . . . . . . Cabccongruent, 162–63, 216,

218, 274 . . . . . . . . . . . . . . . . . . . . . . F =~ Fconjunction, 162–63 . . . . . . . . . . . . . . & difference of sets, 162 . . . . . . . . . . . . . A – Bdisjunction, 162, 164 . . . . . . . . . . . . . .empty set, 316equality, 162–63, 165, 316 . . . . . . . . . A = Bequality by definition, 163, 265 . . . . . r ≡ ABequidistance, quaternary, 357 . . . . . . Dpqrsequidistance, ternary,

280, 357 . . . . . . . . . . . . . . . . . . . Ipqr, Ppqrequivalence, logical, 162–63 . . . . . . . ⇔existential quantifier, 271 . . . . . . . . . . xexistential quantifier string, 271 . . . . . . .following, 163, 249, 253 . . . . A ,B C, O, A ,B Chalf-line, 163, 200 . . . . . . . . . . . . . . . . . . ABhalf-plane, 202 . . . . . . . . . . . . . . . . . . rPhomothety, 233 . . . . . . . . . . . . . . . . . . .

AO

A

′

image by transformation, 269 . . . . . . . . RAimplication, 162–63, 316 . . . . . . . . . .inclusion, 162–63, 165 . . . . . . . . . . . . .inequality, 162–63 . . . . . . . . . . . . . . . A /= Bintersection, 162–63, 165 . . . . . . . . .line joining 2 points, 162, 167 . . . . . . . . ABline reflection, 176 . . . . . . . . . . . . . . . . . . . /rmembership, 162–63, 316 . . . . . . . . . . Y rmidpoint, 162, 174, 357 . . . . . . . A B, Mabcnegation, 162–63 . . . . . . . . . . . . . . . . . . . ¬nonmembership, 162–63 . . . . . . . . . . . Y rorthogonality, 163, 179, 357 . . . . . . . . r sright angle, 357 . . . . . . . . . . . . . . . . . . Rabcplane joining 3 points, 162, 170 . . . . . . ABCplane reflection, 186 . . . . . . . . . . . . . . . . . /point reflection, 162, 174–75 . . . . . . . . . . /Aproduct of transformations, 270 . . Q P, Q Pquantification (Peano’s), 316ray (see symbol: half-line)segment, 163, 195 . . . . . . . . . . . . . . . . ABsemicircle, 252 . . . . . . . . . . . . . . . . . . O(ABsingleton, 162 . . . . . . . . . . . . . . . . . . . . . {X}sphere, 160, 162, 165, 194 . . . . BA, Sph(A,B)substitution of variables,

163, 316 . . . . . . . . . . . . . . . . . . P7§2, ,, ,

D C AA B C

successor, 311, 316, 319, 321 . . . . suc x, x+1

symbol (continued)sum, 226, 228, 318 . . . . . . . . . . . . . +translation, 237 . . . . . . . . . . . . . . . . . . .

AA

′

triangle, 163, 205 . . . . . . . . . A.BC , ABCunion, 162, 164–65 . . . . . . . . . . . . . . .universal quantifier, 271 . . . . . . . . . . . . xuniversal quantifier string, 271 . . . . . . . .See also notation;

Giuseppe PEANO: symbolic notationsymmetric. See under relationsymmetry, 160

over a line, 176over a plane, 187over a point, 174–75

system. See simply infinite systemSZCZERBA, Les aw W odzimierz (1938–),

271, 347–48, 350–51, 354,356, 369, 426, 451

SZEG , Gábor (1895–1985), 451SZMIELEW, Wanda Montlak (1918–1976),

350, 354–57, 360–62, 368, 406, 448

TABARRONI, Giorgio (1921–2001), 100, 451tactile-physical space, 160TAJTELBAUM, Ignacy (1869–1942), 347TAJTELBAUM, Alfred. See Alfred TARSKI

TAJTELBAUM, Rosa Prussak (1879–1942), 347TAJTELBAUM, Waclaw. See Wac aw TARSKI

tangentline and sphere, 181, 200spheres, 167to a variety, 374–75, 383triple (see Mario PIERI: 1889b)

TANNERY, Jules (1848–1910), 74, 325, 448TARSKI, Alfred (1901–1983), v, viii, x, 113,

116, 156, 278, 288, 305, 331, 347,359, 412, 416, 419, 421, 427,431, 448, 451–52, 460

[1927] 1983 Foundations of geometryof solids, 349–50, 368, 451

[1957] 1959 What is elementary geometry,ix, 288, 352–57, 368, 452

birth, 347career at Berkeley, 113change of surname, 348death, 113definability research, 357–58, 360–61doctorate, 348early employment, 348, 351–52emigration to the United States, 113, 352

Index 491

TARSKI, Alfred (continued)family, 113, 347, 352marriage, 351–352and Pieri, 349–50, 353–54, 356–57, 367–70politics, 347schooling, 347See also under elementary geometry;

school; styleTARSKI, Janusz (1934– ), x, 113, 352TARSKI, Krystina. See Krystina Tarski

EHRENFEUCHTTARSKI, Maria Witkowska

(????–1990), 113, 351TARSKI, Wac aw (1903–1944), 347–48TARTUFARI, Luigi (1864–1931), 47, 453Taurinorum, Augusta, 18tautologous point. See fixed pointTAZZIOLI, Rossana (1962–), 34, 40,

73, 81, 88, 92, 94, 108, 453technical institute and school, 7TEITELBAUM. See TAJTELBAUM)TEIXEIRA, Francisco Gomes

(1851–1933), 375, 438TERRACINI, Alessandro (1889–1968),

79, 347, 453tetracyclic. See under coordinatestheorem, 126, 267

Bézout, 52–53Cantor-Bernstein, 435Desargues, 232, 276Heine–Borel, 277intermediate value, 291Miquel, 143Pappus–Pascal, 143, 276, 388Pythagoras, 263Riemann mapping, 141SSS triangle-congruence, 281Staudt (see fundamental theorem

of projective geometry)three perpendiculars, 185

theoryaxiomatic (see hypothetical-

deductive system)categorical, 151, 317complete, 351consistent, 54, 59–61, 290, 306, 382decidable, 351quasi-interpretable, 278, 360

THIEL, Christian (1936–), 399, 453

THOMAE, Johannes Karl (1840–1921),393, 439, 453

THOMSON, William, Lord Kelvin (1824–1907),139, 453

THORUP, Anders (1943–), 424throat cancer, 2, 5–6, 46–47, 97, 391TIMERDING, Heinrich Carl Franz Emil

(1873–1945), 100, 453 TIZIO, 268TOEPELL, Michael-Markus (19??–), x, 148, 453TOEPLITZ, Otto (1881–1940), 376, 439TONELLI, Leonida (1885–1956), 98, 453topology, general, 349–50, 352toponymy, 444totalitarianism, ix, 366Transactions of the American Mathematical

Society. See under journalstransformation, 71, 154, 161, 249, 269, 394

affinity, circular(see transformation: Möbius)

anticongruence, 247–48(see also motion: indirect)

antihomography, 247anti-inversion, 238antiprojectivity, 135antirotation (see reflection: rotary)antitranslation (see reflection: glide)birational, 51, 65, 72, 75, 80, 374, 377–78circular, 57, 139–42collineation, 207, 394 (see also

transformation: homography)conformal, 139–41, 236congruence, 247–48 (see also motion: direct)continuous, 139Cremona (see transformation: birational)group, 56, 57, 81, 86, 133, 146holomorphism (see under function)homographyhomography, 56, 133, 247, 340, 377–78, 380

(see also transformation: collineation)homothety, 233identity, 183, 237, 269inverse, 269, 273inversion, 57, 137–40, 142, 238involution, 137, 176, 270Kreisverwandtschaft

(see transformation: circular)Möbius (see transformation: circular)projectivity, 137

492 Index

transformation (continued)by reciprocal radii

(see transformation: inversion)ribaltamento, 176rotation, 145, 183screw, 246–47as a tool, 157, 278, 283–84See also function; isometry; motion;

permutable; similarity; translation(geometric); and under equality; product

transformational geometrySee under geometry

transitive. See under relationtranslation (geometric), 145, 237, 247

normal, parallel, or oblique, 242translation strategy, ix, 157–58, 162–164TREUTLEIN, Peter (1845–1912),

409, 425, 444, 449triangle, 205

See also under symbolTRICOMI, Francesco Giacomo (1897–1978),

81, 113-115, 453citations, 12, 18, 20, 26, 34,

40, 62, 67, 110, 346effect on Peano’s and Pieri’s

reputations, 346, 363–65, 453unreliability, 42, 115, 364

trigonometric series, 292, 294Turin, xvii–xviii, 5, 17–18, 27

Royal Academy of Sciences,18, 20–21, 27, 148, 335, 346, 460

Royal Military Academy,18, 19, 333, 459, 460

See also Turin, university ofTurin, university of, 16, 18,

20, 23, 31, 123, 332rift between Peano, Segre schools

32, 108, 110, 114, 337, 346Tuscany, xvii, xix, 4, 11, 16, 44type theory, 102, 304

ULLRICH, Peter (19??–), 455ultrasymmetric point, 212UMBERTO I. See Umberto I of SAVOY

undefined concepts. See primitive notionsUNGER, Leo (????–????), 421–22Unification. See under Italyunion, 165, 314

See also under symbolunit of measure, 258

unit point, 257universal sentence. See under sentenceuniversities, Italian. See Italian universitiesUniversity of California at Berkeley, x, 113 USIGLIO (18??–19??), 33Ustica, xviii, 5, 6, 27, 454

VACCA, Giovanni (1872–1953),313, 337, 341, 343, 454

VAI, Gian Battista (19??–), 70, 454VAILATI, Giovanni (1863–1909),

34, 153, 333, 337, 341, 398, 454variations. See calculus: of variations variety

algebraic, 51, 377Jacobian, 12, 50, 375

VARIGNON, Pierre (1654–1722), 418VAROLI, Giuseppe (????–????), 109, 454VASSILIEF, Alexander (1853–1929), 398VEBLEN, Andrew Anderson (1818–1906), 115VEBLEN, Oswald (1880–1960), 43, 115–116,

120, 279, 351, 357, 369, 388, 403,428, 430, 441, 454, 461

1904 System of Axioms for Geometry,161, 271, 277–278, 279, 283,285, 340, 360, 366–67, 448

complex projective geometry, 135, 137on Pieri 131, 133, 370

VEBLEN, Thorstein (1857–1929), 115vector analysis, viii, 1, 46–47, 53,

69, 71, 83–84, 334, 378vector space, 334Vellano, 4Venice, Istituto di Scienze,

Lettere, ed Arti, 460VERGERIO, Attilio (1877–1937), 44, 46–47, 117VERONESE, Giuseppe (1852–1917),

117–18, 132, 447, 449, 490difference with Peano 129, 338, 428, 434on foundations of geometry, 131, 133, 146,

148, 161, 263, 280, 342, 394, 405, 454–55as referee, 22–23, 29, 36, 42

Verschelde, Jan (19??–), 52, 422vertex

angle, 204triangle, 205

Vienna Circle, 361, 366VILLARI, Emilio (1836–1904), 10, 100, 118VILLARI, Pasquale (1827–1917), 118

Index 493

VITALI, Giuseppe (1875–1932),34, 44, 387, 390, 398, 441–42, 455

VITTORIO AMEDEO III. See VittorioAmedeo III of SAVOY

VITTORIO EMMANUELE II. See VittorioEmmanuele II of SAVOY

VITTORIO EMMANUELE III. See VittorioEmmanuele III of SAVOY

VIVANTI, Giulio (1859–1949), 337as reviewer, 375, 382, 407–8,

432–34, 438, 442, 454–56See also BERZOLARI: Encyclopedia

VIVORIO (vice Prefetto, cav. uff.,18??–19??), 48

VOGEL, Wolfgang (1940–), 52, 450VOLTERRA, Edoardo (1904–1987), 119, 455VOLTERRA, Virginia Almagià (18??–19??), 119VOLTERRA, Vito (1860–1940), 10, 12, 30,

114, 118, 119, 123, 398, 410, 455, 460Pieri’s chair quest, 23, 25–27See also under school

volume, 263

WAELSCH, Emil (1863–1927), 377–78, 438–39WAERDEN, Bartel Leendert van der

(1903–1996), 105, 143, 144, 455, 460WALKER, Gay (19??–), 420WALLENBERG, Georg (1864–1924),

421, 431, 438WANG Hao (1921–1995), 294, 304, 455Warsaw, city and university, 347WEBER, Heinrich Martin

(1842–1913), 263, 455WEBER, Wilhelm Eduard(1804–1891), 421WEDBERG, Anders (1913–1978), 113WEDDERBURN, Joseph Henry MacLaghan

(1882–1948), 116WEIERSTRASS, Karl Theodor Wilhelm

(1815–1897), 86, 95, 98, 105,292–94, 300, 305, 419, 455

See also under real-number arithmeticWEINGARTEN, Leonhard Gottfried Johannes

Julius (1836–1910), 374, 437, 455WEINMEISTER, Paul Franz Wilhelm

(1856–1927), 444WEISS, Richard (1948–), 418Well-ordering, 297–98, 313, 348WEYL, Hermann Klaus Hugo

(1885–1955), 445

WHITEHEAD, Alfred North (1861–1947),2, 102, 119–20, 131, 133, 304–5,316, 327, 404, 428, 455–56

1910–1913 Principia Mathematica(with Russell), 102, 120, 304–5, 456

WHITEHEAD, Henry (1853–1947), 120WHITEHEAD, John Henry Constantine

(1904–1960), 116, 120, 454WIELANDT, Helmut (1910–2001), 414WIELEITNER, Heinrich (1874–1931), 398WIENER, Hermann Ludwig Gustav

(1857–1939), 112, 148, 284–85, 456WIENER, Ludwig Christian (1826–1896), 148WIENER, Norbert (1894–1964), 148Williams College, 369WILSON, Edwin Bidwell (1879–1964),

148, 342–43, 422, 447WIRSZUP, Izaak (1915– ), 452WIRTINGER, Wilhelm (1865–1945), 430, 444WITTELSBACH, Luitpold Karl Joseph Wilhelm

Ludwig of, Prinzregent von Bayern(1821–1912), 431

WOJTASIEWICZ, Olgierd Adrian (1916–), 428WOJTOWICZ, W . (????–19??), 415–16women (see female mathematicians)WOODGER, Joseph Henry (1894–1981), 452WOODWARD, Beverly Anne

(1934–), 417, 422, 447World War I

effects, ix, 74, 87, 90, 98, 100,106, 120, 143, 347, 366, 369

service in, 71, 90, 114–16, 119, 347, 351World War II

effects, viii, ix, 92, 109–10, 114,143, 352, 360, 366, 368–70

effects on Jews, 72, 80–81, 91–92,110, 113–14, 116, 352

service in, 116

YOUNG, Grace Chisholm (1868–1944), 398YOUNG, Jacob William Albert

(1865–1948), 454, 456YOUNG, John Wesley (1879–1932), 116, 135,

137, 142–43, 145, 449, 454, 456

n. See integer modulo nZACHARIAS, Max (1873–19??), 404, 407, 416ZAPPULLA, Carmela (19??–), 129, 403ZERMELO, Ernst Friedrich Ferdinand (1871–

1953), 300, 304, 307, 323, 409, 456

494 Index

zero, 258–59, 310ZEUTHEN, Frederik Ludvig Bang

(1888–1959), 121ZEUTHEN, Hieronymous Georg (1839–1920),

120–22, 398, 421, 424, 431, enumerative geometry article,

46, 87, 374, 393, 443 on Staudt’s theorem, 425, 456

ZEUTHEN, Julie Henriette Jespersen(1846–1876), 121

ZEUTHEN, Louise Marie Christine Jespersen(1842–1886), 121

ZEUTHEN, Sophie Christine FrederickeLawaetz (1860–1936), 121

Zurich, ETH, 460