Skip to main content
SpringerLink
Log in

Officers, playing cards, and sheep

On the history of Eulerian squares and of the design of experiments

  • Published:
Metrika Aims and scope Submit manuscript

Abstract

The article presents some aspects of the history of Latin and, especially, Eulerian squares from the times even before Leonhard Euler (1707-1783) to their systematic use in the design of experiments by (Sir) Ronald Aylmer Fisher (1890–c1962).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Bachet CG (1612) Problèmes plaisans et délectables, qui le font par les nombres: Partie re-cueillis de diuers autheurs & inventez de nouveau avec leur demonstration; tres-utiles pour toutes fortes de personnes curieuses qui se servent d’Arithmétiques. Pierre Rigaud, Lyon

    Google Scholar 

  2. Bachet CG (1624) Problèmes plaisans et délectables, qui le font par les nombres: partie re-cueillis de divers autheurs, partie inventez de nouveau avec leur demonstration, 2. édition, revue, corrigée et augmentée. Pierre Rigaud et Associez, Lyon

    Google Scholar 

  3. Bachet CG (1874) Problèmes plaisans & délectables, qui le font par les nombres, troisième édition, revue, simplifiée et augmentée par A. Labosne

  4. Bachet CG (1884) Problèmes plaisans & délectables, qui le font par les nombres, cinquième édition, revue, simplifiée et augmentée par A. Labosne. Gauthier-Villars, Paris, reprinted 1959 with a foreword by Jean Itard

    Google Scholar 

  5. Bose RC, Shrikhande SS (1959) On the falsity of Euler’s conjecture about the nonexistence of two orthogonal Latin squares of order 4t + 2. Proceedings of the National Academy of Sciences of the United States of America 45:734–737

    Article  MATH  MathSciNet  Google Scholar 

  6. Bose RC, Parker ET, Shrikhande SS (1960) Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics 12:189–203

    Article  MATH  MathSciNet  Google Scholar 

  7. Bruck RH, Ryser HJ (1949) The non-existence of certain finite projective planes. Canadian Journal of Mathematics 1:88–93

    Article  MATH  MathSciNet  Google Scholar 

  8. Cochran WG: Early development of techniques in comparative experimentation. In On the History of Statistics and Probability, DB Owen, Ed. Marcel Dekker: New York 1976, pp. 3–25. Also in William G. Cochran: Contributions to Statistics. John Wiley & Sons: New York et al. 1982, pp. 105.3-105.25

    Google Scholar 

  9. Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, I Grattan-Guinness, Ed. 2 vols. Routledge: London, New York 1994

  10. Cretté de Palluel F (1788) Sur les avantages et l’economie que procurent les racines employées a l’engrais des moutons a l’etable. Memoires d’Agriculture, d’Economie rurale & Domestique, publiés par la Société Royale d’Agriculture à Paris, Année 1788. Trimestre d’été, 17–23; English translation (1790): On the advantage and economy of feeding sheep in the house with roots, Annals of Agriculture 14:133-139; German translation of the journal as Arthur Youngs Annalen des Ackerbaus

    Google Scholar 

  11. Dénes J, Keedwell AD (1974) Latin squares and their applications. English Universities Press Limited: London

    MATH  Google Scholar 

  12. Dictionary of Scientific Biography, CC Gillispie, Ed., 18 vols. Charles Scribner’s Sons: New York 1970–1990

  13. Euler L: De quadratis magicis (1776). Here in Leonhardi Euleri: Opera Omnia, Series Prima. B. G. Teubner: Leipzig, Vol. 7 (1923), pp. 441–457

  14. Euler L: Recherches sur une nouvelle espèce de quarrés magiques (1779), Verhandelingen uitgegeven door het zeeuwsch Genootschap der Wetenschappen te Vlissingen, Vol. 9 (1782), pp. 85–239. Here in Leonhardi Euleri: Opera Omnia, Series Prima. B. G. Teubner: Leipzig, Vol. 7 (1923), pp. 291–392

  15. Fisher (Sir) RA: Statistical methods for research workers. Oliver and Boyd: Edinburgh, London, numerous editions from 1925 onwards

  16. Fisher RA (Sir) (1926) The arrangement of field experiments. Journal of Ministry of Agriculture 33:503–513. Also in [18, Vol. 2, pp. 84–94]

    Google Scholar 

  17. Fisher (Sir) RA: The design of experiments. Oliver and Boyd: Edinburgh, London, numerous editions from 1935 onwards

  18. Fisher (Sir) RA: Collected papers of R. A. Fisher, JH Bennett, Ed., 5 vols. Department of Genetics, University of Adelaide: Adelaide 1971–1974

  19. Fisher RA (Sir), Yates F (1934) The 6×6 Latin squares. Proceedings of the Cambridge Philosophical Society 30:492–507. Also in [18, Vol. 3, pp. 146–161]

    Article  Google Scholar 

  20. Fisher (Sir) RA, Yates F: Statistical tables for biological, agricultural and medical research. Oliver and Boyd: Edinburgh, London, numerous editions from 1938 onwards

  21. Fisher Box (1980) R. A. Fisher and the design of experiments, 1922–1926. The American Statistican 34, No. 1:1-7

  22. Holschuh N (1980) Randomization and design: I. In R. A. Fisher: An appreciation, SE Fienberg, and DV Hinkley, Eds. Lecture Notes in Statistics 1. Springer: New York, Heidelberg, Berlin pp. 35–45

    Google Scholar 

  23. Koecher M, Krieg A (2000) Ebene Geometrie. Springer: Berlin et al. 2. edition

    Book  MATH  Google Scholar 

  24. Laugwitz D (1977) Eulersche Quadrate. In Jahrbuch Uberblicke Mathematik 1977. Bibliographisches Institut: Mannheim et al. pp. 174–180

    Google Scholar 

  25. MacMahon PA: Combinatory analysis, 2 vols. Cambridge University Press: Cambridge 1915–1916

    Google Scholar 

  26. Ozanam J (1698) Récréations mathématiques et physiques, qui contiennent plusieurs problèmes utiles & agréables, d’arithmétique, de geometrie, d’optique, de gnomonique, de cosmographie, de mécanique, de pyrotechnie, & de physique, 3. édition revue & corrigée, 2 vols. George Gallet: Amsterdam

    Google Scholar 

  27. Ozanam J (1725) Récréations mathématiques et physiques, qui contiennent plusieurs problèmes d’arithmétique, de géometrie, de musique, d’optique, de gnomonique, de cosmographie, de mécanique, de pyrotechnie, & de physique. Avec un traite des horloges elementaires, Nouvelle édition revue, corrigée & augmentée, 4 vols. Claude Jombert: Paris

    Google Scholar 

  28. Ozanam J: Récréations mathématiques et physiques, qui contiennent les problèmes et les questions les plus remarquables et les plus propres a piquer la curiosité tant des mathéma-tiques que de la physique. Nouvelle édition totalement refondue et considérablement aug-mentée par M. M[ontucla], 4 vols. Firmin Didot: Paris 1790–1791

    Google Scholar 

  29. Parker ET (1959) Orthogonal Latin squares. Proceedings of the National Academy of Sciences of the United States of America 45:859–862

    Article  MATH  MathSciNet  Google Scholar 

  30. Pearce SC (1983) The agricultural field experiment: A statistical examination of theory and practice. Wiley: Chichester

    Google Scholar 

  31. Penfold Street A, Street DJ (1987) Combinatorics of experimental design. Clarendon Press: Oxford

    MATH  Google Scholar 

  32. Stute W (1989) Der historische Streit zwischen R. A. Fisher und J. Neyman oder: Ein Sitten-gemälde aus der Blütezeit der englischen Schule für Statistik. Mathematische Semesterberichte 36:61–84

    MATH  MathSciNet  Google Scholar 

  33. Styan GPH (1987) The emperor. The Institute of Mathematical Statistics Bulletin 16, No. 6:305-306

  34. Tarry G (1900) Le problème des 36 officers. Comptes Rendus de l’Association Fran̜aise pour l’avancement des sciences: Série de mathématiques, astronomie, géodésie et mécanique 29:170–203

    Google Scholar 

  35. Tippett LHC (1934) Applications of statistical methods to the control of quality in industrial production. Manchester Statistical Society

  36. Yates F (1975) The early history of experimental design. In A survey of statistical design and linear models, JN Srivastava, Ed. North Holland: Amsterdam, Oxford, pp. 581–592

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisches Institut, Justus-Liebig-Universität, Arndtstr. 2, 35392, Gießen, Germany

    Peter Ullrich

Authors
  1. Peter Ullrich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Ullrich.

Additional information

The present article enlarges my note “An Eulerian square before Euler and an experimental design before R. A. Fisher: On the early history of Latin squares” Chance 12, No. 1 (1999), 22–26. It is based on lectures that I have given at Bern (1999), Marburg, and Augsburg (2000).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ullrich, P. Officers, playing cards, and sheep. Metrika 56, 189–204 (2002). https://doi.org/10.1007/s001840100172

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s001840100172

Key words

Search

Navigation