Bibliography

This bibliography includes all, and only, works referred to in the present volume. It includes the publications that Pieri cited in the papers whose translations are found in chapter 3 and section 4.2. Its entries for Pieri’s own works duplicate those in chapter 6. The present book mentions more than one author named Pieri; citations that include this surname only are references to Mario Pieri. Alphabetization of the entries ignored diacritical marks, punctuation, and spaces within 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 dates are 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. An article in such a journal is identified by its year of publication; the range is given in parentheses following the volume number.

Book citations often include an International Standard Book Number (ISBN) and a Library of Congress (LC) catalog number. Caution! A book may have similar editions with different ISBNs. Moreover, the LC number is not standard, unless the book was cataloged upon publication and its number included on the copyright page. From one library to the next, the LC number may vary 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: Cornell are provided to help readers locate publications in academic Internet repositories. The word after the colon (:) identifies an Internet http:// address in the following list:

Cornell...... cdl.library.cornell.edu EBSCO ...... ejournals.ebsco.com Göttingen ...... gdz.sub.uni-goettingen.de Gutenberg ...... www.gutenberg.org JSTOR ...... www.jstor.org Michigan . . . . . www.hti.umich.edu/u/umhistmath NUMDAM ...... archive.numdam.org Paris ...... gallica.bnf.fr Poland ...... matwbn.icm.edu.pl ScienceDirect .... www.sciencedirect.com

Many citations contain brief annotations that explain their origin, import, or relationship to other items. An annotation may include a code in the format JFM: 12.0123.01 Surname

1 Most Internet sources are free, but some—for example JSTOR —are free only through libraries with paid subscriptions. 402 Bibliography to indicate, for example, that the work cited was reviewed in the Jahrbuch über die Fortschritte der Mathematik, volume 12, page 0123, item 01, by the named reviewer. This journal may have LC: 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 of Symbolic Logic (Church 1936). That bibliography is available on the Internet via JSTOR; the journal 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 Internet at http://www.ams.org/mathscinet.2

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

As much as possible, journal and book titles appear in the present bibliography as they did on the published title pages. This practice, advocated by current documentation authorities,3 is consistent with major catalogs now accessible and searchable electronically. But readers may encounter 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 original editor; 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 the language—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 in 1938. [Cited in 1.3 and item Pincherle 1954.] Amato, Vincenzo. 1946. Sebastiano Catania. Annuario dell’Università di Catania per l’anno accademico 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 and item Drucker 1991.] Aprile, Giorgio. 1946. Giuseppe Marletta. Bollettino della Unione Matematica Italiana (series 3) 8: 55–56. LC: QA1.U6. [Cited in 1.3.]

2 JFM reviewers are identified because many played other roles in the life and legacy of Mario Pieri. Some indicated JFM and MR reviews refer to earlier editions of the work cited. Some early MR reviews were numbered 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. 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 in 1912. [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’archivio della corrispondenza dei matematici italiani. Milan: ELEUSI, Sezione P.RI.ST.EM. The series title is “Booklets of the Project on Historical and Methodological Research (Progetto Ricerche Storiche e Metodologiche) for archiving the correspondence of Italian mathematicians.” In the present 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. In Drucker 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. Lettera matematica P.RI.ST.EM. 26: 26–51. The acronym stands for Progetto Ricerche Storiche e Metodologiche. [Cited in 2.2.] Avellone, Maurizio, Aldo Brigaglia, and Carmela Zappulla. 2002. The foundations of projective in 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 in 3.10.] Baldwin, James M., editor. 1901–1905. Dictionary of Philosophy and Psychology; Including Many of the Principal Conceptions of Ethics, Logic, Aesthetics, Philosophy of Religion, Mental Pathology, Anthropology, Biology, Neurology, Physiology, Economics, Political and Social Philosophy, 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. [Cited in 1.3 and item Möbius 1885–1887.] Barbagli, Marzio. [1974] 1982. Educating for Unemployment: Politics, Labor Markets, and the School System—Italy, 1859–1973. Translated by Robert H. Ross. New York: Columbia University Press. LC: LA791.B3213. ISBN: 0-231-05284-7. First published in Italian as Disoccupazione 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: Proceedings of the 1964 International Conference. Amsterdam: North-Holland Publishing Company. LC: BC135.I52. [Cited in item Schwabhäuser 1965.] 404 Bibliography

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 deutsche Biographie. Twenty-two volumes to date. Berlin: Duncker & Humblot. LC: DD85.N48. [Cited in 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 der neuhumanistischen 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 sulle curve 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 alla Cattedra 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). [Cited in 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 : 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 dei Lincei. 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 pura ed applicata 3: 336–339. LC: QA1.A62. Reprinted in Betti 1903–1913, volume 2, 11–15. [Cited in 6.1 and items Betti 1903–1913 and Pieri 1886a.] ———. 1871. Sopra gli spazi di un numero qualunque di dimensioni. Annali di matematica pura ed 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–, volume 4, 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 item Carathéodory 1937.] Bibliography 405

Blumberg, Albert Emanuel, and Herbert Feigl. 1931. Logical positivism. Journal of Philosophy 28: 281–296. LC: B1.J6. Internet: JSTOR. This paper introduced the name logical positivism. [Cited in 2.2.] Blumenthal, Otto. [1935] 1970. Lebensgeschichte. In Hilbert [1932–1935] 1970, volume 3, 388–429. [Cited in 1.2 and item Hilbert [1932–1935] 1970.] Bodewig, Ewald. 1930. Eduard Study in memoriam. Giornale di matematiche 68: 233–235. LC: QA1.G5. [Cited in 1.3.] Boi, Luciano. 1990. The influence of the Erlangen program on Italian geometry 1880–1890: N-dimensional geometry in the works of D’Ovidio, Veronese, Segre and Fano. Archives inter- nationales d’histoire des sciences 40: 30–75. LC: Q1.A1.A7. MR: 1137297 (93d:01029). [Cited in 2.3.] Bologna. Istituto di Istruzione Superiore Crescenzi Pacinotti. 1876–1890. Student records for academic years 1876–1877 through 1879–1880. When Mario Pieri attended this school it was called the Reale Istituto Tecnico di Bologna. [Cited in 1.1 and permissions.] Bologna, Provincia di. Istituto Tecnico “Pier Crescenzi.” 1995. Una scuola nel tempo: Documenti e 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.] Bologna, University of. [no date]. Student records for Mario Pieri. [Cited in 1.1, permissions, and items Pieri 1880, 1881.] 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 Studi di 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 la Didactica 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,” starts on 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: 44.0015.03 Salkowski. The title page lists the authors as Wolfgang and Johann Bolyai. [Cited in item Bolyai [1851] 1913.] Bolzano, Bernard. [1817] 1996. Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation. Translated by Steven Russ. In Ewald 1996, volume 1, section 6.C, 225–248. The original German edition was published in Prague by Gottlieb Haase. [Cited in 4.1 and item Ewald 1996.] ———. [1851] 1950. Paradoxes of the Infinite. Edited by Frantisek PĢihonsky. Translated, with a 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 in Leipzig by Reclam, was reprinted in Berlin by Mayer & Müller in 1889. [Cited in 4.1 and item Ewald 1996.] Bompiani, Enrico. 1950. Commemorazione del Socio Luigi Berzolari. Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali: Rendiconti (series 8) 9(2): 396–410. LC: AS222.A24. [Cited in 1.3.] Bonfante, Giuliano. 1969. Silvio Pieri. In S. Pieri 1969, v–x. [Cited in 1.1 and 1.3.] Boolos, George. 1995. Frege’s theorem and the Peano postulates. Bulletin of Symbolic Logic 1: 317–326. LC: QA9.A1.B89. Internet: JSTOR. MR: 1349682 (97g:03007). [Cited in 4.1 and 4.3.] Borga, Marco. 1992. Sulla consistenza delle teorie matematiche: Hilbert e la scuola di Peano a confronto. Epistemologia 15: 41–54. LC: Q174.E64. [Cited in 1.2.] 406 Bibliography

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Included in Peano 2002 as file 1891g.pdf. Reissued in Roero 2003. [Cited in 4.3 and items Peano 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, and items Peano 2002, Roero 2003, and C. Segre 1891, [1891] 1903–1904.] ———. 1891c. Risposta. Rivista di matematica 1: 156–159. LC: QA1.R5. Included in Peano 2002 as file 1891k.pdf. Reissued in Roero 2003. Response to C. Segre 1891. [Cited in 1.3, 5.1 and 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 parts are subtitled “Nota 1” and “Nota 2.” [Cited in 1.2, 4.2, 4.3, and items Peano 1957–1959, 2002 and Roero 2003.] ———. [1891] 1973. The principles of mathematical logic. In Peano 1973, 153–161. Translation of “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 Roero 2003.] ———. 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 items Peano 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 translated in Niddich 1963. Included in Peano 2002 as file 1895 5.PDF. Reissued in Roero 2003. [Cited in 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 items Peano 1957–1959, 2002, and Roero 2003.] ———. [1900] 1901. Les définitions mathématiques. In International Congress of Philosophy 1900–1903, 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 of Philosophy 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 delle Scienze 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 de mathematica 8: 74–83. LC: QA1.R5. JFM: 34.0078.03 Vivanti. Included in Peano 2002 as file 1903d.pdf. Reissued in Roero 2003. This specification starts in Scholastic Latin; as it proposes syntax simplifications, it implements them, so that it ends in Uninflected Latin. [Cited in 5.1 and items Peano 2002 and Roero 2003.] Bibliography 435

———. 1905. Sur les principes de la géométrie selon M. Pieri. Bulletin de la Société Physico- Mathématique de Kasan: ’(*\D ."-**4( " "8(* $& $&- *"&(":  ("’: , &(**’ (series 2)14: 1–4. LC: QA1.K38. Included in Peano 2002 as file 1904b.pdf. [Cited in 1.1 and item Peano 2002.] ———. 1906. Super theorema de Cantor-Bernstein. Rendiconti del Circolo Matematico di Palermo 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 Peano 2002 as file 1906b. Reissued in Roero 2003. Written in Uninflected Latin. [Cited in 1.2 and items Peano 1957–1959, 2002 and Roero 2003.] ———. 1912. Contra gli esame. Torino nuova (17 August): 2. Newspaper article. Included in Peano 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 Peano 2002 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 de interlinguistas. 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 item Peano 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 Peano 1957–1959, volume 3, 389–396, and included in Peano 2002 as file 1915j.pdf. [Cited in the chapter 2 introduction and items Peano 1957–1959, 1973, 2002.] ———. [1916] 2002. Pubblicazioni di Giuseppe Peano. Unpublished memorandum. Included in 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 Peano 1973, 235–246. Translation of “Le definizioni in matematica,” Periodico di matematica per l’insegnamento secondario (series 4)1: 175–189, LC: QA1.P5, JSL: 7152.1, which was reprinted in Peano 1957–1959, volume 2, 423–438, and included in Peano 2002 as file 1921d.pdf. [Cited in 4.1 and items Peano 1957–1959, 1973, 2002.] ———. 1926. Sinense. Academia pro Interlingua 6: 129–130. Included in Peano 2002 as file 1926i.pdf. Reissued in Roero 2003. Written in Uninflected Latin. [Cited in 5.3 and items Peano 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 biographical sketch 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 1884 and 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: Dipartimento di 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.] 436 Bibliography

Peano, Giuseppe, and Corrado Segre. 1905. Relazione sulla memoria del prof. Mario Pieri: “Nuovi principî di Geometria projettiva complessa.” Atti della Reale Accademia delle Scienze di Torino 40(1904–1905): 378–379. LC: AS222.A29. JFM: 36.0613.01 Loria. Review of Pieri 1905c. [Cited in 1.2, 6.4, and item Pieri 1905c.] Peano, Giuseppe, et al. 1895–1908. Formulaire mathématique. Five volumes, the second in three parts and the fifth in two. Turin: Bocca Frères, Ch. Clausen; Paris: Carré et Naud (volume 3). LC: QA41.F6. JFM: 28.0063.03, 30.0073.02, 32.0069.02, 33.0073.04, 36.0087.03, 39.0084.01 Vivanti. JSL: 7117, 21,25,30,35,40, 45. Volume five is titled Formulario mathematico. The material 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 and four are in files 1901b.pdf and 1903f.pdf. The 1906 and 1908 editions of volume five are in 1906g.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 the title 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. American Journal of Mathematics 7: 180–202. LC: QA1.A5. Internet: JSTOR. JFM: 17.0044.02 Michaelis. 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 and Göteborg 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. )# +5!#  ' ! ! - “' + # +'E ! Möbius (Axiomatischer Aufbau der zweidimensionalen Möbiusschen Geometrie). " 6 :  (".(D , &(**:, II. ."-**4( .,**:,    1 (**  .): Annuaire de l’Université de Sofia, II. Faculté Physico-Mathématique, Livre 1 (mathématique et physique) 36(1939–1940): 219–325. Bulgarian, with German abstract. JFM: 66.0708.03 Petkantschin. [Cited in 2.4.] Phillips, Esther R., editor. 1987. Studies in the History of Mathematics. Studies in mathematics, 26. Washington: Mathematical Association of America. LC: QA21.S8851. MR: 0913096 (89b: 01002). [Cited in item G. H. Moore 1987.] Pickert, Günter. 1982. Moritz Pasch. In Gundel, Moraw, and Press 1982, 704–713. [Cited in 1.3, permissions, and item Gundel, Moraw, and Press 1982.] Pieri, Ferruccio. 1901. Nozze Briganti–Luporini. Lucca: Tipografia Giusti. Poetry by Mario Pieri’s bother. [Cited in 1.1.] ———. 1931. Versioncelle da Fedro. Lucca: Scuola Tipografia Artigianelli. Poetry by Mario Pieri’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 present book. [Cited in 1.1 and 6.7.] ———. 1880. Letter to the rector of the University of Bologna, 30 October. In Bologna [no date]. Annotated in section 6.6. [Cited in 1.1, 6.6, and item Bologna [no date].] Bibliography 437

———. 1881. Letter to the rector of the University of Bologna, In Bologna [no date]. Annotated in section 6.6. [Cited in 1.1, 6.6, and item Bologna [no date].] ———. 1881–1882. Integrali delle funzioni di due variabili reali. Undated handwritten lecture notes, probably from the calculus course taught by Ulisse Dini at Pisa that academic year. In possession of Marco Campetti. [Cited in 1.1 and 6.7.] ———. 1884a. Questions for oral defense of the laureate. One signed handwritten sheet preserved in Pieri 1884c. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and item Pieri 1884c.] ———. 1884b. Studi di geometria differenziale. Dissertation 6144A, library of the . Thesis completed in September 1884, evidently under the supervision of Luigi Bianchi for 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. Dissertation 6144, 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 Pieri 1884a. The title page and first paragraph are reproduced on page 14 of the present book. [Cited in 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 1860 and 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 Accademia dei 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 dei Lincei: 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 Conversazioni Scientifiche di Genova 1–15. JFM: 19.0820.01 Loria. Completed in 1884. [Cited in 1.2 and 6.1.] ———. 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. Preceded by 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 in 1.1, 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: 21.0663.01 Kötter. BSM2: 21(1897)196. Presented to the Academy by Enrico D’Ovidio. The opening is reproduced 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 Accademia delle 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. The present authors’ copies of these letters bear the number B69. Annotated in section 6.6. [Cited in 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 spazio a 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 di Torino. 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: 24.0560.01 Loria. Completed in 1891. [Cited in 1.2 and 6.2.] ———. 1892b. Sopra le linee uniformemente illuminate di una superficie qualunque. Atti della Reale 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. [Cited in 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 lineare speciale. 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 Scienze di 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, and items 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 del Circolo 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: 45.0875.05 Loria. First published in 1893 as a pamphlet, Lucca: Tipografia Giusta, JFM: 25.1277.02 Ostrowski. 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 journal was published in Zaragoza, Spain. [Cited in 1.2, 6.1, and 6.2.] Bibliography 439

———. 1894b. Review of Thomae 1894. Rivista di matematica 4: 36–39. LC: QA1.R5. Reissued in Roero 2003. Annotated in section 6.7. [Cited in 5.1, 6.7, and items Thomae 1894 and Roero 2003.] ———. 1894c. Sul problema degli spazi secanti: Nota 2a. Reale Istituto Lombardo di Scienze e 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, and items Pieri 1893d, 1895b.] ———. 1894d. Trasformazione di ogni curva algebrica in altra priva di punti multipli. Rivista di 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 Accademia delle 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 in Pieri 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 Scienze e 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 items Pieri 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. Completed in 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: 27.0420.03 Kötter. 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: 27.0420.03 Kötter. 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. Rivista di matematica 6: 9–16. LC: QA1.R5. JFM: 27.0431.01 Kötter. JSL: 924. Reprinted in Pieri 1980, 83–90. Reissued in Roero 2003. [Cited in 1.2, 2.3, 4.1, 5.1, 6.4, and items Pieri 1980 and Roero 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 1.2 and 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 Accademia delle 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 Pieri 1980, 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. Rendiconti del 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 and permissions.] ———. 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 Istituto Lombardo 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 1898 and 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 del punto 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 Academy by 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, the chapter 3 introduction, 3.3, 3.10, 4.4, 5.1–5.3, 6.4, 6.6, and items D’Ovidio and Segre 1899, Loria 1899, and Pieri 1980.] ———. 1900b. Letter to the rector of the University of Catania, 31 January. In Catania [no date], fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and item Catania [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 Louis Couturat. 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 items Boutroux 1900, Couturat [1900] 1899, International Congress of Philosophy 1900–1903, Lovett 1900–1901, and Pieri 1980.] ———. 1901a. Letter to the rector of the University of Catania, 30 September. In Catania [no date], fascicule Mario Pieri. Translated in section 6.6. [Cited in 1.1, 6.6, permissions, and item Catania [no date].] ———. 1901b. Sui principî che reggono la geometria delle rette. Atti della Reale Accademia delle Scienze 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, and item Pieri 1980.] Bibliography 441

———. [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 item Catania [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, McMaster University, Hamilton, Ontario. Listed in Feinberg et al. 1967, section VI.1, 178. Translated in 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 and item 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, and item 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 item Catania [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 Accademia delle 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 Segre 1905. Reprinted in Pieri 1980, 309–356. [Cited in 1.2, 1.3, 2.3, 5.1, 5.3, 6.4, and items Peano and 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. In Catania [no date], fascicule Mario Pieri. Annotated in section 6.6. [Cited in 6.6 and item Catania [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 Mario Pieri. 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].] 442 Bibliography

———. 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 Pieri 1980, 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 della Accademia Gioenia di Scienze Naturali in Catania 87: 14–22. LC: QH7.A33. JFM: 37.0198.05 Vivanti. Reprinted in Pieri 1980, 367–376. Completed in 1905. [Cited in the chapter 4 introduction, 4.1, 4.4, 5.1, 6.5, and item Pieri 1980.] ———. 1906f. Sulla definizione Staudtiana dell’omografia tra forme semplici reali. Periodico di matematica per l’insegnamento secondario (series 3) 3: 1–5. LC: QA1.P45. JFM: 36.0581.03 Lampe. Reprinted in Pieri 1980, 361–366. Completed in 1905. [Cited in 2.3, 6.4, and item Pieri 1980.] ———. 1906g. Sur la compatibilité des axiomes de l’arithmétique. Revue de métaphysique et de morale, 13: 196–207. LC: B2.R37. JFM: 37.0062.03 Fehr. JSL: 927. Also reviewed in Brown 1906. 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, and 6 November. In Catania [no date], fascicule Mario Pieri. Annotated in section 6.6. [Cited in 1.1, 6.6, and Catania [no date].] ———. 1907a. Sopra gli assiomi aritmetici. Bollettino delle sedute della Accademia Gioenia di Scienze 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 translation of this paper; its original first page is reproduced on page 309. [Cited in the preface, chapter 1 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. In Catania [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.” Memorie di 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 by Corrado Segre. Reprinted in Pieri 1980, 455–560. Chapter 3 of the present book contains an English translation of this memoir; its original first page is reproduced on page 159. Pieri 1914 is a Polish translation. [Cited in the preface, chapter 1 introduction, 1.2, 1.3, 2.5, chapter 3 introduction, 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. In Catania [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: Litògrafo Anghinetti & Giaroli. Approximately 900 pages. Lithographed handwritten course notes from 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. Translated in section 6.6. [Cited in 1.1, 6.6, and permissions.] Bibliography 443

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———. 1905. Abzählende Methoden. In Meyer and Mohrmann 1907–1934, volume 3, part 2, half 1, 257–312. Article III C, 3. JFM: 37.0580.05 Lampe. Pieri [1915] 1991 is an Italian translation and revision. [Cited in 1.3, 6.7, and items Meyer and Mohrmann 1907–1934 and Pieri [1915] 1991.] ———. 1914. Lehrbuch der abzählenden Methoden der Geometrie. Leipzig: Druck und Verlag von B. G. Teubner. LC: QA607.Z4. Internet: Michigan. JFM: 45.0799.01 Ostrowski. [Cited in 1.3.] Permissions

The authors, confident that they speak for the mathematical community, are greatly indebted to 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 Vittorio Campetti for permission to publish them. These include translations of Pieri’s letters 1911b and 1912a in section 6.6, and the following photographs in the indicated sections: Mario Pieri ...... Frontispiece Telescopic projections Pieri & Campetti families ...... 1.1 Silvio Pieri Virginia & Paolo Anastasio Military academy personnel Angiolina Pieri Mario Pieri Pieri’s tomb Gemma Campetti & sons Pieri’s portrait of Klein ...... 1.2 Mario Pieri ...... 2.3 The authors acknowledge gratefully the cooperation of several archival institutions that provided personal access to their resources and permission to publish images or translations of various items in the indicated sections of the present book: Istituto di Istruzione Superiore Pieri’s student records: .... 1.1 Crescenzi Pacinotti, in Bologna, Italy Bologna 1876–1890 Archivio Storico dell’Università degli Pieri’s student records: .... 1.1 Studi di Bologna, in Bologna, Italy Bologna [no date] Photo: Salvatore Pincherle .... 1.1 Biblioteca dell’Università di Pisa, Doctoral thesis: Pieri 1884c .... 1.1 in Pisa, Italy Questions for oral defense of the .... 6.6 laureate: Pieri 1884a Archivio Storico della Scuola Pieri’s student records: .... 1.1 Normale Superiore, in Pisa, Italy Pisa 1884

Archivio Storico dell’Università degli Announcement of Pieri’s promo- .... 1.1 Studi di Catania, in Catania, Italy tion: from Catania 1903b Letters: Pieri 1899a, 1900b, .... 6.6 1901a, 1902a, 1903b, 1904b, 1907b, 1908b Review: Pieri, Lauricella, and .... 6.7 Catania 1903 Archivio Centrale dello Stato, Documents concerning Pieri .... 1.1 in Rome, Italy Bertrand Russell Archives, Photo: Bertrand Russell .... 5.1 at McMaster University, Letter: Pieri 1903a .... 6.6 in Hamilton, Ontario, Canada 460 Permissions

Staats- und Universitätsbibliothek, Letters: Pieri 1897e .... 6.6 in Göttingen, Germany Manuscript Division, Library Letter: Pieri 1905b .... 6.6 of Congress, in Washington DC, United States Archivio Storico dell’Università degli Letter: Pieri 1911a .... 6.6 Studi di Parma, in Parma, Italy Several individuals, publishers, and other organizations graciously extended permission for translating or reprinting images or text from their Internet websites or earlier publications: Mother House of the Salesians of Photo: exterior of the former .... 1.1 Don Bosco, in Turin, Italy military academy in Turin Archivio Storico dell’Accademia delle Photos: Enrico D’Ovidio, Pieri, .... 1.1 Scienze di Torino, in Turin, Italy and Photos: Corrado Segre and .... 1.2 Cesare Burali-Forti Photo: Gino Fano .... 2.3 Photo: Gino Loria .... 2.5 Photo: Giuseppe Peano .... 5.1 Queen’s University, in Kingston, Photo: Federigo Enriques, from .... 1.1 Ontario, Canada Brigaglia and Ciliberto 1995 Pietro Nastasi, in Palermo, Italy Photo: Vito Volterra from .... 1.1 Guerraggio and Nastasi 2005 Section P.RI.ST.EM (Progetto Translation: Francesco .... 1.1 Ricerche Storiche e Metodologiche) of Chizzoni’s 1900 letter to Pieri, the center ELEUSI (L’Elaborazione from Arrighi 1997 Logica e l’Utilizzazione Sistematica dell’Informazione) at the Bocconi University, in Milan, Italy Emilia Levi Resta, in Argentina Photo: Beppo Levi, from .... 1.1 Levi 1999 Historische Kommission für Hessen, Photo: Moritz Pasch, from .... 2.3 in Marburg, Germany Pickert 1982 Archivio dell’Istituto Veneto Photo: Giuseppe Veronese .... 2.3 di Scienze, Lettere ed Arti, in Venice, Italy Image Archive ETH-Bibliothek, Photo: B. L. van der Waerden .... 2.4 in Zürich, Switzerland Springer-Verlag, in Heidelberg, Photo: David Hilbert, . . . 3.10 Germany from Reid 1970 Photos: Alfred Tarski and .... 5.2 Adolf Lindenbaum, from Givant 1991 Bollati Boringhieri Editore, Photo: Giuseppe Peano, .... 4.1 in Turin, Italy from Kennedy 1983 Permissions 461

The authors also included images from the Internet websites of three other organizations: Comitato Guglielmo Marconi Photo: Augusto Righi .... 1.1 International, in Bologna, Italy Archivio Fotografico dell’Ateneo Photos: Luigi Bianchi, Ulisse .... 1.1 dell’Università di Pisa, in Pisa, Italy Dini, and Eugenio Bertini Princeton University, in Princeton, Photo: Oswald Veblen ... 3.10 New Jersey, United States Index

This index lists both persons and subjects. For persons, dates are supplied if possible. Section 1.3 includes biographical sketches of many of them; those names and the page numbers of the sketches are italicized. Page numbers of illustrations are boldfaced. Most entries under “symbol” include examples of usage. For persons surnamed Pieri, the relationship to Mario Pieri is indicated. Italian names with particles—for example De Paolis—are listed with the particle first. Alphabetization ignores spaces, punctuation, and diacritical marks within names. Section 1.3, chapter 6, and the bibliography contain many further cross-references.

1848 revolutions, 66, 75, 82–83, 85, 101 ANASTASIO, Angiolina. See Angiolina PIERI ANASTASIO, Paolo (18??–19??), 37 (see also abscissa, 257, 285 Virginia ANASTASIO; Angiolina PIERI) absolute geometry. See under geometry ANASTASIO, Virginia Pieri (1867–1929?), abstraction 5, 17, 20, 36, 37, 459 definition by (see under definition) marriage, 5, 27, 454 See also under symbol ANDREINI, Angelo (18??–19??), 10, 62 Academia pro Interlingua. See under ANELLIS, Irving H. (19??– ), 302, 402 journals; Giuseppe PEANO ANGELELLI, Ignacio (19??–), 417 ACKERMAN, Michael (19??–), 426 angle, 146, 149, 272 acknowledgments, x, 459 bisector, 220 ADAMSON, Iain T. (19??–), 421 concave, convex, or right, 204 addition. See sum of triangle, 205 affinity, circular. See under transformation See also under symbol Afghanistan, 5 Annuario scientifico ed industriale. AGAZZI, Evandro (1934–), 365, 402, 406 See under journals air power, 6, 119 ANTICHI, Petronilla (18??–????), 4 algebraic curve. See under curve anticongruence. See under transformation algebraic geometry. See under geometry anticongruent figures, 150 algebraic , 76, 84 antihomography. See under transformation algebraic surface. See under surface anti-inversion. See under transformation ALIGHIERI, Dante (1265–1321), 395 antinomies ALMAGIÀ, Virginia. See Virginia VOLTERRA Burali-Forti’s, 69 AMALDI, Ugo (1875–1957), 65, 98, 402 responses to, 302, 304 work with Enriques, 368, 415–16 See also Russell’s antinomy AMATO, Vincenzo (1881–1963), 72, 402 antipolar plane, 238 AMATURO, Enrico (1863–19??), 42 antirotation. See reflection: rotary American Mathematical Society, 43, 388 antitranslation. See reflection: glide AMODEO, Federico (1859–1946), 20, 62, 107 applied mathematics, 87, 365 correspondence, 383, 432, 438 APPOLONIUS of Perga (225–190 B.C.), 121,139 rivalry with Pieri, 22–23, 29, 32, 36 APRILE, Giorgio Antonio (1884–1943), analysis 40, 42, 63, 92, 398, 402, 410 arithmetization, 291–92 arc, 252 functional (see functional analysis) ARCHIMEDES (287–212 B.C.), 41, 121, 403, 421 Peano’s work, 301, 334, 337, 365 principle (see under postulate) use of set theory, 296, 298 archival institutions, ix, 40, 459 464 Index archive. See Mario PIERI: Nachlass BACCARINI, Pasquale (1858–1919), 32 area, 263 BACCELLI, Guido (1832–1916), ARISTOTLE (384–322 B.C.), 266, 305, 347 26–27, 29, 63–64, 456 arithmetic BACHI, Albina. See Albina LEVI. cardinal, 296–97 BACHMANN, Friedrich (1909–1982), complex, 125, 299 284, 403, 427 integer, modulo n, 320 BALBO, Italo (1896–1940), 6 ordinal, 297–98, 387 BALDWIN, James Mark (1861–1934), rational, 77, 289, 292, 294, 299, 334, 392 124, 343, 403 signed-integer, 77, 299, 334 BALTZER, Richard (1818–1887), 93, 403, 430 See also natural-number arithmetic; BALZAC, Honoré de (1799–1850), 97 real-number arithmetic BARBAGLI, Marzio (1938–), 7, 15–17, 403 ARMANO, Tiziana (19??–), vi, 446 BARBARIN, Paul (1855–1931), 41, 64, 403, 407 ARRIGHI, Gino (1906–2001), vi, ix, 6–7, 36, BARBERIS, Bruno (1953?–), 109, 110, 403 42–44, 63, 382, 388, 398–9, 403, 443, 460. BAR-HILLEL, Yehoshua (1915–1975), 403 Citations of individual letters in Arrighi BARIDON, Paolo (18??–19??), 408 1997 are indexed by their authors. BARKER, William A. (????–????), 120, 404 ARZELÀ, Cesare (1847–1912), BARSOCCHINI, Domenico (1777–1862), 10, 63, 98, 109, 398, 426 4, 96, 443 Bologna affair, 25–29 BARTORELLI, Antonio (18??–19??), 46, 64 ASBURGO. See HABSBURG barycentric calculus. See under calculus ascending. See progression BASSO, Giuseppe (1842–1895), ASCHIERI, Ferdinando (1844–1907), 20, 64, 332, 416 22–23, 25, 29 BASTIAN, Beppina. See Beppina PIERI. ÀSCOLI, Graziadio Isaia (1829–1907), 97 BATTAGLINI, Giuseppe (1826–1894), ASKEY, Richard Allen (1933–), 415 78–79, 398 ASPRAY, William (1952–), 116, 403 Giornale di (see journals: Giornale di associativity, 270, 294, 395 matematiche) AUGUST, Friedrich Wilhelm Oscar BAUER-MENGELBERG, Stefan (1927–1996), (18??–????), 374 417, 456 Ausdehnungslehre, 83–4, 334 BEAUFRONT, Louis de (1855–1935), 74, 404 AUSSANT-CARÀ, P. (18??–19??), 426 BECK, Hans (1876–1942), 381, 443, 454 AUSTIN, John Langshaw (1911–1960), 417 Begriffsschrift. See under FREGE , xvii–xix BEKEMEIER, Bernd (19??– ), 292, 404 automorphism, 358, 360 BELLAVITIS, Giusta (1803–1880), AVELLONE, Maurizio (19??–), 127, 129, 403 117, 139, 404 axiom, 54 (see also postulate) BELTRAMI, Eugenio (1835–1900), 65, 97 of choice, 315, 360 BEMAN, Wooster Woodruff (1850–1922), 413 of determinacy, 360 BENZ, Walter (1931–), 143 of foundation, 323 Berkeley. See University of California of infinity, 300, 307 Berlin, Imperial Library, 2 axiomatic method BERNAYS, Paul Isaac (1888–1977), 421–22 in general, v, vii, 87, 95, 291, 350, 370 BERNSTEIN, Felix (1878–1956) in Hilbert, 84, 277 See under theorem in Peano, 300–301, 334–35 BERTINI, Eugenio (1846–1933), in Pieri, 54, 124–27 30, 65, 78, 398, 404 See also hypothetical-deductive system as referee, 22–23, 26, 28–29, 31, 42, 404 axis BERTRAND, Joseph Louis François of rotation, 183 (1822–1900), 78 of symmetry, 176 Index 465

BERZOLARI, Luigi (1863–1949), BOGGIO, Tomasso (1877–1963), 31, 34, 398 22, 65–66, 107, 398, 405 BOGGIO-LERA, Enrico (1862–1956), 10, 67–68 Encyclopedia, 69, 71, 81, 95, 327, 404 BOHLMANN, Georg (1869–1928), 418 at Pavia, 31–32, 36, 94 BOI, Luciano (1957–), 129, 133–34, 405 at Turin, 20, 23, 26, 89, Bollati Boringhieri Editore, 460 BETH, Evert Willem (1909–1964), Bollettino di bibliografia e storia della 300, 304–305, 314–15, 319, 404 scienza. See under journals BETTAZZI, Rodolfo (1861–1941), Bologna, xvii, xviii, 7–8, 11 10, 12, 18, 20, 66, 337, 404 affair, 25–28 BETTI, Enrico (1823–1892), Royal Technical Institute, 7–8, 459 12, 13, 66, 406, 409, 437 See also Bologna, university of works by, 167, 374, 398–99, 404, 416 Bologna, university of, 10–11, 16, 459 betweenness, 146, 334–35, 354, 358, 360–62 faculty renaissance, 8, 63, 70, 75, 98 on arc, 252 Pieri’s chair quest, 25–28 Pieri’s definition, 152, 195, 272, 357 BOLONDI, Giorgio (19??–), 17, 405 undefined, 149, 274, 277, 288, 351, 353 BOLYAI, Farkas Wolfgang (1775–1856), See also under order 209, 405 BÉZOUT, Étienne (1739–1783), 53, 404 See also under parallel postulate See also under theorem BOLYAI, János (1802–1860), 125, 155, 405 BIAGINI, Roderigo (18??–19??), 398 BOLZANO, Bernard Placidus Johann Nepomuk BIANCHI, Francesco Saverio (1827–1908), 67 (1781–1848), 291–92, 300, 405, 450 BIANCHI, Luigi (1856–1928), BOMPIANI, Enrico (1889–1975), 66, 405 13, 34, 67, 398, 409, 461 BONACCORSO, G. (18??–19??), 33 Pieri’s teacher, v, 12, 50, 374–75, 437 BONAPARTE, Napoléon (1769–1821), bibliographic citations, ix, 4, 373, 401–03 xvii, 11, 18, 41 in chapter 3, 62, 157, 167, 416 BONFANTE, Giuliano (1904–1983), BIEBERBACH, Ludwig Georg Elias Moses 4, 5, 8, 20, 49, 97, 405, 444 (1886–1982), 412 BONOLA, Roberto (1874–1911), 327 BIERMANN, Kurt-Reinhold (1919–2002), BOOLE, George (1815–1864), 316, 334 77, 404 BOOLOS, George Stephen (1940–1996), bijection. See under function 299, 318, 405 binary relations. See under primitive notions BORBÓN, BORBONE. See BOURBON biographical sketches, x, 3, 4, 62 BORDIGA, Giovanni (1854–1933), 42 birational BORDONI, Antonio (1789–1860), 75 equivalence (see under surface) BOREL, Félix Edouard Justin Émile Borel See also under geometry; transformation (1871–1956). See under theorem bisector. See under angle BORGA, Marco (????– ), x, 60, 147, 156, 403, BLACK, Max (1909–1988), 404, 432 405–06 BLAIKLOCK, Mary (18??–19??), 446 Borga & Palladino 1992, 54, 58, BLAIR, David E. (1940–), 138–41, 404 153, 301, 316, 326, 335, 406 BLANC, Charles (19??–), 416 BORKOWSKI, Ludwik Stefan (1914–1993), 428 BLANCHET, Marie Parfait Alphonse BORODIN, Alexandr Ivanovich ("&"  , (1813–????), 426 )! '  !# 5, 19??–), 41, 406 BLASCHKE, Wilhelm Johann Eugen BORSUK, Karol (1905–1982), 361, 406 (1885–1962), 425 BOSCHI, Pietro (1833–1918), 8, 10, 68 BLUMBERG, Albert Emanuel (????–19??), BOSIO (cav. uff., 18??–19??), 48 127, 405 BOTTAZZINI, Umberto (1947–), 92, 131, 406 BLUMENTHAL, Otto (1876–1944), 60, 405 Enriques & Castelnuovo, 70, 72–73, 80 BOCCHERINI, Luigi (1743–1805), 4 Enriques & Castelnuovo, Bologna affair, BODEWIG, Ewald (1902–????), 112, 405 23, 25–28, 31–32 466 Index

BOTTAZZINI (continued) BURALI-FORTI (continued) Enriques & Castelnuovo, otherwise life, 12, 20, 69, 398, 429 involving Pieri, 8, 22, 36, 38, 42 logic, 301, 306–307, 316, 321, 326–28 boundary of triangle, 205 Peano school, 22, 337, 341–42 BOURBAKI, Nicolas (pseudonym), vector analysis, 46–47, 53, 378, 392, 428 115, 294, 406 See also under antinomies BOURBON, Carlo Lodovico of (1799–1883), BURAU, Werner (1906–1994), 84, 87, xvii, 41 105, 111, 112, 408, 448 Bourbon dynasty, xvii, 41 BURKHARDT, Heinrich (1861–1914), 430 BOURBON, Ferdinando II of (1810–1859), 41 BURSTALL, Rod M. (1934–), 313, 408 BOURBON, Maria Luisa of (1782–1824), BUSSEY, William Henry (1879–19??), 116 xvii, 41 BÜTZBERGER, Friedrich (18??–19??), 139, 408 BOUTROUX, Émile (1845–1921), 381, 406, 423, 440 CAEMMERER, Hanna von BRANCOLI, Giuseppe (18??–????), 5 See Hanna NEUMANN BRANCOLI, Teresa Pieri (1853–1942?), CAIO, 268 5–6, 390–391 CAJORI, Florian (1859–1930), 313, 315, 408 BRAUNMÜHL, Anton, Edler von (1853–1908), calculus 375, 438, 446 barycentric, 93 Brazil, 6, 17, 36, 44 geometric (see under Giuseppe PEANO) BREDOW, Ilse (????–????), 448 mathematics as uninterpreted, 59 BRESADOLA, Marco (1969–), 100, 406 of variations, 123 BREVEGLIERI, Bruno (1942–), 406 See also differential and integral calculus BRIGAGLIA, Aldo (1945–), and under SCHUBERT 53, 129, 403, 406, 460 CALAPSO, Renato (1901–1976), 92, 408 Brigaglia & Masotto 1982, 1, 42, 131, 406 CALDARERA, Gaetano (1865–19??), Brigaglia & Ciliberto 1995, 52, 80, 406 34, 40, 69–70 Brigaglia, Ciliberto, & Sernesi 2005, CALDARERA, Grazia Macrina (1880–19??), 74, 92, 106, 406 40, 70, 408, 410 BRILL, Alexander Wilhelm von California State University, x (1842–1935), 65 California, University of. See University BRIOSCHI, Francesco (1824–1897), of California 66, 75, 167, 416 CAMBRIA, Maria (1923–), 114 BROADBENT, Thomas Arthur Alan CAMIVI, Mario (18??–19??), 397, 442 (1903–1973), 103, 406 CAMPEDELLI, Luigi (1903–1978), BRODÉN, Torsten (1857–1931), 12, 74, 101, 408 68, 124, 153–154, 407, 436 CAMPETTI, Annarosa (1920–2004), 6 BROWDER, Felix Earl (1928–), 407, 424 CAMPETTI, Beatrice Giusfredi (1894–1995), 6 BROWN, Harold Chapman (1879–1943), Campetti family, 6, 36, 39, 44, 459 290, 382, 407, 442 CAMPETTI, Francesco (1969–), x, 5, 6, 17, 36, BRUNO, Giuseppe (1828–1893), 20, 22–23, 43, 97, 408, 412, 444, 454 68–69, 107, 332, 375, 398, 438, 446 CAMPETTI, Gaetano Enrico (1890–1980), BUCCA, Lorenzo (18??–19??), 32, 37 6, 39, 42, 44, 48, 390, 442 BUEK, Otto (1873–1966), 47, 398 CAMPETTI, Gemma Pieri (1863–1955?), BUHL, Adolphe (1878–1949), 64, 407 5–6, 36, 39, 44, 47, 390, 408, 411, 459 Bulletin des sciences mathématiques CAMPETTI, Ghigo. See Ottorino CAMPETTI See under journals CAMPETTI, Marco (1924–), x, 6, 47, 390–91, BURALI-FORTI, Cesare (1861–1931), 397, 436–37, 442–43, 459 vi, 55, 460 CAMPETTI, Ottorino (1888–1947), 6, 36, 39, geometry, 52, 58, 156 42–44, 46, 48, 63, 390–91, 443 Index 467

CAMPETTI, Pellegrino (1892–1974), CATANIA, Sebastiano (1853–1946), 40, 72, 82, 6, 36, 39, 43 395–96, 402, 410, 419, 441, 443, 459 CAMPETTI, Umberto (1863–1929), Catania, xviii, 1, 32–33, 41 5–6, 17, 36, 44 university, 16, 32–33, 35, 41, 88, 410, 459 CAMPETTI, Vittorio (1930–), x, 4, 6, 411, 459 categorical. See under theory cancer. See throat cancer CAUCHY, Augustin Louis (1789–1857), CANDIDO, Giacomo (1871–1941), 147, 408 167, 410 CANTOR, Georg Ferdinand Ludwig Philipp convergence criterion, 292 (1845–1918), 408–9, 413, 420 CAVINESS, Bob F. (19??–), 352 infinite sets, 69, 117, 296–98, 300, 305–6 CAYLEY, Arthur (1821–1895), 52, 125 logic, 299, 302, 306, 313 center real numbers, 254, 263, 289, 290, 292–94 of anti-inversion, 238 See also under natural-number arithmetic; of arc, 252 real-number arithmetic; theorem of circle, 173 CANTOR, Moritz Benedikt (1829–1920), of circular ordering, 253 281, 409 of homothety, 233 CAPELLINI, Giovanni (1833–1922), of semicircle, 252 8, 29, 70, 414, 454 of symmetry, 175 CARATHÉODORY, Constantin (1873–1950), centered figure, 160 140, 404, 409 central point. See midpoint CARDANI, Pietro (1858–1925), 100, 391, 409 CEPPI, Carlo Giulio (1829–1921), 332 cardinal number, 296, 315 CESÀRO, Ernesto (1859–1906), 20, 73, 432 See also under arithmetic chain CARDUCCI, Giosuè (1835–1907), 96–97 in complex projective line, 56, 134–35, 145 CARLO EMANUELE II. See Carlo in foundations of arithmetic, 314 Emanuele II of SAVOY CHASLES, Michel (1793–1880), CARLO LODOVICO. See Carlo 56, 72–73, 133, 411, 426 Lodovico of BOURBON on enumerative geometry, 52, 104, 121 CARRUCCIO, Ettore (1908–1980), 67, 92, 409 Chicago, University of, 87, 115 Cartesian geometry. See under geometry CHINI, Mineo (1866–1933), 20, 73, 337, 411 CASATI, Gabrio (1798–1873). See under laws CHIÒ, Felice (1813–1871), 82 CASORATI, Felice (1835–1890), 65–66, 91, 97 chirality, 150 CASSINA, Ugo (1897–1964), 71, 139, CHIZZONI, Francesco (1848–1904), 22, 32–34, 319, 409, 436, 450 40, 73, 384, 398, 459–60 on Peano, 337, 345–46, 365 choice. See under axiom on Pieri, 2–3, 22, 31, 34, 42, CHURCH, Alonzo (1903–1995), 116, 348, 284–85, 307, 326, 399 402, 411 CASTELLANO, Filiberto (1860–1919), CHWIAâKOWSKI, Zygmunt (????–????), 20, 71, 79, 334, 337, 410 348, 452 CASTELNUOVO, Enrico (1839–1915), 71 CHWISTEK, Leon (1884–1944), 451 CASTELNUOVO, Guido (1865–1952), CIACCIO, Francesco Paolo (1821–1901), 28 71–72, 119, 263, 410–11, 422 CIAMPINI, Maria Grazia (1942–), algebraic geometry, 52, 107–08, x, 4, 6, 411, 459 concerning Enriques, 25–29, 406, CIANI, Edgardo (1864–1942), concerning Pieri, 44, 49, 157, 12, 42, 74, 114, 408 160, 381, 398, 399, 442 rivalry with Pieri, 22–23, 29, 32, 36 concerning Turin, 20, 22–23, 36, 42, CIESIELSKI, Krzystof (1957– ), 350, 411 See also under BOTTAZZINI;ENRIQUES CIGNA, Gianfrancesco (1734–1790), 18 CATALANI, Alfredo (1854–1893), 4 CILIBERTO, Ciro (1950–). See BRIGAGLIA CIPOLLA, Michele (1880–1947), 40, 42, 94, 411 468 Index circle congruence (continued) in inversive geometry, 57, 138, 141 of figures, 146, 148, 150, 161, in Point and Sphere, 160, 173 249, 272, 278, 338, 348 See also under postulate of point pairs, 155, 216 circular. See under transformation SAS (see under postulate) class, 163–64, 249, 269 of segments, 149, 206, 274, 278, enumerable, 254 280–81, 338, 351, 353, 357, 394 CLAUSIUS, Rudolf Julius Emmanuel of triangles, 220 (1822–1888), 99 See also under equidistance; postulate; CLEBSCH, Rudolf Friedrich Alfred symbol; transformation (1833–1872), 86, 444 conic section, 52, 121, 376, 378, 393 COEN, Salvatore (19??–), x, 8, conjugation. See harmonic conjugation 10, 42, 90, 91, 411 conjunction. See under symbol coherent. See under fundamental sequence consequence, 267 COHN-VOSSON, Stefan (1902–1936), 426 conservation of number, 104, 112 coincidence formulas, 376, 396 consistency. See under natural-number COLAJANNI, Napoleone (1847–1921), 43, 398 arithmetic; projective geometry; theory collegium, 11, 16 contact point, 167, 200 collimation, 169 CONTE, Alberto (1942–), 81, 411, 427 collinearity, definition See also BOTTAZZINI using betweenness, 277, 361 continuity, 278, 289 using equidistance, 152, 154, 169, 280, 357 Dedekind, 137, 141, 146, 149, using motion, 146, 152, 272 155, 273, 276, 336, 355 collineation. See under transformation elementary, 355–56 colonialism. See under Italy Weierstrass, 285 commissione giudicatrice, 16 See also under postulate; transformation communication among mathematicians, continuous magnitude, 291–92 1, 87, 98, 107, 121, 154 CONTRO, Walter Salvator (1937–), commutativity, 143, 226, 270, 294, 298 68, 128, 153–54, 407, 411 compass, geometry of, 263 control-systems theory, vii, 52 complete quadrilateral, 131 convergence criterion. See under CAUCHY completeness convex of real numbers, 151, 293 angle (see under angle) See also under postulate; theory figure, 206 complex COOLIDGE, Julian Lowell (1873–1954), analysis, 125, 140, 142 133, 139–140, 142, 412 Hirstian, 378 coordinates number (see under arithmetic) Cartesian, 57, 276, 278, 282, 288, 354 of lines, 377–78 homogeneous, 60, 93, 128, 277 complexity of axiom systems, 156, 273–74, pentaspherical, 57 276–77, 283, 285, 287–88, 350, 353 tetracyclic, 57, 140 concave. See under angle coplanarity, 146, 154, 173, 277, 280 concorso, 16 CORCORAN, John (1937– ), 305, 347, 412, 452 cones, system of, 377 CORIE, Helen Cullura (1911–2005), x confocal varieties, 51 CORRY, Leo (1956– ), 351, 412 conformal. See under transformation correspondence, 269 congruence algebraic, 52 in algebraic geometry, 53, 376–78 principle, 73, 104, 121, 376 of angles, 149, 274, 278, 280–81, 338 COURANT, Richard (1888–1972), 426 Index 469

COUTURAT, Louis (1868–1914) DANIELE, Pietro Ermenegildo (1875–1949), 1900 Paris congress, 1, 36, 281, 96, 413 321, 341, 381, 423, 440 DANTE. See ALIGHIERI arithmetic, 307 DARBOUX, Gaston (1842–1917), correspondence, 388, 398, 428, 448 121, 137, 139–40, 413, 425 geometry, 58, 145, 327 DARMSTAEDTER, Ludwig (1846–1927), 2, 398 Leibniz, 281, 343, 427 D’ASBURGO. See HABSBURG life, ix, 74, 218, 404, 412, 415 DAUBEN, Joseph Warren (1944– ), logic, 304, 316 296, 406, 413 covariant, 200, 250 DAVINI, Marianna. See Marianna LUPORINI COWAN, Patricia M. See PEARSON DAVIS, Philip J. (1923–), x COXETER, Harold Scott MacDonald (1907– decidability 2003), x, 129, 131, 142–43, 145, 361, 412 of elementary geometry, 351–52 CRELLE, August Leopold (1780–1855) See also under theory See journals: Journal für die reine DEDEKIND, Caroline Marie Henriette und angewandte Mathematik Emperius (1799–1882), 76, 84 CREMONA, Antonio Luigi Gaudenzio Giuseppe DEDEKIND, Julie Marie Sophie (1830–1903), 30, 33, 75, 412–13, 420, 460 (1825–1914), 76 algebraic geometry, 12, 14–15 DEDEKIND, Julius Levin Ulrich Pieri’s chair quest, 27–29 (1795–1872), 76 See also under transformation DEDEKIND, Julius Wilhelm Richard CREMONA, Tranquillo (1837–1878), 75 (1831–1916), 76–77, 295, 306, Crescenzi–Pacinotti, Istituto, 8, 100 404, 409, 413, 415, 419 CRISPI, Francesco (1818–1901), 26 continuity and real numbers, 76, 263, CROCE, Benedetto (1866–1952), 90, 119, 366 273, 289, 293–94, 305 CROWE, Michael J. (19??–), 53, 413 finiteness definition, 290, 296, 300, 314 CULMANN, Karl (1821–1881), 99 logic, 299, 301, 305, 307, 315–16, 319 CURTO, Girolamo (1852–1917), 12 natural numbers, 76, 296, 300–302, 304, CURTZE, Maximilian (1837–1903), 412–13 308, 313–15, 317–20, 322, 324, 326 curvature, 51, 375 See also under continuity; cut; curve real-number arithmetic algebraic, vii, 51, 65, 75, 375–76, 378, 383 DEDEKIND, Karl Adolf (1829–1909), 76 of constant illumination, 51, 375 DEDEKIND, Richard space-filling, 334 See Julius Wilhelm Richard DEDEKIND cut, Dedekind, 76, 293 deduction. See consequence CUTRONE (Procuratore Generale definability, 331, 358, 360 comm., 18??–19??), 48 definiendum, definiens, 265 cyclic order definition, 265, 302, 341, 358 in Euclidean geometry, 252, 278, 285 by abstraction, 392 in inversive geometry, 141 equality by (see under equality) in projective geometry, 128, 130–131 explicit, 265 See also under order hypothesis of, 265 implicit, 127, 265, 267–68, 307–8 DA COLLINA, Mareno (1960–), x nominal, 265, 268, 306–7, 392 DA COLLINA, Mario (1921–), x real, 265 DAGNINO (colonello cav., 18??–19??), 48 recursive, 298–300, 314–15, 319, 324 DAL POGGETTO, Maria. See Maria PIERI DE FRANCHIS, Michele (1875–1946), D’AMICO, Francesco (1880–19??), 40, 75, 36, 42, 44, 77, 89–90, 450 396–97, 410, 413 DEHN, Max (1878–1952), 96, 151, 380, D’ANGIÒ, Giuseppe Gioeni. See GIOENI 414–15, 421, 434, 441, 454, 456 470 Index

DELOGU, Pietro (1857–1932), doctorate, Italian, 7, 11 38, 387, 414, 441 See also under Mario PIERI DEL RE, Alfonso (1859–1921), 36, 42, 77, 430 DONATI, Luigi (1846–1932), 10, 98, 417 rivalry with Pieri, 22, 23, 25, 27–29, 31 D’OVIDIO, Enrico (1843–1933) 20, 21, 49, 78– Denmark, mathematics in, 121–22 79, 97, 332–33, 405, 414, 418, 450, 460 DENTON, William Wells (1882–1961), 456 referee for chair competitions, 22–23, 26 DE PAOLIS, Riccardo (1854–1892), reviewer, 57–58, 131, 145–46, 148, 151, 12, 20, 65, 78, 129, 147, 375–76, 375, 377, 380–81, 437–38, 440 395, 398, 403, 414, 437, 449 D’OVIDIO, Francesco (1849–1926), 79, 97 1892 correspondences paper DRUCKER, Thomas (19??–), 415 See under Mario PIERI DUFLOT, Jeanne (19??–), 406 depression. See economic depression DUGAC, Pierre (1926–2000), 77, 314, 415 DE SANTIS, Maria Gabriela (19??–), 11, 16, 414 DUPORCQ, Ernest (1873–1903), 422 DESARGUES, Girard (1591–1661) DUREN, Peter (1935–), 415 See under theorem descending. See progression École Normale Supérieure, 11 descriptions, theory of, 341 economic depression, ix, 366, 368–70 descriptive geometry. See under geometry economic unrest, 17, 44 design courses, 11, 34, 46, 332 edge of angle or triangle, 163, 204–5 DE STEFANI, Carlo (1851–1924), 70, 414 EDWARDS, Harold Mortimer, Junior determinacy. See under axiom (1936–), 415 diagrams. See under geometry EHRENFEUCHT, Andrzej (1932–), 113 diametrically opposite points, 194 EHRENFEUCHT, Krystina Tarski (1938–), DI BORBONE. See BOURBON 113, 352 DICKSTEIN, Samuel (1851–1939), 124, 398 EISELE, Carolyn (1902–2000), 74, 80, 415 DIEUDONNÉ, Jean Alexandre Eugène elementary function. See under function (1906–1992), 316, 414–415 elementary geometry DI FARNESE. See FARNESE Brodén, 68, 153–154, 407, 412, 436 differential and integral calculus, 333, 346 Hilbert, 58–59, 148–51, foundation for, 76, 291–93, 300, 335 274–77, 288, 350, 421 multidimensional, 53, 397 Pieri, vii, 54, 58, 153–54, 156–270, See also under Giuseppe PEANO 278–87, 349, 355–58, 381, 442 differential equations, 301, 334 Tarski, viii, 58, 288, 350–56, 368, 394, 452 differential geometry. See under geometry elicomozione. See transformation: screw DI LEONARDO, Maria Vittoria (????– ), elliptic geometry. See under geometry 323, 325, 328, 414 elliptic polar system. See polar system dimension, 186, 277, 283, 287–88, EMCH, Arnold (1871–1959), 408, 415 334, 358, 360, 362, 394 Emilia–Romagna, xvii, xix, 7, 11, 16, 44 DINI, Ulisse (1845–1918), 12, 13, empty set, 316 67, 397–98, 450, 461 Encyclopedia of Elementary Mathematics direct. See under isometry; motion (Italian), 65, 327–29, 404 DIRICHLET, Johann Peter Gustav Lejeune See also under BERZOLARI (1805–1859), 76 Encyclopédie des mathématiques pures et DI SAVOIA , DI SAVO JA. See SAVOY appliquées, 46, 87, 430 DI SIENO, Simonetta (19??–), 414, 431 Encyklopädie der mathematischen disjunction. See under symbol Wissenschaften, 2, 46, 87, 325, 429 dissection, 348, 394 end distance, 155, 160, 258, 263, 284–85 of arc or semicircle, 252 distributivity, 294 of segment, 195 Index 471

ENGEL, Friedrich (1861–1941), 96, 105–6, external point 112, 415, 419, 421–22, 427, 447 to circle, 195 ENRIQUES, Federigo (1871–1946), 24, 42, to sphere, 194 72–73, 79–80, 107, 398–99, 410, 460 to triangle, 205 correspondence with Castelnuovo, 23, 72, 80, 406 FAÀ DI BRUNO, Francesco (1825–1888), on elementary geometry, 123, 128–29, 133, 107, 332 153, 160, 278, 285, 287, 368, 415–16 FABBI, Angelo (19??–), 44, 46, 110, 117, 416 rivalry with Pieri, 23, 25–29, 31–32, faculty ranks, Italian, 11 36, 75, 94, 98, 119, 411 FAIFOFER, Aureliano (1843–1909), 71–72 See also under BOTTAZZINI FANO, Gino (1871–1952), 36, 72, 80–81, enumerable. See under class 87, 89, 94, 132, 337, 411, 416, 460 enumerative geometry. See under geometry rivalry with Pieri, 29, 31–32, equality on foundations of geometry, 53, 65, 107–8, by definition, 163, 265 129, 131, 133, 405, 416, 425 in logic, 316–17, 336, 350–52 FÄRBER, Karl (1863–1912), 377, 438, 448, 455 of transformations, 270 FARNESE, Ranuccio I of (1569–1622), 41 See also under symbol Fascism, Italian. See under Italy equidistance, 153 FAVA, Narciso (18??–19??), 48–49, 399, 416 quaternary, 153, 281, 351, 353–54, 357–58, FÀVARO, Antonio (1847–1922), 36, 398, 411 360–62 (see also under congruence: of FAZZI, Carlo (18??–????), 398 point pairs; congruence: of segments) FEFERMAN, Anita Burdman (19??– ), ternary, vii, 152–54, 157, 160–61, 165, 272, 113, 347–48, 416 278, 280, 282, 286, 341–42, 349–50, FEFERMAN, Solomon (1928– ), 357–58, 360–62, 366, 368 113, 347, 348, 416 equilaterality, 362 FEHR, Henri (1870–1954), equinumerous sets, 296 290, 382, 412, 442, 445 equivalence, logical, 163, 348 FEIGL, Georg (1890–1945), See also under symbol 404, 415, 417, 422, 426 ERBA, Giuseppe Bartolomeo (1819–1895), 332 FEIGL, Herbert (1902–1988), 127, 405 Erlanger program. See under KLEIN FEINBERG, Barry (1938–), 386, 416, 441 ERDÉLYI, Arthur (1908–1977), 114 female mathematicians, 40, 365 EUCLID (325?–265? B.C.), 95, 102, 125–26, FERDINANDO II. See Ferdinando II 147, 149–51, 288, 291, 338, 369, 416 of BOURBON in Point and Sphere, 167, 175, 180–83, FERGOLA, Emanuele (1830–1915), 79 185, 187, 192, 199, 206, 209–210, FERMAT, Pierre de (1601–1665), 454 222, 224–28, 230–233, 262, 263 FERMI, Enrico (1901–1954), 114 postulate (see parallel postulate: Bolyai’s) FERRARIS, Galileo (1847–1897), 64, 416 Euclidean. See under field; geometry FERREIRÓS DOMÍNGUEZ, José (19??– ), EULER, Leonhard (1707–1783), 292, 296, 299, 314–15, 323, 416 18, 76, 248, 284, 416 FICHERA, Filadelfo (1850–1909), 33–34 EWALD, Günter (1929–), 143 FICHERA, Gaetano (1922–1996), 115, 417 EWALD, William Bragg (1954–), 291, 409, FIEDLER, Otto Wilhelm (1832–1912), 117 413, 416, 422, 445 field excess-intersection theory, vii, 52 Euclidean, 263, 351, 355 exponentiation, 314, 318–19 formally real, 354 existential. See under sentence ordered, 351, 354, 356 extension Pythagorean, 354 of predicate, 298–99, 302, 304 real-closed, 351–52 of segment, 200 theory, 351 472 Index figure, 164, 249 FRICKE, Robert (1861–1930), 414, 421, 425 finitary methods, 60–61 FRIEDMANN, Sigismondo (1852–1917), 12 finiteness. See under Julius DEDEKIND FRITSCH, Rudolf (1939–), 96, 418 FINZI, Bruno (1899–1974), 110 FUBINI, Guido (1879–1943), 34, 67, FINZI, Cesare (1836–1908), 12, 417 78, 81, 108, 114, 398, 450 FIORINI, Matteo (1827–1901), FUJIWARA, Matsusaburô (1881–1946), 423 8, 10, 28, 81, 398, 423 FULTON, William (1939–), 52–53, 418 FISCHER, Michael J. (1942–), 352, 417 function, 269, 314–15 FISCHER-BENZON, Rudolph Jacob application, 269, 273 Diedrich von (1839–1911), 456 bijection, 269 fixed point, 52, 175–76, 270 composition, 273 FOÀ, Emanuele (1892–1949), 417 elementary, 292, 299 FOLKERTS, Menso (1943–), 412, 417, 453 holomorphism, 140 following, 163, 249, 252 injection, 269, 314 See also under symbol reciprocal (see function: bijection) foot, 187 See also under transformation footnotes, ix, 4, 62, 158 functional analysis, 123 FORDER, Henry George (1889–1981), 286, 417 fundamental sequence or series, 254, 292 formal sum, 292, 294 coherent, 293 formalism, 307 fundamental theorem of projective geometry, formally real. See under field 121, 137, 207, 214, 336, 338, 380, 388 Formolario, Formulaire, formulary FURCH, Robert Otto (1894–1967), 143 See under Giuseppe PEANO FURTH, Montgomery (1933–), 417 foundation. See under axiom fusionism, 147–48, 150, 154–55, 157, 160 foundations of geometry. See under geometry foundations of mathematics, GABELLI, Aristide (1830–1891), 17, 418 85, 116, 128, 298, 327, 343 GALOIS, Evariste (1811–1832) See also under Mario PIERI; RUSSELL; theory, 66, 76 and school: Peano GANDINO, Giovanni Battista (1827–1905), 96 FOURET, Georges François (1845–1921), GARIBALDI, Giuseppe (1807–1882), 41, 65, 80 101, 398, 417 GARIO, Paola (19??–). See BOTTAZZINI France, xvii–xix, 18 GAROSI, Gino (????–????), 444 FRAENKEL, Abraham Adolf (1891–1965), GAUSS, Carl Friedrich (1777–1855), 290, 304, 409, 417, 445 76, 93, 111, 125, 293–94, 418, 421 FRANCESCONI, Stefano (19??–), 10, 63, GAZZANIGA, Paolo (1853–1930), 118, 455 68, 70, 81, 101, 118, 417 GEISER, Carl Friedrich (1843–1934), 100, 418 FRANZ I. See Franz I of HABSBURG GEMINIANI, Francesco (1687–1782), 4 FRÉCHET, Maurice (1878–1973), 124, 398 general topology. See under topology FREGE, Friedrich Ludwig Gottlob (1848– GENOCCHI, Angelo (1817–1889), 1925), 298–307, 316, 347, 405, 417, 20, 82, 91, 107, 332–36, 418 419, 421, 431–32, 434, 447 GENTILE, Giovanni (1875–1944), 366, 453 [1879] 1970 Begriffsschrift, 298–99, 304 See also under laws [1884] 1953 Foundations, 298, 300, 304–5 geometrography, 89 1893–1903 Basic Laws, 301–2 geometry logic, view of, 298–99, 301, 305, 340 absolute, 145, 151, 273, 287, 340 See also under Russell’s antinomy; algebraic, vii, ix, 51–53, 75, successor 84, 103, 107, 123, 406 FREGUGLIA, Paolo (19??–), 156, 402, 406 birational, vii, 52 FREUDENTHAL, Hans (1905–1990), 85, 127, Cartesian, 351, 354 313, 341, 407, 415, 418 Index 473 geometry (continued) Gioenia, Accademia, 1, 34, 41 complex projective (see under See also under journals projective geometry) GIOENI D’ANGIÒ, Giuseppe (1747–1822), 41 descriptive, 11, 29, 31, 120 GIOLITTI, Giovanni (1842–1928), 26, 43 diagrams in, 58 GIORGIO, Gino (avv. 18??–19??), 48 differential, 53, 116, 120, 383 Giornale di matematiche. See under journals elementary (see elementary geometry) GIUDICE, Francesco (1855–1936), 18, 337 elliptic, 356, 358, 360–61 GIUSFREDI, Beatrice. See Beatrice CAMPETTI enumerative, vii, 2, 46–47, 51–52, 73, GIVANT, Steven Roger (1945?– ), x, 113, 156, 104–05, 112, 121, 337, 374, 376, 396 347, 350–55, 357, 360, 369, 419, 452, 460 Euclidean, 145, 149, 151, 157, 161, 274, glide reflection. See under reflection 349, 356–57, 361, 368, 370 (see also GOBLOT, Edmond (1858–1935), 147, 419 elementary geometry) GÖDEL, Kurt (1906–1978), 61, 85, foundations, 53–54, 58, 84, 95, 106, 113, 120, 305, 319, 347, 419–21 117, 123–26, 142, 147, 156, 161, 284, 337, GOHEEN, Harry (19??–????), 149 342, 350, 352, 354, 366, 368, 371, 379, 394 GOLDFARB, Warren (1949– ), 351, 419 fundamentals (complementi), 22 GORDAN, Paul Albert (1837–1912), 112 hyperbolic, 145, 356, 360–61 Göttingen, 84, 87, 107 inversive, 57, 93, 99, 137–43 Library, 460 line, 134 GOW, Roderick Ian (1948–), 103, 419 multidimensional, 53, 108, 117, 129, 133, GRANDJOT REINS, Karl (1900–1979), 319, 409 140, 287, 351, 356, 358, 361, 395 graphical statics, 75, 99, 106 non-Archimedean, 81, 117–18 GRASSI-CRISTALDI, Giuseppe (18??–19??), non-Euclidean, 41, 81, 84, 86, 32, 37, 388, 441 112, 284, 287, 358, 394 GRASSMANN, Hermann Ernst (1857–1922), 83 projective (see projective geometry) GRASSMANN, Hermann Günter (1809–1877), real enumerative, vii 82–84, 295, 300, 408, 419, 421, 448 transformational, 284 on arithmetic, 294, 298, 305, 318, See also compass on multidimensional methods, GÉRARD, Louis (18??–19??), 263, 418 53, 334, 406, 433 GERGONNE, Joseph Diaz (1771–1859), See also under natural-number arithmetic 127, 418 GRASSMANN, Justus Günther (1779–1852), 82 GERHARDT, Karl Immanuel (1816–1899), 427 GRASSMANN, Robert (1815–1901), 83 GERICKE, Helmut (1910–), 417 GRATTAN-GUINNESS, Ivor (1941–), v, x, 291, GERLA, Giangiacomo (????– ), 350, 418 294, 301–2, 341, 364, 424 GIACARDI, Livia (1952–), x, 18, 22, 69, 72, 79, GRAY, Jeremy (1947–), x, 59, 420, 430 81–82, 92, 108, 123, 129, 333, 411, 418 greatest-lower-bound principle, 291–93 GIAMBELLI, Giovanni Zeno (1879–1953), GREITZER, Samuel L. (1905–1988), 75, 420 42, 49, 286, 399, 418 GRIFFIN, Frank Loxley (1881–1969), GIAMPAGLIA, Niccolò (18??–19??), 34, 40, 369–70, 420 82, 393, 395–96, 410, 419, 443 GRIM, 414 GIANNATTASIO, Antonio (19??–), 95, 419 GRIMALDI, Giovan Pietro (1860–1918), GIBSON, George Alexander (1858–1930), 447 32, 37, 387, 389, 420 GIBBS, Josiah Willard (1839–1903), 53, 84 group. See transformation: group GIGLI, Duilio (1878–1933), 65, 419 GRUGNETTI, Lucia (19??–), 406, 430 See also BERZOLARI: Encyclopedia GRUNSKY, Helmut (1904–1986), GILLIES, Douglas Angus (????– ), 409, 415, 417, 426 294, 302, 304, 315, 419 GUCCIA, Giovan Battista (1854–1936), 398 GILLISPIE, Charles Coulston (1918–), 419 GUERRAGGIO, Angelo (1948–), ginnasio, 7 119, 414, 420, 460 474 Index

GUMBEL, Emil Julius (1891–1966), 447 HERMANN, Robert (1931–), 426 GUNDEL, Hans Georg (1912–), 420, 436 HERRE, Paul (1876–1962), 47, 398 GÜNTHER, Siegmund (1848–1923), 398 HERSH, Reuben (1927–), x GUPTA, Haragauri Narayan (19–), Hessen, Historische Kommission, 460 x, 313, 354–57, 420 HEYTING, Arend (1898–1980), 362 Gymnasium. See ginnasio hierarchy, Kleene–Mostowski, 271 HILBERT, David (1862–1943), v, 2, 38, 41, Habsburg dynasty, xvii, 41 59–61, 84–85, 87, 104–5, 118, 127, 271, HABSBURG, Franz I of (1768–1835), 41 275, 300, 369, 378, 405–6, 416, 418, HABSBURG, Leopoldo I of (1747–1792), xvii, 41 420–22, 424, 433, 439, 460 HABSBURG, Leopoldo II of (1797–1870), [1899] 1971 Foundations of Geometry, xvii, 41 58–59, 84, 106, 127, 143, 148–51, 155– 56, HABSBURG, Maria Luigia of (1791–1847), 162, 198, 263, 274, 275, 276–78, 280, xvii, 41 282–83, 285, 288, 342, 348, 350, 354, 357, half-line, 146, 192, 200, 272 360, 366–67, 369, 414, 436, 445, 453 See also under symbol See also under school; style half-plane, 192, 202, 272 HIRST, Thomas Archer Hirst (1830–1892), 378 half-space, 204 history of mathematics, v, vi, viii, 3, 63, half-turn. See reflection: across a line 80–81, 91, 121 HALSTED, George Bruce (1853–1922), Italian, 80, 91, 114, 364, 406, 445 64, 162, 198, 263, 420, 422, 445 HOBSON, Ernest William (1856–1933), 423 HAMBURGER, Meyer (1838–1903), 408, 433 HOFFMAN, Alan Jerome (19??–), 143, 422 HAMEL, Georg (1877–1954), 360, 420 HOLGATE, Thomas Franklin (1859–1945), 445 HAMILTON, William Rowan (1805–1865), 53 Holocaust. See World War II: effects on Jews harmonic conjugation, 134 holomorphic function. See under function harmonic points, 131, 142, 143 homogeneous coordinates HASKELL, Mellen Woodman (1863–1948), 425 See under coordinates HAUSSNER, Robert Karl Hermann homography. See under transformation (1863–1948), 378, 438 homothety. See under transformation HAWKINS, Thomas (1938–), 3, 133, 134, 421 HOPPE, Reinhold Eduard (1816–1900), HEATH, Archie Edward (1887–1961), 84, 421 425, 432 HEATH, Sir Thomas Little (1861–1940), HUBER, Birkett (19??–), 52, 422 403, 416 HUDSON, Hilda Phoebe (1881–1965), 139, 422 HEAVISIDE, Oliver (1850–1925), 53, 84 HUNTINGTON, Edward Vermilye (1874–1952), HEDRICK, Earle Raymond (1876–1943), 327, 340, 350, 422, 445 134, 421, 425 HURWITZ, Adolf (1859–1919), HEEGAARD, Poul (1871–1948), 407 84, 86, 105, 292, 455 HEIBERG, Johan Ludvig (1854–1928), hyperbolic geometry. See under geometry 121, 416, 421 hypermedial point, 212 HEIJENOORT, Jean van (1912–1986), hyperplane reflection. See under reflection 307, 316, 407, 419, 421, 431 hypersurface, 374, 396 HEINE, Heinrich Eduard (1821–1881), 104 classification, 52 See also under theorem quadric, 57 helicoid. See under surface hypersymmetric points, 212 HELLINGER, Ernst David (1883–1950), 425 hypothesis of definition. See under definition HELMHOLTZ, Hermann Ludwig hypothetical-deductive system, v, vii, 54, 56, Ferdinand von (1821–1894), 125 58, 59, 126–28, 141, 147, 151, 154, 157, HEMPEL, Carl Gustav (1905–1997), 452 161–62, 286, 290, 320, 322, 363, 369, 381 HENKIN, Leon Albert(1921–2006), 318–19, See also axiomatic method 324, 421 Index 475 identity. See under transformation Interlingua. See Giuseppe PEANO: Academia ideography. See Giuseppe PEANO: Intermédiaire des mathématiciens symbolic notation See under journals Ido, 74 intermediate value. See under theorem illumination, constant. See under curve internal point imaginary point, 56, 135, 393 to angle, 204 implication. See under symbol to arc or semicircle, 252 incaricato. See professore to circle, 195 incidence, 149, 274, 334 to segment, 195 relation between lines (see under to solid, 349 intersection) to sphere, 194 See also under postulate to triangle, 205 inclusion, 165, 314, 349 International Congress of Mathematicians See also under symbol 1897 Zürich, 89 inconsistent collection, 304 1900 Paris, 60, 85, 104, 148, independence 281, 341–43, 422, 449 of arithmetic axioms, 312, 318, 325–26, 328 1904 Heidelberg, 60, 422 of axioms in general, 54, 59, 61, 131, 148, 1908 Rome, 53, 328, 422 277, 287–88, 320, 342 1912 Cambridge, 2, 47, 328, 423 of primitive notions, 95, 321 1928 Bologna, 98 of Tarski’s geometric axioms, 356 International Congress of Philosophy See also under parallel postulate 1900 Paris, 1, 36, 74, 95, 102, 148, 153, indirect. See under isometry; motion 281, 302, 320, 341–43, 381, 406, 412, induction (mathematical), 60, 294, 423 (see also under RUSSELL) 298–99, 313, 322, 343 1904 Rome, 124 in Dedekind, 314–15 1911 Bologna, 80 Peano’s postulate, 58, 308, 311, Internet repositories, academic, 401 315, 317, 323, 325–26 intersection, 165, 314 Pieri’s postulate, 310, 322–23, 325–26 multiplicity, 51, 53, 378 definition by. See definition: recursive relation between lines, 56, 134 See also minimum principle See also excess-intersection theory inequality. See under symbol and under symbol inference rule, 316 interval. See segment infinite descent, 313 intuition, geometric, 54, 95, 108, 126–27, 131, infinite set. See Julius DEDEKIND: finiteness 135, 161, 266–67, 283, 286–87, 354 infinity inverse. See under transformation point at, 138–40, 277, 393 inversion. See under transformation See also under axiom inversive injection. See under function group, 139, 140 INGOLD, Louis (1872–19??), 134, 421 property, 57 INGRAMI, Giuseppe (18??–19??), See under geometry 394–95, 422, 440 involution. See under transformation Institute for Advanced Study irrational number, 289, 293, 382 See under Princeton isomer, 218 institute. See technical institute isometric figures, 218 instruction, minister of See also under congruence See minister of instruction isometry, 154–55, 218, 271 integer modulo n. See under arithmetic classification, 241, 247–48 integral calculus. See differential See also motion and integral calculus 476 Index isomorphic journals simply infinite systems, 300 Academia pro Interlingua, 345 well-ordered sets, 297 Annuario scientifico ed industriale, 90 ISRAEL, Giorgio (1945– ), 291, 423 Atti della Accademia Gioenia di Scienze Italian Mathematical Union, 98 Naturali in Catania, 34, 35 Italian universities, 11 Battaglini (see journals: Giornale di hiring policies, 16, 25 matematiche) private, 16 Bollettino di bibliografia e storia della public, xviii, 16 scienza, 92 Italy Bulletin des sciences mathématiques, 402 cities, xviii Crelle’s (see journals: Journal für die reine Chamber of Deputies, 18 und angewandte Mathematik) colonialism, 43 Geisteswissenschaften, 47 corruption, 26, 43 Giornale di matematiche, 402 economy, 15, 23 Intermédiaire des mathématiciens, 89 Fascism, 72, 80–81, 90–92, 119, 366 Jahrbuch über die Fortschritte der government, 75, 366 Mathematik, 299, 329, 333, 368, 402 history, xvii Journal für die reine und angewandte regions, xvii, xix Mathematik, 93 school system, 7, 15, 366 Journal of Symbolic Logic, 402 Senate, 18, 27 Mathematicae Notae, 91 Unification, xvii, 4, 11, 18, 41, Mathematical Reviews, 402 63, 65–66, 70, 78, 85, 109 Mathematische Annalen, 86 See also Italian universities; literacy Revista de mathematica, 336, 343 IZUMI, Shin-ichi (1904–1990), 143, 423 Revue de métaphysique et de morale, 341 Revue des mathématiques, 336 Jacobian. See under variety Rivista di matematica, 71, 301, 336–37, JADANZA, Nicodemo (1847–1920), 342–43 20, 81, 85, 423, 432 Rivista populare di politica, lettere Jahrbuch über die Fortschritte der e scienze sociali, 43 Mathematik. See under journals Transactions of the American JAMES, Glenn (1882–1961), 412, 423 Mathematical Society, 388 JANELLI, Angiolina. See Angiolina PIERI JESPERSEN, Julie. See Julie ZEUTHEN KAGAN, Benjamin Fedorovich (1869–1953), JESPERSEN, Louise. See Louise ZEUTHEN 124, 385, 398, 441 JEVONS, William Stanley (1835–1882), 316 KALMÁR, László (1905–1976), 319 JOHNSON, Jeremy R. (196?–), 352, 411, 417 KANNENBERG, Lloyd Chambers (1939–), join 419, 433 2 points, 54, 56, 167 (see also symbol: line) KANT, Immanuel (1724–1804), 266, 305, 3 points, 167, 170 (see also symbol: plane) 412, 416 JOLLES, Stanislaus (1857–19??), 456 philosophy of arithmetic, 84 JORDAN, Marie Ennemond Camille philosophy of geometry, 102 (1838–1922), 74, 86 KANTOR, Seligmann (1857–19??), 398 JORDAN, Zbigniew (1911–1977), 351, 423 KARZEL, Helmut (1928–), 143, 145, 424 JOURDAIN, Philip Edward Bertrand KASNER, Edward (1878–1955), 142–43, 424 (1879–1919), 289–90, 292–94, 304–5, KELVIN, Lord. See William THOMSON 316, 327–28, 409, 423–24, 431, 456 KENNEDY, Hubert Collings (1931–), Journal für die reine und angewandte 27, 41, 69, 73, 82, 95–96, 301–2, 316, Mathematik. See under journals 320, 332–338, 341–42, 345–47, Journal of Symbolic Logic. See under journals 364–65, 399, 424, 433, 435, 460 Index 477

KEYSER, Cassius Jackson (1862–1947), LAMPE, Emil (1840–1918), 377, 379–80, 408, 127–28, 424 414–16, 418–19, 422–27, 430–32, 439, KILLING, Wilhelm (1847–1923), 441–42, 445, 448–50, 454–57 41, 133, 147, 424 LANDAU, Edmund Georg Hermann KIRKOR, Andrzej (19??–), 426 (1877–1938), 290, 294, 319, 426 KIST, Günter (1943–), 145, 424 LANDI, Claude P. (18??–19??), 446 KLEENE, Stephen Cole (1909–1994) LANTELME, Giuseppe (????–????), 332 See hierarchy larger-than relation KLEIMAN, Steven Lawrence (1942–), for angles, 223 x, 51–52, 105, 122, 424, 448 for cardinals, 296 KLEIN, Christian Felix (1849–1925), 55, 56, for natural numbers, 311, 313–14, 319 67, 80, 84, 85–87, 105, 107, 117, 121, for ordinals, 297 125, 133, 135, 137, 143, 146, 340, 365, for real numbers, 293 383, 398, 408, 425–26, 430–31, for segments, 220, 272 440, 444, 446, 449, 459 Latin, 316, 343 1872 Erlanger program, 56–58, 80, 86–87, Uninflected, 42, 49, 74, 301, 107–8, 116, 133–34, 139–42, 145, 336–37, 343, 345, 365 154, 421, 425 lattice theory, 143, 350 KLINE, Morris (1908–1992), 291, 426, 456 LAUGEL, Leonce (18??–19??), 421 KNEEBONE, Geoffrey Thomas (1918–2003), laureate. See doctorate 294, 301, 305, 426 LAURICELLA, Giuseppe (1867–1913), KNOBLOCH, Eberhard (1943–), 446 34, 37, 40, 63, 88, 393, 395–96, KOHN, Gustav (1859–1921), 148, 426 413, 419, 426, 443, 450, 459 KOLLÁR, János (19??–), 426, 450 LAVAGGI, Luigi (18??–19??), 46, 88 KOPPELMAN, Elaine (1937–), 73, 426 LAVRENKO, N. I. (& ", . ., 19??–), KOSTANT, Ann (1937–), x 41, 406 KOTARBIN´SKI, Tadeusz Marian LAWAETZ, Sophie. See Sophie ZEUTHEN (1886–1981), 429 laws KÖTTER, Fritz Wilhelm Ferdinand Casati and Mamiani, 16, 25 (1857–1912), 375–77, 379, 415, Gentile, 90 426, 434, 437–39, 445, 455 LAX, Anneli (1922–1999), x KOWALEWSKI, Gerhard Waldemar LAX, Peter David (1926–), x Hermann (1876–1950), 454 LAZARSFELD, Robert (19??–), 426, 450 KRATZER, Adolf (1858–), 422 LAZZERI, Giulio (1861–1935), 99, 239, 426 Kreisverwandtschaft LEBESGUE, Henri Léon (1875–1941), 89 See transformation: circular LEGENDRE, Adrien-Marie (1752–1833), KROLL, Hans-Joachim (1942–), 185, 227, 426 143, 145, 424 LEIBNIZ, Gottfried Wilhelm von (1646–1716), KRONECKER, Leopold (1823–1891), 95 74, 83, 154, 167, 280–81, 284, 307, KRYNICKI, Michaã (19??–), 271, 426 340–43, 361, 427, 454 KUMMER, Ernst Eduard (1810–1893), 104–5 LEJA, Franciszek (1885–1979), 427, 445 KURATOWSKI, Kazimierz (1896–1980), LEMMERMEYER, Franz (1962–), 421 350, 426 LEMOINE, Émile Michel Hyacinthe KUZAWA, Mary Grace (1918–), 367, 426 (1840–1912), 41, 88–89, 426, 450 KWIETNIEWSKI, Stefan (1874–1940), length of segment, 259 381, 415, 443 LENNES, Nels Johann (1874–1951), 116, 454 LEOPOLDO I. See Leopoldo I of HABSBURG LAGRANGE, Joseph Louis (1736–1813), 18 LEOPOLDO II. See Leopoldo II of HABSBURG LAISANT, Charles-Ange (1841–1920), 89, 426 LES´NIEWSKI, Stanisãaw (1886–1939), 347, 349 less-numerous set, 296 478 Index

LESSONA, Michele (1823–1894), 332 logic (continued) letters. See under Mario PIERI elementary or first-order, viii, 156, 288, 316, LEVI, Albina Bachi (1886–1951), 90 350–51, 353, 355–356, 368, 370–71 LEVI, Beppo (1875–1961), 2, 10, 12, 15, 20, 22, See also under Georg CANTOR; Julius 34, 36, 41–42, 45, 46–49, 61, 65, 79, DEDEKIND; FREGE; Giuseppe PEANO; 89–91, 98, 107, 119, 124, 161, 254, Mario PIERI; RUSSELL 373, 398–99, 411, 427, 460 logical constants and terms, 265, 386 LEVI, Decio (1885–1917), 90 See also under symbol LEVI, Donato (1834–1885), 332 logical positivism, 127 LEVI, Emilia. See Emilia RESTA logicism, 102, 120, 298, 307 LEVI, Eugenio Elia (1883–1917), 90 logistic, 348 LEVI, Giulio (19??–????), 90–91 LORIA, Achille (1857–1943), 92 LEVI, Laura (1915–2003), 42, 90–91, 427 LORIA, Gino (1862–1954), 31, 91–92, 105, LEVI-CIVITA, Tullio (1873–1941), 114, 342 107–108, 152, 333, 398, 409, 418, 447, 460 libera docenza, 11 as reviewer, 92, 123, 150–51, 335, 374–78, liberal-arts education, 369 380, 393, 397, 407–8, 410, 414, 416, 427, Library of Congress, 460 433, 436–41, 443, 447, 449–50, 455 licenza, 12 LOVE, Augustus Edward Hough liceo, 7 (1863–1940), 423 LIE, Sophus (1842–1899), 41, 56, LOVETT, Edgar Odell (1871–1957), 86, 105, 121, 146, 427, 446 341, 381, 423, 427, 440 limit, inferior or superior, 255, 273, 285 LOWE, Victor (1907–1988), 120, 428 Lincei, Reale Accademia dei, 5, 11, 342 LÖWENHEIM, Leopold (1878–1957), 423, 428 LINDEMANN, Carl Louis Ferdinand von LRM3D2, 405 (1852–1939), 84 Lucca, xvii, xviii, 1, 4, 6, 44, 49 LINDENBAUM, Adolf (1904–1941?), Royal Academy of Science, Letters, 358, 359, 360, 362, 427, 452, 460 and Arts, 1, 3, 4, 11, 49 LINDGREN, Uta (19??–), 417, 453 LUCCHESI, Camillo (18??–????), 398 line LUCIANO, Erika (19??–), vi, 74, 341, 428 defined, 133–34, 138, 146, 154, LUGLI, Aurelio (1853–1896), 18 170, 272, 281, 340–41 LUITPOLD, Prinzregent von Bayern. geometry (see under geometry) See Luitpold of WITTELSBACH) at infinity, 277 âUKASIEWICZ, Jan (1878–1956), reflection (see under reflection) 294, 313, 420, 428 undefined, 128–29, 149, 274, 338 LUPORINI, Erminia. See Erminia PIERI See also under symbol LUPORINI, Gaetano (18??–????), 7 linguistics, 83, 97, 123 LUPORINI, Luigi (????–18??), 4 See also under Giuseppe PEANO LUPORINI, Marianna Davini (????–18??), 4 LIOUVILLE, Joseph (1809–1882), 140 LÜROTH, Jakob (1844–1910), 425 LIPSCHITZ, Rudolf Otto Sigismund (1832–1903), 86 MACCAFERRI, Eugenio (1870–1953), literacy, 7, 15 2, 392, 398, 407, 428, 443 LOBACHEVSKY, Nikolai Ivanovich MACCOLL, Hugh (1836–1909), 316 ("4(, #  !# 5, MACHOVER, Moshé (1936–), 313, 428 1792–1856), 41, 125 MACK SMITH, Denis (1920–), 26, 44, 428 Prize, v, 2, 38, 41, 64, 84, 89, MAC LANE, Saunders (1909–2005), 116, 428 106, 112, 342, 406, 445 Magnitude, continuous geometric, 291–92 logic, 22, 59, 95, 123, 126, 267, 269, 271, MAGNUS, Heinrich Gustav (1802–1870), 104 316, 320, 325, 335–37, 341–42, 348, MAKOWIECKA, Helena (19??–), 362, 428 352, 368, 381–82, 387, 402, 411 Index 479

MAMIANI, Terenzio, Conte della Rovere MCKENZIE, Ralph Nelson Whitfield (1799–1885). See under laws (1941–), 452 MANARA, Carlo Felice (1916–), 129, 428 MCKINSEY, John Charles Chenoweth MANFREDI, Bianca (1920?–1999), (1908–1953), 352, 357, 452 47, 71, 88, 106, 428 mean point. See midpoint MANN, Thomas (1948–), 402, 428 mechanics, 12, 15, 383 MANSION, Paul (1844–1919), 167 median of triangle, 233 MARACCHIA, Silvio (19??–), 366, 428 medio-symmetric point, 212, 285 MARCHISOTTO, Elena Anne Corie (1945–), MELDRUM, John (19??–), 406 iv–v, x, 146, 328–29, 336, membership. See under symbol 357, 411, 416, 429 MENEGHINI, Giuseppe (1811–1889), 70 MARCHISOTTO, Joseph Anthony, Jr. (1942–), x MENGER, Karl (1902–1985), MARCOLONGO, Roberto (1862–1943), 350, 361, 366–67, 429 46, 53, 69, 378, 392, 407, 429, 443 MÉRAY, Hugues Charles Robert (1835–1911), MARCONI, Guglielmo (1874–1937), 100 218, 429 Comitato International, 461 mereology, 349, 350 MARIA LUIGIA. See Maria Luigia of HABSBURG MERIANO (18??–1935?), 5 MARIA LUISA. See Maria Luisa of BOURBON MERIANO, Gemma Pieri (????–1961?) 5, 9, 97 MARIE-LOUISE. See Maria Luigia of MERZBACH, Uta Caecilia (1933–), 415 HABSBURG MEYER, Friedrich Wilhelm Franz MARINO, Teresa (????– ), 323, 325, 328, 414 (1856–1934), 87, 105, MARLETTA, Giuseppe (1878–1944), 34, 40, Encyklopädie, 2, 87, 429, 92, 388, 398, 402, 408, 410, 429 Encyclopédie, 87, 325, 393, 430 Mars. See under Mario PIERI reviewer, 413, 424, 427, 455 MARSILI, Luigi (????–18??), 70 MIANOWSKI, Józef (1804–1879), 367, 381, 443 MARTIN, George Edward (1932–), Foundation, 367–68 145, 147, 151, 429 MICHAELIS, G. J. (18??–19??), MARTINETTI, Vittorio (1859–1936), 22, 36, 411 412, 431, 434, 436 MARTINI, Eligio (1828–1896), 332 middle school, 7, 20 MARTINI, Ferdinando (1841–1928), 26 midpoint, 146, 155, 164, 174, MARTOLINI, Guglielmo (18??–????), 12 272, 283, 357, 362 MASOTTO, Guido (19??–). See under See also under symbol BRIGAGLIA Milan, Istituto Tecnico Superiore, 66 Mathematicae Notae. See under journals minimum principle, 313, 323, 327–28 mathematical induction. See induction weak form, 58, 310, 313, 322–23, Mathematical Reviews. See under journals 325–26, 328 mathematics See also induction education (see pedagogy) minister of instruction, 16, 26–27, 36, 38, 43, history (see history of mathematics) 63, 66, 118 See also under RUSSELL MINKOWSKI, Hermann (1864–1909), 84 Mathematische Annalen. See under journals MIQUEL, Auguste (????–18??) Mathesis, 18, 20, 49, 66, 71, 326–27 See under theorem MAXWELL, James Clerk (1831–1879), 75 MITCHELL, Ulysses Grant (1872–19??), 456 MAY, Kenneth Ownsworth (1915–1977), 429 MITTELSTRASS, Jürgen (1936–), 430, 453 MAYER, Christian Gustav Adolph MÖBIUS, August Ferdinand (1790–1868) (1839–1907), 418, 445 57, 93, 139, 140, 144, 403, 430, 436 MCCALL, Ronald Storrs (1930–), 423, 429 See also under transformation MCCLEARY, John (1952–), 446 modus ponens, 316 MCCONNELL, Albert Joseph (1903–1993), MOESE, Henryk (????–19??), 451–52 103, 429 480 Index

MOHRMANN, Hans (1881–1941), 2, 410, NAPOLITANI, Pier Daniele (1951–), x 415–16, 429–30, 449, 457 NARDI DEI, Angiolo (1833–1913), 12, 74, 94 MOIGNO, Abbé François Napoléon Marie NARDINI, Rodolfo (18??–19??), 385 (1804–1884), 410 NASTASI, Pietro (1941–), 63, 65, 73, MOLK, Jules (1857–1914), 87, 393, 429–30 88, 101, 119, 414, 420, 431, 460 MOLLAME, Vincenzo (1848–1911) NATORP, Paul Gerhard (1854–1924), 32, 34, 75, 93–94, 411, 430 328, 423, 431 MOLLWEIDE, Karl Brandan (1774–1825), 93 natural-number arithmetic, 291, 294 MONGE, Gaspard (1746–1818), 33 Cantor’s theories, 297 monoid, 377 consistency, 60, 85, 290, 306 (see also monotonic sequence, 254 under Mario PIERI) MONTALDO, Oscar (19??–), 406, 430 Dedekind’s theory (see under MONTANARI, Enrica. See Enrica PIERI Julius DEDEKIND) MONTESANO, Domenico (1863–1930), foundation, 290, 302, 313 36, 40, 42, 94, 430, 448 Frege’s theory, 298, 305 and Pieri’s chair quest, 22, 25–27, 29, 398 Grassmann’s work, 84, 294 MONTGOMERY, Deane (1909–1992), 116, 430 Padoa’s reduction, 321, 322, 325 MOORE, Eliakim Hastings (1862–1932), Peano’s system, 84, 301, 307, 116, 342 316–22, 370, 405, 409, 449 MOORE, Gregory H. (1944–), Pieri’s axioms (see under Mario PIERI) 59, 351, 420, 430 natural-number definition 290, 298, 306–7 MOORE, Robert Lee (1882–1974), negation. See under symbol 116, 342, 351, 431 NERVO, Natalia (19??–), vi, 446 MORAW, Peter (1935–), 420, 436 NEUGEBAUER, Otto (1899–1990), 426 MORRICE, George Gavin (1859–1936), 425 NEUMANN, Hanna von Caemmerer MORRISON, David Robert (1955–), 426, 450 (1914–1971), 455 MOSSOTTI, Ottaviano Fabrizio (1791–1863), NEWSON, Mary Francis Winston 66, 78 (1869–1959), 421 MOSTOWSKI, Andrzej (1913–1975) NEWTON, Sir Isaac (1643–1727), 316 See hierarchy NIDDITCH, Peter Harold (1928–1983), motion, 139, 161, 248, 271, 394 299, 302, 431, 434 direct, 57, 145–47, 150, 152, NIELSEN, Niels (1865–1931), 448 271, 278, 280–81, 338, 340 NIERI, Idelfonso (1853–1920), 101 indirect, 145–46, 271 NOBLE, Charles Albert (1867–1962), 425 MÜLLER, Felix (1843–1928), 416, 423, 455 NOETHER, Emmy (1882–1935), 414 multidimensional. See under differential NOETHER, Max (1844–1921), 65, 111, 122, 431 and integral calculus; geometry; non-Archimedean. See under geometry projective geometry non-Euclidean. See under geometry multiplication. See product normal multiplicity. See under intersection to a variety, 374–75 music, 4, 7, 70, 76, 83, 88, 92–93, 101, 103 See also orthogonality MUSSOLINI, Benito (1883–1945), 119, 366 normal school, 7 NORTH, John David (1934–), 146, 431 Nachlass. See under Mario PIERI notation NAGEL, Ernest (1901–1985), 128, 418, 431 in Point and Sphere, 158, 162–64, 270 NANNEI, Enrico (1864–1961), 398 Pieri’s usage in general, ix, 59, 87, Naples, xvii, xviii, 41 126, 306, 323, 340, 366–67, 383 university, 5, 16, 22, 25 See also under Giuseppe PEANO; symbol NAPOLEON. See BONAPARTE number theory, 313 Napoleonic wars, xvii, 4, 11, 18, 41 Index 481 obituaries. See under Mario PIERI PALLADINO, Franco (1945– ), x, 20, 22, 27, O’CONNOR, John J. (1945–), 87, 431 34, 62, 73, 147, 156, 383, 432, 438 OETTINGEN, Arthur Joachim von See also under BORGA (1836–1920), 444 PAMBUCCIAN, Victor (1959– ), 362, 432 OHM, Martin (1792–1872), 291–92, 300, PANETTI, Modesto (1875–1957), 85, 432 404, 431 PANNELLI, Marino (1855–1934), 36 OHRTMANN, Carl (18??–????), 409, 450 PANNWITZ, Erika (1904–1975), 409 one, 258 Papal States, xvii, 11 operation on points. See transformation PAPPUS (290–350). See under theorem opposite. See diametrically opposite parallel postulate and under order Bolyai’s form, 155, 206, 209, 278, 285 order independence, 125, 287 on a circle, 252 Playfair’s form, 149, 155, 274, 277 in elliptic geometry, 356, 360–61 parallelism, 229 on a line, 251 of line and plane, 232 opposite, 252, 254 of lines, 155, 229 See also betweenness; cyclic order; larger- of planes, 231 than; and under field; postulate PARDUCCI, Amos (1877–1949), 5, 79, 97, 432 ordinal number, 297, 313, 323 Parma, xvii, xviii, 11, 41, 44 See also under arithmetic university, 16, 41, 44, 90, 460 ordinario (see professore) PARSONS, Charles Dacre (1933– ), ORE, Øystein (1899–1968), 414 300, 305, 432 orientation, 145, 247, 271 PASCAL, Blaise (1623–1662), 117 origin, 257 See also under theorem of half-plane, 202 PASCAL, Ernesto (1865–1940), 110, 398 of ray, 200 PASCH, Moritz (1843–1930), 54, 95–96, orthogonality, 154–55, 280 132, 161, 197, 398, 415, 418, of line and plane, 147, 183, 188, 272 424, 432–33, 436, 449, 460 of lines, 147, 155, 179, 272, 280, 362 1882b Lectures on Geometry, 58, 126–28, of planes, 192 146–48, 338, 351, 394 See also under symbol on real numbers, 289–90, 293–94, 305 OSTROWSKI, Alexander Markowich See also under postulate; real- (1873–1986), 374, 410, 438, 457 number arithmetic) OTTE, Michael (1938–), 404 PÀSCOLI, Giovanni (1855–1912), 96 PEANO, Bartolomeo (????–1888), 332, 333 pacifism, 74, 102 PEANO, Bartolomeo (????–????), 332, 333 PADOA, Alessandro (1868–1937) vi, 94–95, PEANO, Carola Crosio (????–1940), 333 167, 281, 316, 318, 320–27, 337, 341–42, PEANO, Francesco (????–????), 332, 333 392, 398, 419, 424, 431, 446 PEANO, Giuseppe (1858–1932), v–ix, 20, [1900] 1901 Essay on an Algebraic 53–54, 59–60, 69, 71, 73–74, 82, 84, 87, Theory of Integers, 320–22, 360 95, 96, 102, 107–8, 114–15, 127, 129, [1900] 1902 New System of Definitions for 156, 167, 294, 299–305, 303, 315–22, Geometry, 148, 161, 278, 281, 341–42 327–28, 331–47, 339, 363–68, 380, 392, 1902 Theory of Natural Numbers, 395–96, 398, 409, 412, 417, 419, 424, 308, 310, 314, 321–23 428, 431–36, 446, 450, 453, 460 method, 321, 360–62 [1888] 2000 Geometric Calculus, Padua, xviii viii, 53, 84, 334, 433 university, 16 1889 Principles of Geometry Presented PAGLI, Paolo (19??–), 424 Logically, 128, 334–35, 338, 340, 433 PAGLIERO, Giuliano (18??–19??), 398 482 Index

PEANO, Giuseppe (continued) pedagogy, mathematical, 40, 98, 108, [1889] 1973 Principles of Arithmetic, 327, 336, 346, 351, 365, 367 58, 294, 301, 303, 308, 313–14, 316, for arithmetic, 322–23, 326, 395–96 318–20, 324, 334–35, 343, 433 for geometry, 29, 117, 127, 147–48, 1891d Concept of Number, 150, 153–54, 157, 160, 162, 317–18, 320–21, 336–37, 434 266, 286–88, 348, 393–94 1894 Fundamentals of Geometry, See also Mathesis 127, 147, 338, 340, 351, 434 PEDRAZZI, Maino (19??–), 156, 436 1895–1908 Mathematical Formulary, 66, PEIRCE, Charles Sanders (1839–1914), 69, 71, 269, 301, 306, 317–18, 321, 336– 296, 316, 334, 413, 436 37, 342–43, 344, 345, 364, 409, 436 PEJLARE, Johanna (19??–), 1903 Geometry Based on Ideas of Point and 68, 153–54, 407, 436 Distance, 161, 280, 341, 351, 394, 434 PELLEGRINI, Aldo (????–1940), 6 Academia pro Interlingua, 42, 49, 345 PELLEGRINI, Geminiano (18??–????), 5–6 birth, 332 PELLEGRINI, Paolina Livia Pieri calculus text, 82, 333, 336, 418 (1865–1959?), 5–6 death, 346 PENNACCHIETTI, Giovanni (1850–1916), differential equations research, 334 32, 34, 37, 96, 413 early career, 333 PENSA, Angelo (18??–19??), 286, 436 family, 332–33 pentaspherical. See under coordinates Lingua (see Latin: Uninflected) periphery. See boundary linguistics research, 345, 364–65 permutable transformations, 270 logic, view of, 59, 301, 305–6, 308, perpendicularity. See orthogonality 315, 318–19, 327, 340, 342, 386 PETERSEN, Julius Peter Christian Pieri, relationship with (1839–1910), 120–22 (see under Mario PIERI) PETKANTSCHIN, Boyan Lazarov ($* 4 :, politics, 333, 346, 365 #E! '# , 1907–1987), 143, 436 postulates (see under natural- PETRI, Cesare (18??–????), 398 number arithmetic) PFAFF, Johann Friedrich (1765–1825), 93 praising or supporting Pieri, viii, 27, 38, PFANNER, Pietro (1864–1935), 48–49 43, 56, 124, 131, 135, 153, 325, 329, PHILLIPS, Esther Rodlitz (1933–), 430, 436 378, 399, 414–15, 439–41, 443 philosophy of mathematics and science, professorship at Turin, 335, 346 80, 125–28, 147, 160, 162, 266–67, Segre polemic, 108, 337, 449 286, 341, 343, 381–82, 394 schooling, 332 physics, mathematical, 123, 394 students, 71, 80, 89, 94, 365, 396 PICARD, Charles Émile (1856–1941), 64, 74 symbolic notation, ix, 59, 87, 126, 301, PICKERT, Günter (1917–), 96, 436, 460 306, 316, 319, 325, 334, 336, 338, 340, Piedmont, xvii, xix, 11, 16, 18 342–45, 364–367, 383, 395–96 PIERI, Alfonso (nephew, 1890–1890), 5 space-filling curve (see under curve) PIERI, Angiolina Anastasio Janelli (wife, teaching, 108, 333, 345–46, 364, 395 18??–19??), 5–6, 36, 45, 385, 391, 459 university studies, 332 PIERI, Beppina Bastian (sister-in-law, Veronese polemic, 117 ????–????), 5 See also school: Peano and under PIERI, Enrica Montanari (sister-in-law, real-number arithmetic 18??–????), 5, 27, 97 PEANO, Michele (????–????), 332–33 PIERI, Erminia Luporini (mother, 1826–1914), PEANO, Rosa (????–????), 332 4–5, 39 PEANO, Rosa Cavallo (????–1910), 332 PIERI, Felice Ettore Pacifico Giovanni PEARSON, Patricia M. Cowan (1928?–), (brother, 1866–1920?), 5–6 x, 145, 151, 369–70, 436 Index 483

PIERI, Ferruccio Fabio (brother, PIERI, Mario (continued) 1864–1933?), 5–6, 39, 390, 436 1908a Point and Sphere, viii, 3, 58, 84, 90, PIERI, Gemma (niece). See Gemma MERIANO 145, 153–271, 278, 280–87, 340, 349–51, PIERI, Gemma (sister). See Gemma CAMPETTI 355–58, 361, 366–68, 371, 381 PIERI, Maria Dal Poggetto (sister-in-law, 1911d, 1912c Geometry of Inversions, 18??–19??), 5–6, 39 137, 141, 145, 381 PIERI, Mario (1860–1913), i, 19, 24, 130 on algebraic geometry, viii, 1, 12, 38, 50– 53, 1884b,c dissertations, 73, 75, 99, 103, 343, 363–64, 370, 374–78 12, 14, 15, 50, 374–75, 459 archive (see Mario PIERI: Nachlass) 1889a translation of Staudt 1847, 20, 107, on arithmetic and logic, vii–viii, 1, 3, 38, 111, 123, 137, 335–336, 338, 393 61, 77, 290, 305–13, 320, 322–29, 1889b Triple Tangents, 20, 21, 375 342, 363, 370, 381–382 1895a, 1896a,b Principles Governing birth, 4 Geometry of Position, 124, 128, burial, 48–50, 90, 459 137, 305–6, 338, 379 chair quest, 22– 32, 384 1896c Postulates for Projective Geometry character, 2, 23, 42–43, 47, 50, of hyperspaces, 128, 379 124, 131, 363, 383, 390–92 1897c Primitive Entities of Projective death, 2, 6, 47 Geometry, 128–29, 380 on differential geometry, 1898b New Method for Developing viii, 1, 15, 50–51, 53, 374 Projective Geometry, 133, 380 early career, 5, 17, 20, 22, 27, 31, 383 1898c Principles of Geometry of Position, editing De Paolis 1892, 78, 123–24, 393, 414 38, 57, 59, 102, 120, 129, 131, 134, 137, election to academies, 34 306–307, 327, 338, 340, 343, 367, family, 3–8, 9, 17, 27, 36, 37, 39, 42–44, 45, 371, 380 46–49, 96, 382–83, 385, 388, 390–91, 459 1900a Point and Motion, 57–58, 84, 124, finances, 8, 34, 43–44, 382, 388, 390 126, 134, 145, 148–57, 271–74, formulas, ix, 1, 52 276–78, 280–82, 285–87, 327, on foundations of geometry, vii, ix–x, 1, 32, 340–343, 367, 369–71, 381 38, 54–58, 61, 86, 118, 123–25, 145–56, [1900] 1901 Geometry as a Purely Logical 305–6, 327–28, 331, 340–42, 347–51, 363, System, 1, 36, 74, 126–27, 148, 153, 278, 366–70, 374, 381, 385, 394–95 306, 320–22, 327, 341–43, 379, 381 on foundations of projective geometry, 1901b Principles of Line Geometry, vii, 99, 111, 121, 128–37, 287, 124, 134, 380 328, 336, 367, 370, 379–80 1904a Staudt’s Fundamental Theorem, health, 8, 43, 46–47, 117, 388–89, 391–92 137, 306, 336, 343, 380 interest in pedagogy, 20, 49, 147, 154, 1905c, 1906a Complex Projective 160, 162, 265–67, 286–87, 322, Geometry, 134–35, 380 326–27, 368, 392–96 1906d New Logico-Mathematical Direction, on inversive geometry, vii, 46, 137, 139, 289, 304, 307, 343, 382 141–42, 145, 363, 370, 381, 443 1906e Definition of the Irrationals, knighthood, 2, 42, 389 289, 328, 343, 382 lecture notes, published, 1906f Staudtian Homography ix, 20, 22, 46, 397–98 Definition, 137, 380 lecture notes taken by, 12, 392, 397–99 1906g Consistency of Arithmetic, legacy, 2, 61, 115, 363–71 60, 74, 289–90, 306–308, 326–27, letters from, ix, 382–92, 399 340, 343, 363, 382 letters to, vi, ix, 382, 398, 403 1907a Axioms of Arithmetic, viii, 3, 58, 289, libera docenza, 20, 22 300–301, 308–13, 321–28, 343, 370, 382 licenza, 12 484 Index

PIERI, Mario (continued) PIERI, Teresa Ricci (grandmother, logic, view of, 59, 102, 272–73, 289–90, ????–18??), 4 306–8, 327, 340, 342–43, 363, 386 PIERI, Teresa (sister). See Teresa BRANCOLI marriage, 5–6, 36, 385 PIERI, Virginia (sister) Mars drawing, 8, 9, 397, 459 See Virginia ANASTASIO music, 7, 42 PIERPONT, James (1866–1932), 342, 444 Nachlass, ix, 6, 398–99 PIESYK, Zbigniew (????– ), 355, 420 obituaries of, 49, 61, 90, 399 PINCHERLE, Salvatore (1853–1936),10, 25–29, Peano, relationship with, ix, 22, 27, 42, 49, 30, 46, 91, 97–98, 101, 108, 292, 398, 59, 338, 340, 342–43, 345, 363, 386 402, 444, 449, 453, 455, 459 philosophy of mathematics, viii, x, PINL, Maximilian (1897–1981), 454 102, 147, 160–62, 266–67, 289, PIRONDINI, Geminiano (1857–1914), 306–8, 343, 363, 370, 386 10, 98–99, 426 physical stature, 8, 43 Pisa, xvii, xviii, 11 Poincaré dispute, 58, 74, 290, 326, 343 Scuola Reale Normale Superiore, 4, 8, professorship at Catania, 4, 32, 10–12, 13, 17, 23, 50, 66–67, 78, 459 34, 38, 384–85, 387, 390 university, 10–12, 15–16, 50, 459, 461 professorship at Parma, 4, 44, 46–47, 390 PITTARELLI, Giulio (1852–1934), 36, 411 promotion, 36, 37, 38, 386, 459 PLANA, Giovanni (1781–1864), 82 as referee, 34, 42, 388–89 plane relation (see equidistance: ternary) antipolar, 238 reviews by, ix, 38, 393–97 defined, 129, 146, 154, 167, schooling, 5, 7–8, 100, 382, 397 170, 272, 281, 341 student records, 8, 10, 12, 382–83, 459 at infinity, 277 students, 34, 38, 40, 70, 75, 82, 92, of symmetry, 187 104, 388–89, 393, 395–97 undefined, 149, 274 university studies, 5, 8, 10, reflection (see under reflection) 12–15, 382–83, 397 See also under polar plane; symbol on vector analysis, viii, 1, PLATO (427?–347? B.C.), 436 46, 53, 343, 364, 378 PLAUMANN, Peter Klaus (19??–), 424, 444 workload, 33–34, 44, 384, 387–90 PLAYFAIR, John (1748–1819). See under works (collected or listed), vi, ix, 3, 50, parallel postulate 329, 373–74, 379, 392, 398, 401 PLÜCKER, Julius (1801–1868), See also under notation; style; successor 53, 86, 134, 139, 444 PIERI, Nisedi (mother of sister-in-law, poetry, 5, 6, 8, 97, 101, 337 18??–????), 6 POGGENDORFF, Johann Christian PIERI, Paolina (sister). See Paolina (1796–1877), 399, 444 PELLEGRINI POGODA, Zdzisãaw (????– ), 350, 411 PIERI, Pellegrino (father, 1826–1882), POINCARÉ, Jules Henri (1854–1912), 4, 5, 17, 96, 443 38, 84, 86, 116, 133, 421, 445 PIERI, Pellegrino [Rinuccio] (nephew, See also under Mario PIERI 18??–19??), 5, 8, 97, 398 Point and Motion. See under Mario PIERI PIERI, Piero (nephew, 1893–1979), 5, 97 Point and Sphere. See under Mario PIERI PIERI, Rinuccio point (defined), 349 See Pellegrino [Rinuccio] PIERI point reflection. See under reflection PIERI, Romano (grandfather, ????–18??), 4 Poland under Russia, 347, 367 PIERI, Silvio Dante (brother, 1856–1936), polar plane 5, 7–8, 9, 17, 34, 39, 49, 79, 96–97, for sphere, 238 398, 405, 432, 444, 459 for two points, 188 polar system, elliptic, 278 Index 485 pole-placement problem, 52 projective geometry, vii, 8, 11, 20, 22, 51, poles of a sphere, 194 56, 71–73, 99, 116, 121, 125, 143, polyhedra, 17 277–78, 283, 379, 393, 397 PONCELET, Jean-Victor (1788–1867), 104 analytic, 60, 93, 140 PONZI, L. (18??–19??), 397, 443 complex, vii, 52, 56, 111, positivism. See logical positivism 134–35, 145, 370, 380 postulate, 54, 95, 162, 266–68 consistency, 59–60 American theorists, 116 finite, 116 Archimedes, 117, 137, 141, 213, 273, foundations, vii, 1, 53–54, 62, 78, 80, 276, 278, 284–85, 287, 336, 355 95, 106, 111, 116, 120, 128–33, circle, 355, 358, 361–62 284, 338, 371, 379–80 completeness, 149, 151, 276, 350 multidimensional, vii, 51–54, 57, 65, congruence, 146, 149, 274 80, 107, 117, 128–29, 135, 145, continuity, 141, 146–47, 149, 151, 155, 263– 338, 374, 376, 379, 396 64, 273, 277–78, 285, 287, 353, 355, 362 protothetic, 348, 350 Euclid (see parallel postulate: Bolyai) provinces, xvii incidence, 11, 95, 146, 149, 274 See also Emilia–Romagna; Piedmont; order, 95, 149, 274 Tuscany; Sicily Pasch, 95, 196, 198, 274, 283, 353, 356, 451 36 sentence. See under sentence parallel (see parallel postulate) PUCCINI, Giacomo (1858–1924), 4 SAS congruence, 149, 150, 274 PYTHAGORAS (569? B.C.–475? B.C.), 41, 263 See also axiom Pythagorean. See under field postulates, choice of, 126, 267, 282, 286, 394 postulates, Peano. See under quadric. See under surface. natural-number arithmetic quadrilateral. See complete quadrilateral POTTS, Donald (1921–2001), x quantifier, 271, 348 preceding, 251–52 elimination, 351, 361 predicate, 298–99 prefix, 271 prefix. See under quantifier over sets, 315 PRESBURGER, Mojz˙esz (1904–1943?), 445, 351 quasi-interpretable. See under theory PRESS, Volker (1939–1993), 420, 436 Queen’s University, 460 PRIHONSKY, Frantisek (1788–1859), 405 primitive notions, v, vii, 54, 126–27, 147, 149, RABIN, Michael Oser (1931–), 352, 417 266–68, 320–21, 350, 358, 360–63, 394 RANUCCIO I. See Ranuccio I of FARNESE binary relations as, 360–363 RASCHI, Leonida (1831–1917), 46, 99 Princeton rational number, 392 Institute for Advanced Study, 116 See also under arithmetic university, 116, 461 RAUSENBERGER, Otto (1852–19??), 263, 445 PRISTEM, 403, 445, 460 ray. See half-line Privatdozent, 11 RAZZABONI, Amilcare (1855–1920), 26 product real-closed. See under field of cardinals, 297 real enumerative. See under geometry of natural numbers, real-number arithmetic, 257–58, 263–64, 299–300, 314, 318–19, 344 276, 278, 285, 288, 289–92 of ordinals, 297 Cantor, 293 of scalars, 354 Dedekind, 76, 293–94, 299 of transformations, 270, 273 Pasch, 293 (see also under symbol) Peano, 293, 334 professore (all ranks), 11, 16 Russell, 293 progression, ascending or descending, 254–55 Weierstrass, 292, 305 486 Index

Realschule, 7 RIEMANN, Georg Friedrich Bernhard reciprocal (1826–1866), 66, 76, 86, 98–99, 125 radii (see transformation: inversion) sphere, 140 See also function: bijection See also under theorem recursive definition. See under definition RIGHI, Augusto (1850–1920), 8, 9, 25, recursive function theory, 315 100, 397, 409, 445, 451, 461 recursive proof. See induction Right angle (see under angle) Reed College, 369–70 RINDI, Scipione (1859–1952), 3, 4, 6–8, 20, reflection, 154 22, 34, 36, 47, 49, 61, 101, 123–24, hyperplane, 140 375, 398–99, 408, 438, 446 line, 139, 147, 155, 164, 176 (see also RINOW, Willi (1907–1979), 425 transformation: ribaltamento) Rivista di matematica. See under journals plane, 140, 145, 147, 155, 178, 186–87, 248 Rivista populare di politica, lettere e scienze point, 145, 155, 164, 174–75, 280 sociali. See under journals glide, 245, 248 RIZZI, Fortunato (1880–1965), 99, 446 rotary, 243, 248 ROBERTSON, Edmund Frederick (1943–), See also under symbol 87, 431 reflexive. See under relation ROBINSON, Raphael Mitchel (1911–1995), region. See under Italy 361–62, 446 regular open set, 349 RODRIGUEZ-CONSUEGRA, Francisco A. (1951–), REID, Constance (1917–), 85, 100, 445, 460 x, 294, 307, 328, 341, 386, 429, 446 REINHARDT, Curt (1855–1938?), 430 ROERO, Clara Silvia (1952–), vi, 11, 18, relation, 249, 269 32, 69, 74, 95, 341, 418, 428, 446 Pieri’s (see equidistance: ternary) Peano CDs, vi, 336, 345, 347, reflexive, 165 377–79, 393–94, 446 symmetric, 165 ROGERS, Reginald Arthur Percy transitive, 165 (1874–19??), 447 univocal, 268 (see also function) ROHN, Karl (1855–1921), 451 See also betweenness; cyclic order; ROLLAND, Romain (1866–1944), 88–89, 446 equality; equidistance; equilaterality; ROMAGNOLI, Raffaello (19??–), x larger-than; order Roman Catholic Church, xvii, 4 RENSCHUCH, Bodo (19??–), 105, 408 Rome, xvii, xviii, 11 representation, 266 archives, 459 See also transformation university, 16, 22–23, 75 RESTA, Emilia Levi (1921–), 90–91, 460 Rosario, university, 91 Revista de mathematica. See under journals ROSENTHAL, Arthur (1887–1959), 451 reviews, sources of, 402 ROSS, Robert Hemphill (1949–), 403 revolutions. See 1848. ROSSI, Luigi (18??–????), 398 Revue de métaphysique et de morale. ROTA, Gian-Carlo (1932–1999), 148, 446 See under journals rotary reflection. See under reflection Revue des mathématiques. rotation. See under transformation See under journals ROTHE, Rudolf Ernst (1873–1942), 378, 443 REYE, Carl Theodor Reye (1838–1919), ROWE, Charles Henry (1893–1943), 447 57, 71, 72, 99–100, 106, 136, ROWE, David E. (1950–), x, 56, 446 137, 139, 418, 445, 448, 453 ROYDEN, Halsey Lawrence (1928–1993), REYES Y PRÓSPER, Ventura (1863–1922), 398 361, 447 ribaltamento. See under transformation RUFFINI, Ferdinando Paolo (1823–1908), RICCÒ, Annibale (1844–1919), 8, 28, 101, 444 17, 32, 37, 398, 445 RUFFINI, Paolo (1765–1822), 101 RICHARD, Jules (1862–1956), 326 ruled. See under surface Index 487

RUSS, Steven (19??–), 405 SCACCIANOCE, Rosario (1880–19??), RUSSELL, Bertrand Arthur William (1872– 40, 43, 104, 389, 410, 447 1970), viii, x, 47, 74, 102–3, 120, 125, scale determined by two points, 155, 212, 354 300, 304–5, 327, 339, 341–42, 392, 402, Scandinavian Congress of Mathematicians, 406, 417, 419–20, 428, 446–48, 456 1911, 153, 448 1900 Paris epiphany, 74, 102, 302, 341 SCANLAN, Michael (1957?–), 116, 278, 360, 448 1903 Principles of Mathematics, SCHAPPACHER, Norbert (19??–), 421 38, 102, 129, 249, 290, 294, 302, SCHAYER, W. (????–????), 348, 452 304, 328, 338, 386, 392 SCHEIBNER, Wilhelm (1826–1908), 430 Archives, 416, 459 SCHEPP, Adolf (1837–????), 418 letter from Pieri, 307–8, 386, 398, 441, 459 SCHIFF, Hugo Josef (1834–1915), 332 on real numbers, 289–90, 293–94, SCHILLING, Friedrich Georg (1868–1950), 425 regard for Pieri, v, 2, 38, 129, 328, 338, 343 SCHLEGEL, Viktor (1843–1905), view of logic, 386, 307, 327, 386 380, 408–9, 433–34, 440, 448 view of mathematics, 307 SCHLESINGER, Ludwig (1864–1933), 41 See also under real-number arithmetic; SCHMID, Anne-Françoise (1949– ), 305, 448 Russell’s antinomy; A. N. WHITEHEAD SCHNEEWIND, Jerome Borges (1930–), 428 Russell’s antinomy, 102, 302, 304–5, 447 SCHOENEBERG, Bruno (1906–1995), 87, 408 Frege’s response, 302, 304, 417 SCHOLZ, Heinrich (1884–1956), 454, 456 RUSSO, Achille (1866–1955), 37 SCHÖNFLIES, Arthur Moritz (1853–1928), RUZIEWICZ, Stanisãaw (1888–1941), 450 377, 398, 438–39 school SAINT-SAËNS, Camille (1835–1921), 88 Hilbert, vii, ix, 84–85, 143, 284 Salesians of Don Bosco, 460 Peano, vii, 1, 2, 22, 27, 32, 40, 54, 58– SALKOWSKI, Erich (1881–1943), 59, 66, 69, 71–74, 80, 84, 95, 102, 405, 412, 431–32, 448, 456 108, 110, 114, 80, 123, 126–27, 131, Salmon, George (1819–1904), 153, 300–301, 305, 308, 320, 326, 331, 52, 103, 117–18, 419, 429, 447 335–38, 341–42, 346, 363–66, 369, 405 SALUTA, Andrea (18??–19??), Segre, vii, ix, 1, 22, 32, 51, 72, 107, 40, 104, 410, 447 123, 131, 333, 337–38, 346 SALUZZO, Giuseppe Angelo, Count of system (see under Italy) Monesiglio (1734–1810), 18 Tarski, viii, 58, 113, 352–53, SANNIA, Achille (1823–1892), 79, 85, 93 356, 361, 367–68, 370 SANTAGATA, Domenico (1812–1901), 8, 10 Volterra, 123 Sant’Andrea di Compito, 6, 44, 47, 49, 459 See also logical positivism; middle school; Sardinia normal school; postulate: American kingdom, xvii, 18 theorists; technical school; Vienna Circle region, xvii, xix, 16 Schraubenbewegung SAS. See under postulate See transformation: screw SAVI, Paolo (1798–1871), 70 SCHRÖDER, Friedrich Wilhelm Karl Ernst SAVOIA, SAVOJA. See SAVOY (1841–1902), 104, 316, 334, 341 SAVOY, Carlo Emanuele II of (1634–1675), 18 SCHRÖTER, Heinrich Eduard (1829–1892), Savoy dynasty, xvii, 18 95, 105 SAVOY, Umberto I of (1844–1900), 70 SCHUBERT, Hermann Cäsar Hannibal SAVOY, Vittorio Amedeo III of (1725–1796), 18 (1848–1911), vii, 52, 104–5, 112, 325, SAVOY, Vittorio Emmanuele II of 396, 398, 408, 417, 424, 448, 453, 456 (1820–1878), xvii, 18 calculus, 52, 104, 112 SAVOY, Vittorio Emmanuele III of SCHUBRING, Gert (19??–), 406, 448 (1869–1947), 423 SCHUMANN, Richard (1864–1945), 456 488 Index

SCHUR, Friedrich (1856–1932), sequence, fundamental 41, 86, 99–100, 105–6, 124, See fundamental sequence 148, 263, 398, 448, 453 sequent of natural number, 311, 324 SCHUR, Issai (1875–1941), 148 series, fundamental SCHWABHÄUSER, Wolfram (1931–1985), See fundamental sequence 350, 354–57, 360–62, 448 SERNESI, Edoardo (19??–) SCHWARTZE, Heinz (1924–2006), 84, 448 See under BRIGAGLIA SCORZA, Bernardino Gaetano (1876–1939), SERRET, Joseph Alfred (1819–1885), 78 46–47, 94, 106, 448 set theory, SCOTT, Charlotte Angas (1858–1931), antinomies, 69, 304 343, 422, 449 axiomatic, 290, 300, 304 screw. See under transformation Cantor’s development, 296–97, 299, 305 SCRIBA, Cristoph J. (1929–), 84, 406, 408, 413 in Dedekind, 77, 293, 299, 305, 314–15 Scuola Reale Normale Superiore. general usage, 313, 316, 324, 360 See under Pisa in Point and Sphere, 157, 294, secant space, 377, 395–96 305, 350, 352, 360 segment, 128, 146, 195, 272, 335, 394 Pieri’s natural-number model, 61, 290 open, 277 Tarski’s work, 113, 348 projective, 54, 128, 130, 131, 134–5, 338 See also class See also length and under symbol SETON-WATSON, Christopher (1918–), SEGRE, Beniamino (1903–1977), 98, 449 43–44, 449 SEGRE, Corrado (1863–1924), 20, 36, 42, SEVERI, Francesco (1879–1961), 55, 69, 107–8, 118, 383, 393, 398–99, 42, 77, 104, 114, 290, 398, 450 410, 428, 440, 449–50, 460 SEVERINI, Carlo (1872–1951), 34, 98, 108, 450 algebraic geometry, v, vii, ix, 51, 53, SFORZA, Giuseppe (18??–1927), 306, 398 72, 78–79, 87, 89, 91, 105, 123–24, shadow 133, 333, 405, 414, 418, 449 of line, 202 and Pieri’s chair quest, 23, 26, 28–29 of plane, 204 projective geometry, 53, 80, 123, 335, 393 of point, 200 rift with Peano, 108, 114, 337–38, SIACCI, Francesco (1839–1907), 346, 428, 434, 449 20, 27, 91, 109, 332, 403 supporting Pieri, 56–58, 131, 135, 145–46, SIBIRANI, Filippo (1880–1957), 157, 160, 376, 380–81, 436–37, 441–42 46, 63, 98, 109, 454 See also under school Sicilies, Kingdom of Two, xvii, 41 SEGRE, Michael (1950–), 291–92, 301, Sicily, xvii, xix, 16, 41 320, 332, 335–337, 342, 346, 449 side SEIDENBERG, Abraham (1916–1988), 96, 449 of line in plane, 203 semicircle, 252 of plane, 204 SEMPRONIO, 268 of point on line, 202 Senate. See under Italy signed integer. See under arithmetic sense, 247, 251–52 SIGNORINI, Antonio (1888–1963), 110, 450 See also cyclic order; order SILLA, Lucio (1872–1959), 88, 450 sentence similar figures, 207, 160 classification by quantifier prefix, similarity, 57, 139, 154, 161, 206–7, 281 271, 276–77, 282–83 classification, 278, 284 existential, 271 in hyperbolic geometry, 361 36, 271, 373 simply infinite system, 300, 314 universal, 271 existence, 300, 307, 314 separation. See cyclic order SINACEUR, Hourya (1940–), 351, 450 singleton. See under symbol Index 489 singular point, vii, 12, 50–51, 375 STEINER, Jakob (1796–1863), SINIGALLIA, Luigi (1864–1944), 46, 110 83, 101, 105, 139, 450 SINSHEIMER, Thomas (1943–), x STEINHARDT, Fritz (1922–1993), 426 skew lines, 246 STEINITZ, Ernst (1871–1928), SKOF, Fulvia (1932–), 71, 399, 450 404, 431, 448, 454 SKOLEM, Thoralf Albert (1887–1963), 452 STÉPHANOS, Cyparissos (1857–1918), 398 smaller-than. See larger-than stereographic projection, 138 SMID, Lucas Johannes (????–????), 143, 455 STOLZ, Otto (1842–1905), 291, 408, 425, 450 SMITH, David Eugene (1860–1944), 89, 450 STRAMBACH, Karl (1939–), 424, 444 SMITH, Helen Marie Patteson (1940–), x STRANEO, Paulo (1874–1968), 108, 450 SMITH, James Thomas (1939– ), iv, x, 357, 450 straordinario. See under professore solid, 349 STÜCKRAD, Jürgen (1948–), 52, 450 SOMIGLIANA, Carlo (1860–1955), STUDY, Eduard (1862–1930), 41, 105, 10, 79, 110, 403, 417, 450 111–12, 133, 405, 408, 415, 436, 451 SOMMER, Julius (1871–1943), 380, 440, 451 STURM, Rudolf (1841–1919), 445 SONCINI, A. (18??–19??), 397, 443 sequence, 351 SOTTILE, Frank (1963–), x, 52, 422, 450 STURMFELS, Berndt (1962–), 52, 422 SPERANZA, Francesco (1932–1998), x style, SPEZIA, Giorgio (1842–1911), 332 Hilbert, 148–51, 276, 283, 354 SPEZIALI, Pierre (1913–1995), 78, 81, 108, 450 Pieri, 58, 126, 141, 147, 150–51, 156, sphere, 160, 349 158, 162, 171, 272–73, 278, 283–86, concentric, 349 323, 340, 350, 356, 368, 370 fundamental, of anti-inversion, 238 Tarski, 350, 354, 356 in inversive geometry, 137, 140–42 successor, 58, 299, 308, 315–6, 321–22 in Point and Sphere, 58, 157, 165, 281 Frege’s definition, 298 in Point and Motion, 146, 272 Pieri’s definition, 290, 307 polar, 194, 272 Zermelo’s construction, 300 as tool, 154, 157, 283 See also under symbol tangent, 167 sum See also under Riemann; symbol of angles, 220, 228, 395 SPOGLIANTI, Maria (19??–), 129, 428 of cardinals, 297 Springer-Verlag, 460 in mereology, 349 SSS triangle-congruence. See under theorem of natural numbers, 299–300, STÄCKEL, Paul (1862–1919), 451, 455 311, 314–15, 318–19, 324 STAHL, Wilhelm (1846–1894), 432, 449, 455 of ordinals, 297 STAMM, Edward (????–1940), 345, 450 of scalars, 354 statics. See graphical statics of segments, 210, 220, 226, 272, 392 STAUDT, Georg Karl Christian von sundial, 52, 376 (1798–1867), 54, 110–11, superposition, 218 136, 408, 429, 431, 444 argument, 150, 338 1847 Geometrie der Lage, vii, 57, SUPPES, Patrick (1922– ), 297, 421, 431, 451 95, 99, 129, 160–61, 269, 450 surface collineation or homography, 207, 380 algebraic, vii, 12, 15, 20–21, complex projective geometry, 56, 134–36 51, 72, 75, 375–76, 383 Pieri’s 1889a translation of Staudt 1847, birationally equivalent, 79 20, 107–8, 123, 335, 338, 379, 393, 437 helicoidal, 51, 374 See also fundamental theorem quadric, 51, 57, 140, 375 of projective geometry ruled, vii, 51–52 STEELE, Donald A. (19??–????), 405 system, 378 STEFANINI, Annibale (18??–????), 398 Sweden, mathematics in, 68, 153–54 490 Index symbol symbol (continued) abstraction, 316 sum, 226, 228, 318 ...... D+E ­½A′ ˇ ®¾ angle notation, 163, 204 ..... Â.BC, A.BC translation, 237 ...... ¯¿A arc, 252 ...... O(AB) triangle, 163, 205 ...... A.BC , ABC between, 272, 351 ...... q-p-r, Bqpr union, 162, 164–65 ...... M F \ collinear, 357 ...... Cabc universal quantifier, 271 ...... ~x) congruent, 162–63, 216, universal quantifier string, 271 ...... 3 218, 274 ...... F =~ FU See also notation; conjunction, 162–63 ...... ) & < Giuseppe PEANO: symbolic notation difference of sets, 162 ...... A – B symmetric. See under relation disjunction, 162, 164 ...... ) Z < symmetry, 160 empty set, 316 over a line, 176 equality, 162–63, 165, 316 ...... A = B over a plane, 187 equality by definition, 163, 265 .....r ≡ AB over a point, 174–75 equidistance, quaternary, 357 ...... Dpqrs system. See simply infinite system equidistance, ternary, SZCZERBA, Lesãaw Wãodzimierz (1938–), 280, 357 ...... Ipqr, Ppqr 271, 347–48, 350–51, 354, equivalence, logical, 162–63 ...... ) ⇔ < 356, 369, 426, 451 existential quantifier, 271 ...... } x) SZEGĝ, Gábor (1895–1985), 451 existential quantifier string, 271 ...... 6 SZMIELEW, Wanda Montlak (1918–1976),

following, 163, 249, 253 .... VA,B C, VO, A,B C 350, 354–57, 360–62, 368, 406, 448 half-line, 163, 200 ...... AB half-plane, 202 ...... rP TABARRONI, Giorgio (1921–2001), 100, 451 ­½A′ homothety, 233 ...... ®¾O tactile-physical space, 160 ¯¿A image by transformation, 269 ...... R A TAJTELBAUM, Ignacy (1869–1942), 347 implication, 162–63, 316 ...... ) _ < TAJTELBAUM, Alfred. See Alfred TARSKI inclusion, 162–63, 165 ...... M I \ TAJTELBAUM, Rosa Prussak (1879–1942), 347 inequality, 162–63 ...... A =/ B TAJTELBAUM, Waclaw. See Wacãaw TARSKI intersection, 162–63, 165 ...... M  \ tangent line joining 2 points, 162, 167 ...... AB line and sphere, 181, 200 line reflection, 176 ...... /r spheres, 167 membership, 162–63, 316 ...... Y  r to a variety, 374–75, 383 midpoint, 162, 174, 357 ...... A B, Mabc triple (see Mario PIERI: 1889b) negation, 162–63 ...... ¬) TANNERY, Jules (1848–1910), 74, 325, 448 nonmembership, 162–63 ...... Y Õ r TARSKI, Alfred (1901–1983), v, viii, x, 113, orthogonality, 163, 179, 357 ...... r ] s 116, 156, 278, 288, 305, 331, 347, right angle, 357 ...... Rabc 359, 412, 416, 419, 421, 427, plane joining 3 points, 162, 170 ...... ABC 431, 448, 451–52, 460 plane reflection, 186 ...... /S [1927] 1983 Foundations of geometry point reflection, 162, 174–75 ...... /A of solids, 349–50, 368, 451 product of transformations, 270 .. Q P, Q $P [1957] 1959 What is elementary geometry, quantification (Peano’s), 316 ix, 288, 352–57, 368, 452 ray (see symbol: half-line) birth, 347 segment, 163, 195 ...... AB career at Berkeley, 113 semicircle, 252 ...... O(AB change of surname, 348 singleton, 162 ...... {X} death, 113

sphere, 160, 162, 165, 194 ....BA, Sph(A,B) definability research, 357–58, 360–61 substitution of variables, doctorate, 348 §·DC,, A 163, 316 ...... ¨¸P7§2 early employment, 348, 351–52 ©¹ABC,, successor, 311, 316, 319, 321 ....suc x, x+1 emigration to the United States, 113, 352 Index 491

TARSKI, Alfred (continued) THOMAE, Johannes Karl (1840–1921), family, 113, 347, 352 393, 439, 453 marriage, 351–352 THOMSON, William, Lord Kelvin (1824–1907), and Pieri, 349–50, 353–54, 356–57, 367–70 139, 453 politics, 347 THORUP, Anders (1943–), 424 schooling, 347 throat cancer, 2, 5–6, 46–47, 97, 391 See also under elementary geometry; TIMERDING, Heinrich Carl Franz Emil school; style (1873–1945), 100, 453 TARSKI, Janusz (1934– ), x, 113, 352 TIZIO, 268 TARSKI, Krystina. See Krystina Tarski TOEPELL, Michael-Markus (19??–), x, 148, 453 EHRENFEUCHT TOEPLITZ, Otto (1881–1940), 376, 439 TARSKI, Maria Witkowska TONELLI, Leonida (1885–1956), 98, 453 (????–1990), 113, 351 topology, general, 349–50, 352 TARSKI, Wacãaw (1903–1944), 347–48 toponymy, 444 TARTUFARI, Luigi (1864–1931), 47, 453 totalitarianism, ix, 366 Taurinorum, Augusta, 18 Transactions of the American Mathematical tautologous point. See fixed point Society. See under journals TAZZIOLI, Rossana (1962–), 34, 40, transformation, 71, 154, 161, 249, 269, 394 73, 81, 88, 92, 94, 108, 453 affinity, circular technical institute and school, 7 (see transformation: Möbius) TEITELBAUM. See TAJTELBAUM) anticongruence, 247–48 TEIXEIRA, Francisco Gomes (see also motion: indirect) (1851–1933), 375, 438 antihomography, 247 TERRACINI, Alessandro (1889–1968), anti-inversion, 238 79, 347, 453 antiprojectivity, 135 tetracyclic. See under coordinates antirotation (see reflection: rotary) theorem, 126, 267 antitranslation (see reflection: glide) Bézout, 52–53 birational, 51, 65, 72, 75, 80, 374, 377–78 Cantor-Bernstein, 435 circular, 57, 139–42 Desargues, 232, 276 collineation, 207, 394 (see also Heine–Borel, 277 transformation: homography) intermediate value, 291 conformal, 139–41, 236 Miquel, 143 congruence, 247–48 (see also motion: direct) Pappus–Pascal, 143, 276, 388 continuous, 139 Pythagoras, 263 Cremona (see transformation: birational) Riemann mapping, 141 group, 56, 57, 81, 86, 133, 146 SSS triangle-congruence, 281 holomorphism (see under function) Staudt (see fundamental theorem homography of projective geometry) homography, 56, 133, 247, 340, 377–78, 380 three perpendiculars, 185 (see also transformation: collineation) theory homothety, 233 axiomatic (see hypothetical- identity, 183, 237, 269 deductive system) inverse, 269, 273 categorical, 151, 317 inversion, 57, 137–40, 142, 238 complete, 351 involution, 137, 176, 270 consistent, 54, 59–61, 290, 306, 382 Kreisverwandtschaft decidable, 351 (see transformation: circular) quasi-interpretable, 278, 360 Möbius (see transformation: circular) THIEL, Christian (1936–), 399, 453 projectivity, 137 492 Index transformation (continued) unit point, 257 by reciprocal radii universal sentence. See under sentence (see transformation: inversion) universities, Italian. See Italian universities ribaltamento, 176 University of California at Berkeley, x, 113 rotation, 145, 183 USIGLIO (18??–19??), 33 screw, 246–47 Ustica, xviii, 5, 6, 27, 454 as a tool, 157, 278, 283–84 See also function; isometry; motion; VACCA, Giovanni (1872–1953), permutable; similarity; translation 313, 337, 341, 343, 454 (geometric); and under equality; product VAI, Gian Battista (19??–), 70, 454 transformational geometry VAILATI, Giovanni (1863–1909), See under geometry 34, 153, 333, 337, 341, 398, 454 transitive. See under relation variations. See calculus: of variations translation (geometric), 145, 237, 247 variety normal, parallel, or oblique, 242 algebraic, 51, 377 translation strategy, ix, 157–58, 162–164 Jacobian, 12, 50, 375 TREUTLEIN, Peter (1845–1912), VARIGNON, Pierre (1654–1722), 418 409, 425, 444, 449 VAROLI, Giuseppe (????–????), 109, 454 triangle, 205 VASSILIEF, Alexander (1853–1929), 398 See also under symbol VEBLEN, Andrew Anderson (1818–1906), 115 TRICOMI, Francesco Giacomo (1897–1978), VEBLEN, Oswald (1880–1960), 43, 115–116, 81, 113-115, 453 120, 279, 351, 357, 369, 388, 403, citations, 12, 18, 20, 26, 34, 428, 430, 441, 454, 461 40, 62, 67, 110, 346 1904 System of Axioms for Geometry, effect on Peano’s and Pieri’s 161, 271, 277–278, 279, 283, reputations, 346, 363–65, 453 285, 340, 360, 366–67, 448 unreliability, 42, 115, 364 complex projective geometry, 135, 137 trigonometric series, 292, 294 on Pieri 131, 133, 370 Turin, xvii–xviii, 5, 17–18, 27 VEBLEN, Thorstein (1857–1929), 115 Royal Academy of Sciences, vector analysis, viii, 1, 46–47, 53, 18, 20–21, 27, 148, 335, 346, 460 69, 71, 83–84, 334, 378 Royal Military Academy, vector space, 334 18, 19, 333, 459, 460 Vellano, 4 See also Turin, university of Venice, Istituto di Scienze, Turin, university of, 16, 18, Lettere, ed Arti, 460 20, 23, 31, 123, 332 VERGERIO, Attilio (1877–1937), 44, 46–47, 117 rift between Peano, Segre schools VERONESE, Giuseppe (1852–1917), 32, 108, 110, 114, 337, 346 117–18, 132, 447, 449, 490 Tuscany, xvii, xix, 4, 11, 16, 44 difference with Peano 129, 338, 428, 434 type theory, 102, 304 on foundations of geometry, 131, 133, 146, 148, 161, 263, 280, 342, 394, 405, 454–55 ULLRICH, Peter (19??–), 455 as referee, 22–23, 29, 36, 42 ultrasymmetric point, 212 Verschelde, Jan (19??–), 52, 422 UMBERTO I. See Umberto I of SAVOY vertex undefined concepts. See primitive notions angle, 204 UNGER, Leo (????–????), 421–22 triangle, 205 Unification. See under Italy Vienna Circle, 361, 366 union, 165, 314 VILLARI, Emilio (1836–1904), 10, 100, 118 See also under symbol VILLARI, Pasquale (1827–1917), 118 unit of measure, 258 Index 493

VITALI, Giuseppe (1875–1932), WHITEHEAD, Alfred North (1861–1947), 34, 44, 387, 390, 398, 441–42, 455 2, 102, 119–20, 131, 133, 304–5, VITTORIO AMEDEO III. See Vittorio 316, 327, 404, 428, 455–56 Amedeo III of SAVOY 1910–1913 Principia Mathematica VITTORIO EMMANUELE II. See Vittorio (with Russell), 102, 120, 304–5, 456 Emmanuele II of SAVOY WHITEHEAD, Henry (1853–1947), 120 VITTORIO EMMANUELE III. See Vittorio WHITEHEAD, John Henry Constantine Emmanuele III of SAVOY (1904–1960), 116, 120, 454 VIVANTI, Giulio (1859–1949), 337 WIELANDT, Helmut (1910–2001), 414 as reviewer, 375, 382, 407–8, WIELEITNER, Heinrich (1874–1931), 398 432–34, 438, 442, 454–56 WIENER, Hermann Ludwig Gustav See also BERZOLARI: Encyclopedia (1857–1939), 112, 148, 284–85, 456 VIVORIO (vice Prefetto, cav. uff., WIENER, Ludwig Christian (1826–1896), 148 18??–19??), 48 WIENER, Norbert (1894–1964), 148 VOGEL, Wolfgang (1940–), 52, 450 Williams College, 369 VOLTERRA, Edoardo (1904–1987), 119, 455 WILSON, Edwin Bidwell (1879–1964), VOLTERRA, Virginia Almagià (18??–19??), 119 148, 342–43, 422, 447 VOLTERRA, Vito (1860–1940), 10, 12, 30, WIRSZUP, Izaak (1915– ), 452 114, 118, 119, 123, 398, 410, 455, 460 WIRTINGER, Wilhelm (1865–1945), 430, 444 Pieri’s chair quest, 23, 25–27 WITTELSBACH, Luitpold Karl Joseph Wilhelm See also under school Ludwig of, Prinzregent von Bayern volume, 263 (1821–1912), 431 WOJTASIEWICZ, Olgierd Adrian (1916–), 428 WAELSCH, Emil (1863–1927), 377–78, 438–39 WOJTOWICZ, Wã. (????–19??), 415–16 WAERDEN, Bartel Leendert van der women (see female mathematicians) (1903–1996), 105, 143, 144, 455, 460 WOODGER, Joseph Henry (1894–1981), 452 WALKER, Gay (19??–), 420 WOODWARD, Beverly Anne WALLENBERG, Georg (1864–1924), (1934–), 417, 422, 447 421, 431, 438 World War I WANG Hao (1921–1995), 294, 304, 455 effects, ix, 74, 87, 90, 98, 100, Warsaw, city and university, 347 106, 120, 143, 347, 366, 369 WEBER, Heinrich Martin service in, 71, 90, 114–16, 119, 347, 351 (1842–1913), 263, 455 World War II WEBER, Wilhelm Eduard(1804–1891), 421 effects, viii, ix, 92, 109–10, 114, WEDBERG, Anders (1913–1978), 113 143, 352, 360, 366, 368–70 WEDDERBURN, Joseph Henry MacLaghan effects on Jews, 72, 80–81, 91–92, (1882–1948), 116 110, 113–14, 116, 352 WEIERSTRASS, Karl Theodor Wilhelm service in, 116 (1815–1897), 86, 95, 98, 105, 292–94, 300, 305, 419, 455 YOUNG, Grace Chisholm (1868–1944), 398 See also under real-number arithmetic YOUNG, Jacob William Albert WEINGARTEN, Leonhard Gottfried Johannes (1865–1948), 454, 456 Julius (1836–1910), 374, 437, 455 YOUNG, John Wesley (1879–1932), 116, 135, WEINMEISTER, Paul Franz Wilhelm 137, 142–43, 145, 449, 454, 456 (1856–1927), 444

WEISS, Richard (1948–), 418 n. See integer modulo n Well-ordering, 297–98, 313, 348 ZACHARIAS, Max (1873–19??), 404, 407, 416 WEYL, Hermann Klaus Hugo ZAPPULLA, Carmela (19??–), 129, 403 (1885–1955), 445 ZERMELO, Ernst Friedrich Ferdinand (1871– 1953), 300, 304, 307, 323, 409, 456 494 Index zero, 258–59, 310 ZEUTHEN, Frederik Ludvig Bang (1888–1959), 121 ZEUTHEN, 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 Fredericke Lawaetz (1860–1936), 121 Zurich, ETH, 460