Skip to main content
SpringerLink
Log in

On “Pre-historic” Linear Programming and the Figure of the Earth

  • Forum
  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

An important scientific problem of the seventeenth and eighteenth centuries was to determine the figure of the earth. This problem attracted the attention of many notable people, one of whom was Roger Joseph Boscovich a Jesuit priest and natural scientist who formulated—and gave an algorithm for solving—linearly constrained optimization problems whose minimand is the sum of the absolute values of the errors, i.e., a special linear regression model. After first recounting some of the rich history of this development, the present expository note focuses on the method of Boscovich, exploring its interpretation as a variant of the simplex algorithm for linear programming applied to problems of that form.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. A contradictory view was held by the aforementioned Cassini who claimed that the earth has the shape of a prolate spheroid, i.e., is flatter at the equator than at the poles. The controversy over this issue, which lasted for decades, was settled in Newton’s favor.

  2. In the language of earlier centuries, the word “nothing” is used in place of “zero.”

References

  1. Fourier, J.: Solution d’une question particulière du calcul des inégalités, Nouveau Bulletin des Sciences par la Société philomathique de Paris, 99-100 (1826)

  2. Gauss, C.F.: Theoria Motus Corporum Celestium. Perthes et Besser, Hamburg (1809)

    Google Scholar 

  3. Chvátal, V.: Linear Programming. W.H. Freeman and Company, New York (1983)

    MATH  Google Scholar 

  4. Gass, S.I., Assad, A.A.: An Annotated Timeline of Operations Research: An Informal History. Kluwer, New York (2005)

    MATH  Google Scholar 

  5. Eisenhart, C.: Boscovich and the combination of observations. In: Whyte, L.L. (ed.) Roger Joseph Boscovich, pp. 200–212. Fordham University Press, New York (1961)

    Google Scholar 

  6. Sheynin, O.B.: R.J. Boscovich’s work on probability. Arch. Hist. Exact Sci. 9, 306–324 (1973)

    MathSciNet  MATH  Google Scholar 

  7. Heyde, C., Seneta, E.: Statisticians of the Centuries. Springer, New York (2001)

    Book  MATH  Google Scholar 

  8. Farebrother, R.W.: Rogerius Josephus Boscovich. In: Heyde, C., Seneta, E. (eds.) Statisticians of the Centuries. Springer, New York (2001)

    Google Scholar 

  9. Farebrother, R.W.: Fitting Linear Relationships. Springer, New York (1999)

    Book  MATH  Google Scholar 

  10. Grattan-Guinness, I.: “A new type of question”: on the prehistory of linear and non-linear programming 1770–1940. In: Knobloch, E., Rowe, D.E. (eds.) The History of Modern Mathematics Volume III: Images, Ideas and Communities, pp. 43–89. Academic, Boston (1994)

    Google Scholar 

  11. Stigler, S.M.: Laplace, Fisher, and the discovery of the concept of sufficiency. Biometrika 60, 439–445 (1973)

    MathSciNet  MATH  Google Scholar 

  12. Stigler, S.M.: Studies in the history of probability and statistics XL: Boscovich, Simpson and a 1760 manuscript note on fitting a linear relation. Biometrika 71, 615–620 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  13. Stigler, S.M.: The History of Statistics. Harvard University Press, Cambridge (1986)

    MATH  Google Scholar 

  14. Stigler, S.M.: Statistics on the Table. Harvard University Press, Cambridge (1999)

    MATH  Google Scholar 

  15. Maire, C., Boscovich, R.J.: De litteraria expeditione per Pontificiam ditionem ad dimetiendos duos meridiani gradus, et corrigendam mappam geographicam, jussu, et auspicilis Benedicti XIV Pont. Max. suscepta. Romæ, (1755). [For a French translation, see Voyage Astronomique et Géographique dans l’État de l’Église, entrepris par l’Ordre et sous les Auspices du Pape Benoît XIV, pour mesurer deux degrés du méridien, et corriger la Carte de l’État ecclesiastique. Paris: (1770)]

  16. Stay, B.: Philosophiæ Recentioris, a Benedicto Stay in Romano Archiggynasis Publico Eloquentare Professore, versibus traditæ Libri X, cum adnotationibus et Supplementis p.Rogerii Josephi Boscovich S.J., Tomus II, Romæ (1760)

  17. Laplace, P.S.: Sur quelques points du système du monde. Mém. l’Acad. Sci. 1–87 (1789) [Reprinted in Laplace, P.S.: Oeuvres Complètes, vol. 11, pp. 447–558. Gauthier-Villars et Fils, Paris (1893)]

  18. Laplace, P.S.: Traité de Mécanique Céleste, vol. 2. Duprat, Paris (1799) [Available in English translation by Bowditch N. as Celestial Mechanics, Vol. 2, originally published in 1832 (in Boston), reprinted by Chelsea (in New York) in 1966]

  19. Whyte, L.L. (ed.): Roger Joseph Boscovich. Fordham University Press, New York (1961)

    Google Scholar 

  20. Greenberg, J.L.: The Problem of the Earth’s Shape from Newton to Clairaut. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  21. Hoare, M.R.: The Quest for the True Figure of the Earth. Ashgate, Aldershot (2005)

    Google Scholar 

  22. Smith, J.R.: From Plane to Spheroid. Landmark Enterprises, Rancho Cordova (1986)

    Google Scholar 

  23. Smith, J.R.: Introduction to Geodesy. Wiley, New York (1997)

    Google Scholar 

  24. Mazzucato, M.T.: La Figura della Terra. Libreria Clup, Milano (2003)

    Google Scholar 

  25. Gleich, J.: Isaac Newton. Pantheon Books, New York (2003)

    Google Scholar 

  26. Encyclopædia Britannica, Micropædia, vol. VI, 15th edn., p. 842 (1973)

  27. Favre, A.: Les Origines du Système Métrique. Les Presses Universitaires de France, Paris (1931)

    MATH  Google Scholar 

  28. Mayer, J.T.: Abhandlung über die Umwälzung des Monds um seine Axe und die scheinbare Bewegung der Mondflecten, Kosmographische Nachrichten und Sammlungen auf das Jahr 1748, pp. 52–183 (1750)

  29. Legendre, A.M.: Nouvelles méthodes pour la détermination des orbites des comètes. Courcier, Paris(1805) [Reissued with a supplement in 1806 and with a second supplement in 1820]

  30. Todhunter, I.: A History of the Mathematical Theories of Attraction and the Figure of the Earth. Dover, New York (1962) [Two volumes published in one. Originally published by Macmillan in 1873]

  31. Singleton, R.R.: A method for minimizing the sum of absolute values of deviations. Ann. Math. Stat. 11, 301–310 (1940)

    Article  MathSciNet  MATH  Google Scholar 

  32. Sharpe, W.F.: Mean-absolute deviation characteristic lines for securities and portfolios. Manag. Sci. 18, B1–B13 (1971)

    Article  MATH  Google Scholar 

  33. Rao, M.R., Srinivasan, V.: A note on Sharpe’s algorithm for minimizing the sum of absolute deviations in a simple regression algorithm. Manag. Sci. 19, 222–225 (1972)

    Article  MATH  Google Scholar 

  34. Zukhovitskiy, S.I., Avdeyeva, L.I.: Linear and Convex Programming. W.B. Saunders, Philadelphia (1966) [Translation of the Russian edition published in 1964]

  35. Charnes, A., Cooper, W.W., Ferguson, R.O.: Optimal estimation of executive compensation by linear programming. Manag. Sci. 1(2), 138–151 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  36. Dantzig, G.B.: Maximization of a linear function of variables subject to linear inequalities. In: Koopmans, T.C. (ed.) Activity Analysis of Production and Allocation, pp. 339–347. Wiley, New York (1951)

    Google Scholar 

  37. Bejar, J.: Regresión en mediana y la programación lineal. Trab. Estad. 7, 141–58 (1956)

    Article  MathSciNet  Google Scholar 

  38. Bejar, J.: Cálcolo prátical de la regresión en mediana. Trab. Estad. 8, 157–73 (1958)

    Article  MathSciNet  Google Scholar 

  39. Wagner, H.M.: Linear programming techniques for regression analysis. J. Am. Stat. Assoc. 54, 206–212 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  40. Gass, S.I., Saaty, T.L.: The computational algorithm for the parametric objective function. Nav. Res. Log. Q. 2, 39–45 (1955)

    Article  MathSciNet  Google Scholar 

  41. Čubranić, N.: Geodestiski Rad Rudera Boskovica. Institute of Higher Geodesy, Zagreb (1961)

    Google Scholar 

  42. Harris, T.E.: Regression using minimum absolute deviations. Am. Stat. 4, 14–15 (1950). [Answer to Question 25]

    Google Scholar 

  43. Edgeworth, F.Y.: On the use of medians. Philos. Mag. 46(Ser. 6), 1074–1088 (1923)

    Article  MATH  Google Scholar 

  44. Rhodes, E.C.: Reducing observations by the method of minimum deviations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 9(60), 974–992 (1930)

    Article  MATH  Google Scholar 

  45. Karst, O.J.: Linear curve fitting using least deviations. J. Am. Stat. Assoc. 53, 118–132 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  46. Fisher, W.D.: A note on curve fitting with minimum deviations by linear programming. J. Am. Stat. Assoc. 56, 359–362 (1961)

    Article  MathSciNet  Google Scholar 

  47. Edgeworth, F.Y.: On a new method of reducing observations relating to several quantities. Philos. Mag. 25(Ser. 5), 184–191 (1888)

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Stanford University, Stanford, CA, USA

    Richard W. Cottle

Authors
  1. Richard W. Cottle
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Richard W. Cottle.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cottle, R.W. On “Pre-historic” Linear Programming and the Figure of the Earth. J Optim Theory Appl 175, 255–277 (2017). https://doi.org/10.1007/s10957-017-1165-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-017-1165-5

Keywords

Mathematics Subject Classification

Search

Navigation