Advertisement

Some Comments on Fisher's α Index of Diversity and on the Kazwini Cosmography

  • Oskar Maria Baksalary
  • Ka Lok Chu
  • Simo Puntanen
  • George P.H. Styan

Abstract

Biodiversity, or biological diversity, is “the variety of life on our planet, measurable as the variety within species, between species, and the variety of ecosystems” [12, 41] and the most widely applied index of biodiversity is surely Fisher's α, defined implicitly by S = αloge{1+(n/α)}, where n is the number of individuals and S is the number of species. This index αwas first proposed over 60 years ago by R. A. Fisher in a three-part joint paper with A. Steven Corbet and C. B. Williams [14]. We also present some comments on the diversity of the paintings by Johannes Vermeer (1632–1675) depicted on postage stamps updating our findings in [3]. The earliest study of biodiversity seems to be that reported in the Kazwini Cosmography c. 1283; this study involved 72 different kinds of papillons that were collected in what we believe was Baghdad in c. 900 AD. We also found some magic squares in the Kazwini Cosmography. Our list of references is annotated and contains hyperlinks to open-access and restricted-access files on the internet.

Keywords

National Library Postage Stamp Rothamsted Experimental Station Hawk Moth Edited Translation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Julie Anne Oeming Badiee (1978). An Islamic Cosmography: The Illustrations of the Sarre Qazwīnī. Ph. D. dissertation42, University of Michigan, Ann Arbor, xvii + 466 pp.Google Scholar
  2. 2.
    Martin Bailey (1995). Vermeer. Phaidon, London.Google Scholar
  3. 3.
    Oskar Maria Baksalary & George P. H. Styan (2008). Some comments on the diversity of Vermeer paintings depicted on postage stamps. Discussiones Mathematicae—Probability and Statistics, 28, 65–83.Google Scholar
  4. 4.
    BiStamp.com (2008). Stamps & paper money from ex-USSR states. Online open-access: web archive (http://www.bistamp.com/inter/BiStamp.nsf/ Title!OpenPage).
  5. 5.
    Christian Boyer (2008). Personal communication, 14 February 2008.Google Scholar
  6. 6.
    Christopher Brown (1996). Washington and The Hague: Vermeer (exhibition review). The Burlington Magazine, 138 (1117), 281–283. Full-text pdf on JSTOR (http://www.jstor.org/stable/pdfplus/887007.pdf).
  7. 7.
    Sir Richard Francis Burton (2001). The Arabian Nights: Tales from a Thousand and One Nights. Translated, with a preface and notes, by Sir Richard Francis Burton (1821–1890) (http://en.wikipedia.org/wiki/Richard_Francis_ Burton); introduction by A. S. Byatt. Modern Library, New York, 2001, xxxvi + 872 pp., ISBN 0-375-75675-2. [Reprint edition (2004), 1104 pp., ISBN 0-812-97214-7.]Google Scholar
  8. 8.
    Tracy Chevalier (1999). Girl with a Pearl Earring. Dutton, New York.Google Scholar
  9. 9.
    A.-L. de Chézy (1827). Extraits du livre des Merveilles de la nature et des singularités des choses créés par Mohammed Kazwini, Fils de Mohammed. Traduits par Antoine Leonard de Chézy (1773–1832) (http://en. wikipedia.org). In Chrestomathie arabe [45, pp. 387–516]. [Italicization in title copied from Contents page vii (vij [sic]).]Google Scholar
  10. 10.
    A. Steven Corbet & H. M. Pendlebury [Henry Maurice Pendlebury (1893– 1945)] (1934). The Butterflies of the Malay Peninsula. [First published in 1934; 2nd edition revised by A. Steven Corbet and edited by N. D. Riley, pub. Oliver & Boyd, Edinburgh, 1956; 3rd edition, revised and enlarged by Lt. Col. J. N. Eliot, 1978; 4th edition revised by Lt. Col. J. N. Eliot with plates by Bernard d'Abrera, pub. Malayan Nature Society, Kuala Lumpur, 1992.]Google Scholar
  11. 11.
    Dailymotion: Upload Videos (2008). LaLaitière-Ravaillac. Online open-access: video (http://www.dailymotion.com/related/1292171/video/xse64_ lalaitiereravaillac_ads).
  12. 12.
    Dandelion watch (2008). Glossary. Online open-access: article (http://www. naturewatch.ca/english/plantwatch/dandelion/glossary.html). Website developed in partnership with Environment Canada.
  13. 13.
    The Encyclopaedia of Islam (1960/2008). New edition prepared by a number of leading orientalists; H. A. R. Gibb et al., editorial committee. Brill Academic Publishers, Leiden, 12 vols., ISBN 90-04-09239-0, 90-04-12803-4. Online edition edited by P. Bearman, Th. Bianquis, C.E. Bosworth, E. van Donzel & W.P. Heinrichs, Brill Online, 2008. {http://www.encislam.brill.nl}{Online restricted access:} Brill Online. [Includes “al-Mu'tad.id Bi'llāh, Abu'l-'Abbās Aḥmad b. Ṭalḥa” by H. Kennedy [25], “al-Mu'tad.id Bi'llāh, Abū Ámr Ἀbbād b. Muh.ammad b. Ἀbbād.” by E. Lévi-Provençal [27], and “al-Qazwīnī, Zakariyya' ibn Muḥammad” by L. Richter-Bernburg [38].]Google Scholar
  14. 14.
    R. A. Fisher [Sir Ronald Aylmer Fisher (1890–1962). (http://www-groups. dcs.st-and.ac.uk/history/Biographies/Fisher.html)], A. S. Corbet [Alexander Steven Corbet (1897–1948). (http://www.nhm.ac.uk/research-curation/collections-library/collections-management/collections-navigator/transform.jsp? rec=/ead-recs/nhm/uls-a340905.xml)] & C. B. Williams [Carrington Bonsor Williams (1889–1981) (http://www.jstor.org)] (1943). The relation between the number of species and the number of individuals in a random sample of an animal population. Journal of Animal Ecology, 12 (1), 42–58. Full-text pdf on JSTOR (http://www.jstor.org/stable/pdfplus/1411.pdf).Google Scholar
  15. 15.
    Walther Hinz (1955). Islamische Maße und Gewichte: Umgerechnet insmetrische System, Handbuch der Orientalistik herausgegeben von Bertold Spuler, Ergänzungsband 1, Heft 1, E. J. Brill, Leiden, [v] + 66 pp. [Reprinted 1970 & translated into English as Marcinkowski [31].]Google Scholar
  16. 16.
    The Internet Movie Database (2008). Girl with a Pearl Earring, 2003, movie: United Kingdom/Luxembourg. Screenplay adapted by Olivia Hetreed from the novel [8] by Tracy Chevalier and starring Colin Firth43 as Vermeer. Online open-access: photos and reviews (http://imdb.com/title/tt0335119/).
  17. 17.
    Jonathan Janson (2007). The Complete Vermeer Catalogue. Online open-access: website (http://www.essentialvermeer.com/), Essential Vermeer Resources, updated December 7, 2007.
  18. 18.
    Kazwini (1973). Ἀja'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muḥammad ibn Maḥmūd Abū Yahyā al-Ḳazwīnī (c. 1203–1283) (http://en.wikipedia.org/wiki/Zakariya_al-Qazwini), In Arabic, edited translation by Faruq Sa'd. pub. Dar al-Afaq al-Jadidah, Bayrut [Beirut, Lebanon], 526 pp.Google Scholar
  19. 19.
    Kazwini (1996). Ἀjā'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muḥammad ibn Mahṃd Abū Yaḥya al-Ḳazwīnī (c. 1203–1283). In Persian, with citations in Arabic, edited translation by Ja'far Mudarris Sadiqi. Bazkhvani-i mutun 6, pub. Nashr-i Markaz, Tihran [Tehran, Iran], 37 + 567 pp.Google Scholar
  20. 20.
    Kazwini (2001). Miniatury arabskie [Arabic miniatures]: Al-Kazwini Kosmografia Irak 1280 r. Illustration from Ἀjā'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muḥammad ibn Maḥmud Abū Yah0323;yā al-Ḳazwīnī (c. 1203-1283). Online open-access: Krakow Pedagogical University, Krakow, Poland: © Stanislaw Skórka, updated 25 August 2001 (http://www.wsp.krakow.pl/whk/galeriar/alkazwini3.html).
  21. 21.
    Kazwini (2006). Illustrations from Ἀjā'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muḥammad ibn Mahmūd Abū Yah0323;yā al-Ḳazwīnī (c. 1203-1283), MS P 1 [44, P 1, p. 330]. Online open-access: National Library of Medicine [35], Bethesda, updated: 23 February 2006 (http://www.nlm.nih.gov/hmd/arabic/natural_hist3.html). [The manuscript consists of 335 leaves.]
  22. 22.
    Kazwini (2006). Illustrations from Ἀjā'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muḥammad ibn Mahmūd Abū Yah0323;yā al-Ḳazwīnī (c. 1203-1283), MS P 2 [44, P 2, p. 330]. Online open-access: National Library of Medicine [35], Bethesda, updated: 23 February 2006 (http://www.nlm.nih.gov/hmd/arabic/natural_hist4.html). [Manuscript dated 1788-1789.]
  23. 23.
    Kazwini (2006). Illustrations from Ἀjā'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muḥammad ibn Mahmūd Abū Yah0323;yā al-Ḳazwīnī (c. 1203-1283), MS P 3 [44, P 3, p. 330]. Online open-access: National Library of Medicine [35], Bethesda, updated: 23 February 2006 (http://www.nlm.nih.gov/hmd/arabic/natural_hist5.html). [Folio 310a on image page 8.]
  24. 24.
    Kazwini (2006). Illustrations from Ἀjā'ib al-makhlūqāt wa-gharāib al-mawjūdā t [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muh.ammad ibn Mah.mūd Abū Yaḥ.yā al-K. azwīnī (c. 1203–1283)], MS P 29 [44, P 29, p. 330]. Online open-access: National Library of Medicine [35], Bethesda, updated: 23 February 2006 (http://www.nlm.nih.gov/hmd/arabic/natural_hist6.html). [Manuscript signed by the scribe, Sayyid H. usayn Yazdī. Undated, but probably completed by 1546.]
  25. 25.
    H. Kennedy (1960/2008). al-Mu'tad.id Bi'llāh, Abu'l-Ἀbbās Aḥ.mad b. Ṭ. alḥ.a. In The Encyclopaedia of Islam [13, vol. 7, p. 760, col. 1]&online restricted-access: Brill OnLine (http://www.encislam.brill.nl/subscriber/uid =; 1417/en-try?entry =; islam_SIM-5640).
  26. 26.
    Alexander Kheyfits&Dmitry Kheyfits (2005). Fisher's α revisited: closed-form representation and computation. Journal of Interdisciplinary Mathematics, 8 (2), 215–225.MATHMathSciNetGoogle Scholar
  27. 27.
    E. Lévi-Provençal (1960/2008). al-Mu'tad.id Bi'llāh, Abū Ámr Ἀbbād b. Muh.ammad b. Ἀbbād. %u'l-Ἀbbās Aḥ.mad b. Ṭ. alh#.a. In The Encyclopaedia of Islam [13, vol. 7, p. 761, column 1] and {http://www.encislam. brill.nl/subscriber/uid=1417/entry?entry=islam_SIM-5641 online restricted-access:} Brill OnLine.
  28. 28.
    T. Lewicki (1960/2008). al-K. azwīnī, Zakariyyā' b. Muh.ammad b. Maḥmūd Abū Yah.yā. In The Encyclopaedia of Islam [13, vol. 4, pp. 865–867] and online restricted access: Brill OnLine (http://www.encislam.brill.nl/ subscriber/uid = 1417/entry?entry = islam_SIM-4093).
  29. 29.
    The MacTutor History of Mathematics Archive (2008). Online open-access: web archive created by John J. O'Connor and Edmund F. Robertson, School of Mathematics and Statistics, University of St Andrews, Scotland, UK (http://www-groups.dcs.st-and.ac.uk/ history/BiogIndex.html).Google Scholar
  30. 30.
    Anne E. Magurran (2004). Measuring Biological Diversity. Blackwell Science, Oxford.Google Scholar
  31. 31.
    M. Ismail Marcinkowski (2003). Measures and Weights in the Islamic World: An English Translation of Walther Hinz's Handbook “Islamische Maße und Gewichte” (Foreword by Professor C. E. Bosworth, F.B.A.) International Institute of Islamic Thought and Civilization (ISTAC), International Islamic University Malaysia (IIUM), Kuala Lumpur xxii + 98 pp., ISBN 983-9379-27-5. [Translation into English of the 1970 reprint of Hinz [15, (1955)].]Google Scholar
  32. 32.
    Julie Scott Meisami&Paul Starkey, eds. (1998). Encyclopedia of Arabic Literature. Routledge, London, 1998, 2 vols., xvii + 857 pp., ISBN 0-415-06808-8 (set).Google Scholar
  33. 33.
    Merriam-Webster's Collegiate Dictionary, 11th edition (2003). Merriam-Webster, Springfield, Massachusetts. Available as paper copy, on CD-ROM, and restricted-access online (https://member.collegiate.com/subscribe.php).
  34. 34.
    Sir William Muir (2001). The Caliphate: Its Rise, Decline, and Fall. Adamant Media Corporation, 626 pp., ISBN 1-40219327-0.Google Scholar
  35. 35.
    National Library of Medicine (2006). Islamic medical manuscripts. Online open-access: Catalogue: Natural History (http://www.nlm.nih.gov/hmd/ arabic/natural_hist2.html) and Glossary of terms: National Library of Medicine, Bethesda (http://www.nlm.nih.gov/hmd/arabic/glossary.html).
  36. 36.
    Oxford English Dictionary: The Definitive Record of the English Language (2002), edited by John Simpson&Edmund Weiner. 20 volume set in 5 boxes, 22000 pp.&online restricted-access: Oxford University Press (http://www.oed.com/).Google Scholar
  37. 37.
    C.G. Rice&S. Demarais (1996). A table of values for Fisher's α log series diversity index. Texas Journal of Science, 48, 147–158. [See also online open-access table: USACERL, Champaign, Illinois (http://nhsbig. inhs.uiuc.edu/general_stats/alpha.lst).]Google Scholar
  38. 38.
    L. Richter-Bernburg (1998). al-Qazwīnī, Zakariyya' ibn Muḥammad (C.600-82/C.1203-83). In Encyclopedia of Arabic Literature [32, vol. 2, pp. 637–638].Google Scholar
  39. 39.
    George Ripley&Charles A. Dana, eds. (1858). The New American Cyclopædia: A Popular Dictionary of General Knowledge, Volume IV: Brownson—Chartres, D. Appleton, New York, 16 vols., 1858–1863. [vol. IV pub. 1858. Revised as [40]. Online open-access: Google Books (http:// books.google.com/books?).Google Scholar
  40. 40.
    George Ripley&Charles A. Dana, eds. (1881–1883). The American Cyclopæ-dia: A Popular Dictionary of General Knowledge, D. Appleton, New York, 17 vols. [vol. 17 = Index by T.J. Conant. Revised version of [39] and apparently not digitized. Earlier version [39] has “New” in title!].Google Scholar
  41. 41.
    Michael L. Rosenzweig (1995). Species Diversity in Space and Time. Cambridge University Press. [Reprinted (with corrections) 1996–2002.]Google Scholar
  42. 42.
    Stuart Rossiter&John Flower (1986). The Stamp Atlas: A Unique Assembly of Geography, Social and Political History, and Postal Information. W.H. Smith, London.Google Scholar
  43. 43.
    Norbert Schneider (2000). Vermeer 1632–1675: Veiled Emotions. Benedikt Taschen Verlag GmbH, Köln.Google Scholar
  44. 44.
    Dorothy M. Schullian&Francis E. Sommer (1950). A Catalogue of Incunabula and Manuscripts in the Army Medical Library. Schuman, New York, xiii + 361 pp.Google Scholar
  45. 45.
    Baron Silvestre de Sacy (1827). Chrestomathie arabe, ou Extraits de divers écrivains arabes, tant en prose qu'en vers, avec une traduction française et des notes, A l'usage des Élèves de l'École royale et spéciale des Langues orientales vivantes, Tome III, seconde édition, corrigée et augmentée [by http://en.wikipedia.org/wiki/Silvestre_de_SacyAntoine Isaac, baron Silvestre de Sacy (1758–1838)], imprimé par autorisation du roi, á l'Imprimerie royale [Paris]. [See “Extraits du livre des Merveilles de la nature et des singularités des choses créés par Mohammed Kazwini, Fils de Mohammed” by A.-L. de Chézy [9] on pp. 387–516.]Google Scholar
  46. 46.
    George P.H. Styan (2006). Some notes on an early Islamic study of biodiversity and on some early Islamic magic squares. Unpublished manuscript, 33 pp.Google Scholar
  47. 47.
    G.P.H. Styan&S. Puntanen (2006). Fisher's alpha index of biodiversity: 1943–2005. Talk presented (by G. P. H. Styan) at the XXIIIernational Biometric Conference (IBC-2006): Montréal (Québec), Canada: 17 July 2006 [abstract #453.pdf in IBC2006-CD].Google Scholar
  48. 48.
    Sir Vincent Wigglesworth [Sir VincentBrianWigglesworth(1899–1994)(http:// en.wikipedia.org/wiki/Sir_Vincent_Brian_Wigglesworth)] (1982).Carrington Bonsor Williams: 7 October 1889–12 July 1981. Biographical Memoirs of Fellows of the Royal Society, 28, 666–684. (1982). Full-text pdf on JSTOR (http://www.jstor.org/stable/pdfplus/769914.pdf).
  49. 49.
    Wikimedia Foundation (2008). Wikipedia, The Free Encyclopedia Open-access web archive (in English) (http://en.wikipedia.org).
  50. 50.
    Wikimedia Foundation (2008). Wikipédia, l'encyclopédie libre. Open-access web archive (in French) (http://fr.wikipedia.org).
  51. 51.
    C.B. Williams (1964). Patterns in the Balance of Nature and Related Problems in Quantitative Ecology. Academic Press, London. [Biography of Carrington Bonsor Williams (1889–1981) full-text pdf on JSTOR [48] (http://www.jstor.org/stable/pdfplus/769914.pdf)].Google Scholar
  52. 52.
    Kenneth A. Wood (1983/1985). Where in the World? An Atlas for Stamp Collectors. Van Dahl Publications, Albany, Oregon. [1st printing © 1983, 2nd printing 1985].Google Scholar
  53. 53.
    Ferdinand Wüstenfeld (1848). Zakarija Ben Muhammed Ben Mahmud el-Cazwini's Kosmographie, Erster Theil: Die Wunder der Schöpfung. Aus den Handschriften der Bibliotheken zu Berlin, Gotha, Dresden und Hamburg. In German, edited translation of Part 1 of Ἀja'ib al-makhlūqāt wa-gharāib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā' ibn Muhammad ibn Mahmūd Abū Yahyā al-Kazwīnī (c. 1203–1283)], Verlag der Dieterichschen Buchhandlung, Göttingen. [Part 2 published in 1849 [54], reprinted in 1967 [55].]Google Scholar
  54. 54.
    Ferdinand Wüstenfeld (1849). Zakarija Ben Muhammed Ben Mahmud el-Cazwini's Kosmographie, Zweiter Theil: Die Denkmäler der Länder. Aus den Handschriften des Hn. Dr. Lee und der Bibliotheken zu Berlin, Gotha, und Leyden. In German, edited translation of Part 2 of Ἀja'ib al-makhlūqāt wa-gharib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyya' ibn Muḥammad ibn Maḥmūd Abū Yahyā al-Ḥazwīnī (c. 1203–1283), Verlag der Dieterichschen Buchhandlung, Göttingen. [Part 1 published in 1848 [53], reprinted in 1967 [55].]Google Scholar
  55. 55.
    Ferdinand Wüstenfeld (1967). Zakarija Ben Muhammed Ben Mahmud el-Cazwini's Kosmographie, Erster Theil: Die Wunder der Schöpfung&Zweiter Theil: Die Denkmäler der Länder. In German, edited translation of Ἀja'ib al-makhlūqāt wa-gharaib al-mawjūdāt [Book of the Marvels of Nature and the Singularities of Created Things], by Zakariyyā’ ibn Muḥammad ibn Maḥmūd Abū Yaḥyā al-Kazwīnī (c. 1203–1283), Dr. Martin Sändig oHG, Wiesbaden. [“Genehmigter Neudruck der Ausgabe von 1848–1849 des Ver-lages der Dieterichschen Bunchhandlung (Titel-Nummer 1941).” Reprint edition of [53, 54] published in 1848/1849, two volumes in one].Google Scholar
  56. 56.
    Richard A. Young&Thomas J. Glover (1996). Measure for Measure. Sequoia Publishing, Littleton, Colorado. [7th printing: May 2006].Google Scholar
  57. 57.
    YouTube: Broadcast Yourself (2008). Groupies pour La Laitière. Online video (http://fr.youtube.com/watch?v=;rcyW3bHgQbE). [The audio for this video is the song Twist and Shout (http://en.wikipedia.org/wiki/Twist_and_Shout) written by Phil Medley and Bert Russell. It was originally recorded by the Topnotes and then covered by The Isley Brothers and later by The Beatles (http://en.wikipedia.org/wiki/The_Beatles), with John Lennon (1940–1980) (http://en.wikipedia.org/wiki/John_Lennon) on the lead vocals, and originally released on their first album Please Please Me (http://en.wikipedia.org/wiki/Please_Please_Me).]

Copyright information

© Physica-Verlag Heidelberg 2009

Authors and Affiliations

  1. 1.Faculty of PhysicsAdam Mickiewicz University, ul. Umultowska 85Pozna’nPoland
  2. 2.Department of MathematicsDawson College, 3040 ouest, rue SherbrookeWestmountCanada
  3. 3.Department of Mathematics and StatisticsUniversity of TampereTampereFinland
  4. 4.Department of Mathematics and StatisticsMcGill UniversityMontréalCanada

Personalised recommendations