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Tullio Levi-Civita

March 12, 2023

This article is about the mathematician. For the mathematical symbol, see Levi-Civita symbol.

Tullio Levi-Civita, ForMemRS[1] (English: /ˈtʊlioʊ ˈlɛvi ˈtʃɪvɪtə/, Italian: [ˈtulljo ˈlɛːvi ˈtʃiːvita]; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation)[2] and hydrodynamics.[3][4]

Tullio Levi-Civita
Tullio Levi-Civita
Born(1873-03-29)29 March 1873
Padua, Italy
Died29 December 1941(1941-12-29) (aged 68)
Rome, Italy
Alma materUniversity of Padua
Known for
AwardsSylvester Medal (1922)
FRS (1930)
Scientific career
FieldsMathematics
InstitutionsUniversity of Rome
Doctoral advisorGregorio Ricci-Curbastro
Doctoral students

Biography

 
Tullio Levi-Civita

Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Mechanics (left uncovered by death of Ernesto Padova) where he met and, in 1914, married Libera Trevisani, one of his pupils.[5] He remained in his position at Padua until 1918, when he was appointed to the Chair of Higher Analysis at the University of Rome; in another two years he was appointed to the Chair of Mechanics there.

In 1900 he and Ricci-Curbastro published the theory of tensors in Méthodes de calcul différentiel absolu et leurs applications,[6] which Albert Einstein used as a resource to master the tensor calculus, a critical tool in the development of the theory of general relativity. In 1917 he introduced the notion of parallel transport[7][8] in Riemannian geometry, motivated by the will to simplify the computation of the curvature of a Riemannian manifold.[9] Levi-Civita's series of papers on the problem of a static gravitational field were also discussed in his 1915–1917 correspondence with Einstein. The correspondence was initiated by Levi-Civita, as he found mathematical errors in Einstein's use of tensor calculus to explain the theory of relativity. Levi-Civita methodically kept all of Einstein's replies to him; and even though Einstein had not kept Levi-Civita's, the entire correspondence could be re-constructed from Levi-Civita's archive. It is evident from this that, after numerous letters, the two men had grown to respect each other. In one of the letters, regarding Levi-Civita's new work, Einstein wrote "I admire the elegance of your method of computation; it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot". In 1933 Levi-Civita contributed to Paul Dirac's equations in quantum mechanics as well.[10]

His textbook on tensor calculus, The Absolute Differential Calculus (originally a set of lecture notes in Italian co-authored with Ricci-Curbastro), remains one of the standard texts almost a century after its first publication, with several translations available.

In 1936, receiving an invitation from Einstein, Levi-Civita traveled to Princeton, United States and lived there with him for a year. But when the risk of war in Europe again rose, he returned to Italy. The 1938 race laws enacted by the Italian Fascist government deprived Levi-Civita of his professorship and of his membership of all scientific societies. Isolated from the scientific world, he died in his apartment in Rome in 1941.

Among his PhD students were Octav Onicescu, Attilio Palatini, Giovanni Lampariello and Gheorghe Vrânceanu.

Later on, when asked what he liked best about Italy, Einstein said "spaghetti and Levi-Civita".[11]

Other studies and honors

Analytical dynamics was another aspect of Levi-Civita's studies: many of his articles examine the three-body problem. He wrote articles on hydrodynamics and on systems of differential equations. He is credited with improvements to the Cauchy–Kowalevski theorem, on which he wrote a book in 1931. In 1933, he contributed to work on the Dirac equation. He developed the Levi-Civita field, a system of numbers that includes infinitesimal quantities.

The Royal Society awarded him the Sylvester Medal in 1922 and elected him as a foreign member in 1930. He became an honorary member of the London Mathematical Society, of the Royal Society of Edinburgh, and of the Edinburgh Mathematical Society, following his participation in their colloquium in 1930 at the University of St Andrews. He was also a member of the Accademia dei Lincei and the Pontifical Academy of Sciences.

Works

All his mathematical works, except for the monographs, treatises and textbooks, were posthumously gathered in the six volumes of his "Collected works", in a revised typographical form amending both typographical errors and author's oversights.

Articles

  • Ricci, Gregorio; Levi-Civita, Tullio (1900), "Méthodes de calcul différentiel absolu et leurs applications" [Methods of the absolute differential calculus and their applications], Mathematische Annalen (in French), 54 (1–2): 125–201, doi:10.1007/BF01454201, JFM 31.0297.01, S2CID 120009332.
  • Levi-Civita, Tullio (1904), "Sulla integrazione della equazione di Hamilton-Jacobi per separazione di variabili" [On the integration of the Hamilton-Jacobi equation by separation of variables], Mathematische Annalen (in Italian), 59 (3): 383–397, doi:10.1007/bf01445149, JFM 35.0362.02, S2CID 123144759.
  • Levi-Civita, Tullio (1917), "Nozione di parallelismo in una varietà qualunque e conseguente specificazione geometrica della curvatura riemanniana" [Notion of parallelism in any variety and consequent geometric specification of the Riemannian curvature], Rendiconti del Circolo Matematico di Palermo (in Italian), 42: 173–205, doi:10.1007/BF03014898, JFM 46.1125.02, S2CID 122088291.

Books

  • Tullio Levi-Civita and Ugo Amaldi Lezioni di meccanica razionale (Bologna: N. Zanichelli, 1923)
  • Tullio Levi-Civita Questioni di meccanica classica e relativistica (Bologna, N. Zanichelli, 1924)
  • Tullio Levi-Civita Lezioni di calcolo differenziale assoluto (Roma: Alberto Stock Editore 1925)
    • The Absolute Differential Calculus (London & Glasgow, Blackie & Son 1927) (edited by Enrico Persico, trans. by Marjorie Long)[12]
  • Tullio Levi-Civita and Enrico Persico Fondamenti di meccanica relativistica (Bologna : N. Zanichelli, 1928)
  • Tullio Levi-Civita Caratteristiche dei sistemi differenziali e propagazione ondosa (Bologna, N. Zanichelli 1931)
  • Tullio Levi-Civita and Ugo Amaldi Nozioni di balistica esterna (Bologna: N. Zanichelli, 1935)
  • Tullio Levi Problème des N Corps en relativité générale (Gauthier-Villars, Paris, 1950, Mémorial des sciences mathématiques ISSN 0025-9187)
  • Levi-Civita, Tullio (1954), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. primo (1893−1900), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. XXX, 564.
  • Levi-Civita, Tullio (1956), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. secondo (1901−1907), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 636.
  • Levi-Civita, Tullio (1957), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. terzo (1908−1916), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 600.
  • Levi-Civita, Tullio (1960), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (PDF) (in French and Italian), vol. quarto (1917−1928), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 608.
  • Levi-Civita, Tullio (1970), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (in French and Italian), vol. quinto (1929−1937), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 670.
  • Levi-Civita, Tullio (1970), Opere Matematiche. Memorie e Note [Collected mathematical works. Memoirs and notes] (in French and Italian), vol. sesto (1938−1941), Pubblicate a cura dell'Accademia Nazionale dei Lincei, Roma: Zanichelli Editore, pp. VI, 502.
  • Levi-Civita, Tullio (2007) [1895], Pamphlets, mathematics, University of Michigan, retrieved 14 January 2017. A collection of some of his published papers (in their original typographical form), probably an unordered uncorrected collection of offprints.

See also

Notes

  1. ^ Tullio Levi-Civita. Nndb.com. Retrieved on 2011-08-14.
  2. ^ (Levi-Civita 1904)
  3. ^ O'Connor, John J.; Robertson, Edmund F., "Tullio Levi-Civita", MacTutor History of Mathematics archive, University of St Andrews
  4. ^ Tullio Levi-Civita at the Mathematics Genealogy Project
  5. ^ Goodstein, Judith R. (2018). Einstein's Italian mathematicians : Ricci, Levi-Civita, and the birth of general relativity. American Mathematical Society. pp. 115–117. ISBN 978-1470428464.
  6. ^ (Ricci & Levi-Civita 1900).
  7. ^ (Levi-Civita 1917)
  8. ^ Levi-Civita, Tullio (2022). "Notion of Parallelism on a Generic Manifold and Consequent Geometrical Specification of the Riemannian Curvature". arXiv:2210.13239 [gr-qc].
  9. ^ Iurato, Giuseppe (2016). "On the history of Levi-Civita's parallel transport". arXiv:1608.04986. Bibcode:2016arXiv160804986I. {{cite journal}}: Cite journal requires |journal= (help)
  10. ^ C Cattani and M De Maria, Geniality and rigor: the Einstein – Levi-Civita correspondence (1915–1917), Riv. Stor. Sci. (2) 4 (1) (1996), 1–22; as cited in MacTutor archive.
  11. ^ Jackson, Allyn (1996). "Celebrating the 100th Annual Meeting of the AMS". In Case, Bettye Anne (ed.). A Century of Mathematical Meetings. Providence, RI: American Mathematical Society. pp. 10–18. ISBN 0-8218-0465-0.
  12. ^ Rainich, G. Y. (1928). "Levi-Civita on Tensor Calculus" (PDF). Bull. Amer. Math. Soc. 34: 775–777. doi:10.1090/s0002-9904-1928-04644-x.

References

Biographical references

  • "Professor T. Levi-Civita, Member of Vatican Academy," The Jewish Chronicle (UK), February 6, 1942.

General references

Scientific references

Publications dedicated to his memory

External links

  •   Media related to Tullio Levi-Civita at Wikimedia Commons
  • Tullio Levi-Civita at the Mathematics Genealogy Project
  • Scienceworld biography
  • An Italian short biography of Tullio Levi-Civita in Edizione Nazionale Mathematica Italiana online.

March 12, 2023
tullio, levi, civita, this, article, about, mathematician, mathematical, symbol, levi, civita, symbol, formemrs, english, italian, ˈtulljo, ˈlɛːvi, ˈtʃiːvita, march, 1873, december, 1941, italian, mathematician, most, famous, work, absolute, differential, calc. This article is about the mathematician For the mathematical symbol see Levi Civita symbol Tullio Levi Civita ForMemRS 1 English ˈ t ʊ l i oʊ ˈ l ɛ v i ˈ tʃ ɪ v ɪ t e Italian ˈtulljo ˈlɛːvi ˈtʃiːvita 29 March 1873 29 December 1941 was an Italian mathematician most famous for his work on absolute differential calculus tensor calculus and its applications to the theory of relativity but who also made significant contributions in other areas He was a pupil of Gregorio Ricci Curbastro the inventor of tensor calculus His work included foundational papers in both pure and applied mathematics celestial mechanics notably on the three body problem analytic mechanics the Levi Civita separability conditions in the Hamilton Jacobi equation 2 and hydrodynamics 3 4 Tullio Levi CivitaTullio Levi CivitaBorn 1873 03 29 29 March 1873Padua ItalyDied29 December 1941 1941 12 29 aged 68 Rome ItalyAlma materUniversity of PaduaKnown forTensor calculus Levi Civita symbol Levi Civita connection Levi Civita field Levi Civita parallelogramoidAwardsSylvester Medal 1922 FRS 1930 Scientific careerFieldsMathematicsInstitutionsUniversity of RomeDoctoral advisorGregorio Ricci CurbastroDoctoral studentsEvan Tom Davies Albert Joseph McConnell Octav Onicescu Attilio Palatini Antonio Signorini Libera Trevisani Giovanni Lampariello Gheorghe Vranceanu Contents 1 Biography 2 Other studies and honors 3 Works 3 1 Articles 3 2 Books 4 See also 5 Notes 6 References 6 1 Biographical references 6 2 General references 6 3 Scientific references 6 4 Publications dedicated to his memory 7 External linksBiography Edit Tullio Levi Civita Born into an Italian Jewish family in Padua Levi Civita was the son of Giacomo Levi Civita a lawyer and former senator He graduated in 1892 from the University of Padua Faculty of Mathematics In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher s college in Pavia In 1898 he was appointed to the Padua Chair of Rational Mechanics left uncovered by death of Ernesto Padova where he met and in 1914 married Libera Trevisani one of his pupils 5 He remained in his position at Padua until 1918 when he was appointed to the Chair of Higher Analysis at the University of Rome in another two years he was appointed to the Chair of Mechanics there In 1900 he and Ricci Curbastro published the theory of tensors in Methodes de calcul differentiel absolu et leurs applications 6 which Albert Einstein used as a resource to master the tensor calculus a critical tool in the development of the theory of general relativity In 1917 he introduced the notion of parallel transport 7 8 in Riemannian geometry motivated by the will to simplify the computation of the curvature of a Riemannian manifold 9 Levi Civita s series of papers on the problem of a static gravitational field were also discussed in his 1915 1917 correspondence with Einstein The correspondence was initiated by Levi Civita as he found mathematical errors in Einstein s use of tensor calculus to explain the theory of relativity Levi Civita methodically kept all of Einstein s replies to him and even though Einstein had not kept Levi Civita s the entire correspondence could be re constructed from Levi Civita s archive It is evident from this that after numerous letters the two men had grown to respect each other In one of the letters regarding Levi Civita s new work Einstein wrote I admire the elegance of your method of computation it must be nice to ride through these fields upon the horse of true mathematics while the like of us have to make our way laboriously on foot In 1933 Levi Civita contributed to Paul Dirac s equations in quantum mechanics as well 10 His textbook on tensor calculus The Absolute Differential Calculus originally a set of lecture notes in Italian co authored with Ricci Curbastro remains one of the standard texts almost a century after its first publication with several translations available In 1936 receiving an invitation from Einstein Levi Civita traveled to Princeton United States and lived there with him for a year But when the risk of war in Europe again rose he returned to Italy The 1938 race laws enacted by the Italian Fascist government deprived Levi Civita of his professorship and of his membership of all scientific societies Isolated from the scientific world he died in his apartment in Rome in 1941 Among his PhD students were Octav Onicescu Attilio Palatini Giovanni Lampariello and Gheorghe Vranceanu Later on when asked what he liked best about Italy Einstein said spaghetti and Levi Civita 11 Other studies and honors EditAnalytical dynamics was another aspect of Levi Civita s studies many of his articles examine the three body problem He wrote articles on hydrodynamics and on systems of differential equations He is credited with improvements to the Cauchy Kowalevski theorem on which he wrote a book in 1931 In 1933 he contributed to work on the Dirac equation He developed the Levi Civita field a system of numbers that includes infinitesimal quantities The Royal Society awarded him the Sylvester Medal in 1922 and elected him as a foreign member in 1930 He became an honorary member of the London Mathematical Society of the Royal Society of Edinburgh and of the Edinburgh Mathematical Society following his participation in their colloquium in 1930 at the University of St Andrews He was also a member of the Accademia dei Lincei and the Pontifical Academy of Sciences Works EditAll his mathematical works except for the monographs treatises and textbooks were posthumously gathered in the six volumes of his Collected works in a revised typographical form amending both typographical errors and author s oversights Articles Edit Ricci Gregorio Levi Civita Tullio 1900 Methodes de calcul differentiel absolu et leurs applications Methods of the absolute differential calculus and their applications Mathematische Annalen in French 54 1 2 125 201 doi 10 1007 BF01454201 JFM 31 0297 01 S2CID 120009332 Levi Civita Tullio 1904 Sulla integrazione della equazione di Hamilton Jacobi per separazione di variabili On the integration of the Hamilton Jacobi equation by separation of variables Mathematische Annalen in Italian 59 3 383 397 doi 10 1007 bf01445149 JFM 35 0362 02 S2CID 123144759 Levi Civita Tullio 1917 Nozione di parallelismo in una varieta qualunque e conseguente specificazione geometrica della curvatura riemanniana Notion of parallelism in any variety and consequent geometric specification of the Riemannian curvature Rendiconti del Circolo Matematico di Palermo in Italian 42 173 205 doi 10 1007 BF03014898 JFM 46 1125 02 S2CID 122088291 Books Edit Tullio Levi Civita and Ugo Amaldi Lezioni di meccanica razionale Bologna N Zanichelli 1923 Tullio Levi Civita Questioni di meccanica classica e relativistica Bologna N Zanichelli 1924 Tullio Levi Civita Lezioni di calcolo differenziale assoluto Roma Alberto Stock Editore 1925 The Absolute Differential Calculus London amp Glasgow Blackie amp Son 1927 edited by Enrico Persico trans by Marjorie Long 12 Tullio Levi Civita and Enrico Persico Fondamenti di meccanica relativistica Bologna N Zanichelli 1928 Tullio Levi Civita Caratteristiche dei sistemi differenziali e propagazione ondosa Bologna N Zanichelli 1931 Tullio Levi Civita and Ugo Amaldi Nozioni di balistica esterna Bologna N Zanichelli 1935 Tullio Levi Probleme des N Corps en relativite generale Gauthier Villars Paris 1950 Memorial des sciences mathematiques ISSN 0025 9187 Levi Civita Tullio 1954 Opere Matematiche Memorie e Note Collected mathematical works Memoirs and notes PDF in French and Italian vol primo 1893 1900 Pubblicate a cura dell Accademia Nazionale dei Lincei Roma Zanichelli Editore pp XXX 564 Levi Civita Tullio 1956 Opere Matematiche Memorie e Note Collected mathematical works Memoirs and notes PDF in French and Italian vol secondo 1901 1907 Pubblicate a cura dell Accademia Nazionale dei Lincei Roma Zanichelli Editore pp VI 636 Levi Civita Tullio 1957 Opere Matematiche Memorie e Note Collected mathematical works Memoirs and notes PDF in French and Italian vol terzo 1908 1916 Pubblicate a cura dell Accademia Nazionale dei Lincei Roma Zanichelli Editore pp VI 600 Levi Civita Tullio 1960 Opere Matematiche Memorie e Note Collected mathematical works Memoirs and notes PDF in French and Italian vol quarto 1917 1928 Pubblicate a cura dell Accademia Nazionale dei Lincei Roma Zanichelli Editore pp VI 608 Levi Civita Tullio 1970 Opere Matematiche Memorie e Note Collected mathematical works Memoirs and notes in French and Italian vol quinto 1929 1937 Pubblicate a cura dell Accademia Nazionale dei Lincei Roma Zanichelli Editore pp VI 670 Levi Civita Tullio 1970 Opere Matematiche Memorie e Note Collected mathematical works Memoirs and notes in French and Italian vol sesto 1938 1941 Pubblicate a cura dell Accademia Nazionale dei Lincei Roma Zanichelli Editore pp VI 502 Levi Civita Tullio 2007 1895 Pamphlets mathematics University of Michigan retrieved 14 January 2017 A collection of some of his published papers in their original typographical form probably an unordered uncorrected collection of offprints See also EditLevi Civita connection Levi Civita crater Levi Civita field Levi Civita parallelogramoid Levi Civita symbolNotes Edit Tullio Levi Civita Nndb com Retrieved on 2011 08 14 Levi Civita 1904 O Connor John J Robertson Edmund F Tullio Levi Civita MacTutor History of Mathematics archive University of St Andrews Tullio Levi Civita at the Mathematics Genealogy Project Goodstein Judith R 2018 Einstein s Italian mathematicians Ricci Levi Civita and the birth of general relativity American Mathematical Society pp 115 117 ISBN 978 1470428464 Ricci amp Levi Civita 1900 Levi Civita 1917 Levi Civita Tullio 2022 Notion of Parallelism on a Generic Manifold and Consequent Geometrical Specification of the Riemannian Curvature arXiv 2210 13239 gr qc Iurato Giuseppe 2016 On the history of Levi Civita s parallel transport arXiv 1608 04986 Bibcode 2016arXiv160804986I a href Template Cite journal html title Template Cite journal cite journal a Cite journal requires journal help C Cattani and M De Maria Geniality and rigor the Einstein Levi Civita correspondence 1915 1917 Riv Stor Sci 2 4 1 1996 1 22 as cited in MacTutor archive Jackson Allyn 1996 Celebrating the 100th Annual Meeting of the AMS In Case Bettye Anne ed A Century of Mathematical Meetings Providence RI American Mathematical Society pp 10 18 ISBN 0 8218 0465 0 Rainich G Y 1928 Levi Civita on Tensor Calculus PDF Bull Amer Math Soc 34 775 777 doi 10 1090 s0002 9904 1928 04644 x References EditBiographical references Edit Professor T Levi Civita Member of Vatican Academy The Jewish Chronicle UK February 6 1942 General references Edit Segre Beniamino 1975 Parole introduttive al Convegno in Segre Beniamino Cattaneo Carlo Bompiani Enrico Colombo Giuseppe Finzi Bruno Graffi Dario Radicati di Brozolo Luigi Tricomi Francesco Giacomo eds Tullio Levi Civita Convegno internazionale celebrativo del centenario della nascita Roma 17 19 dicembre 1973 Tullio Levi Civita International congress for the celebration of the centenary of his birth Rome 17 19 December 1973 Atti dei Convegni Lincei in Italian vol 8 Roma Accademia Nazionale dei Lincei pp 171 177 ISSN 0391 805X The Inaugural address English translation of the contribution title of Beniamino Segre a commemoration describing briefly many aspects of the life and the work of Levi Civita Scientific references Edit Aczel Amir D 1999 God s Equation New York MJF Books pp 236 ISBN 1 56858 139 4 Graffi Dario 1975 L Elettromagnetismo in Levi Civita in Segre Beniamino Cattaneo Carlo Bompiani Enrico Colombo Giuseppe Finzi Bruno Graffi Dario Radicati di Brozolo Luigi Tricomi Francesco Giacomo eds Tullio Levi Civita Convegno internazionale celebrativo del centenario della nascita Roma 17 19 dicembre 1973 Tullio Levi Civita International congress for the celebration of the centenary of his birth Rome 17 19 December 1973 Atti dei Convegni Lincei in Italian vol 8 Roma Accademia Nazionale dei Lincei pp 171 177 ISSN 0391 805X Electromagnetism in the work of Levi Civita English translation of the contribution title is a survey of some of the works of Levi Civita on the theory of electromagnetism Loinger Angelo 2007 Einstein Levi Civita and Bianchi relations arXiv physics 0702244 Publications dedicated to his memory Edit Segre Beniamino Cattaneo Carlo Bompiani Enrico Colombo Giuseppe Finzi Bruno Graffi Dario Radicati di Brozolo Luigi Tricomi Francesco Giacomo eds 1975 Tullio Levi Civita Convegno internazionale celebrativo del centenario della nascita Roma 17 19 dicembre 1973 Tullio Levi Civita International congress for the celebration of the centenary of his birth Rome 17 19 December 1973 Atti dei Convegni Lincei in English French Italian and Russian vol 8 Roma Accademia Nazionale dei Lincei p 316 ISSN 0391 805X External links Edit Media related to Tullio Levi Civita at Wikimedia Commons Tullio Levi Civita at the Mathematics Genealogy Project Scienceworld biography Another short biography An Italian short biography of Tullio Levi Civita in Edizione Nazionale Mathematica Italiana online Retrieved from https en wikipedia org w index php title Tullio Levi Civita amp oldid 1133160670, wikipedia, wiki, book, books, library,

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