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Mathematics of Computation

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Math. Comp. 5 (1951), 133-160 Request permission
  • K. Inkeri, Über den Euklidischen Algorithmus in quadratischen Zahlkörpern, Ann. Acad. Sci. Fennicae Ser. A. I. Math.-Phys. 1947 (1947), no. 41, 35 (German). MR 25498
  • Harald Bergström, Die Klassenzahlformel für reelle quadratische Zahlkörper mit zusammengesetzter Diskriminante als Produkt verallgemeinerter Gaussscher Summen, J. Reine Angew. Math. 186 (1944), 91–115 (German). MR 13402
  • Not only the publications of Goodwyn to which we refer, but others, have been discussed in thorough fashion by J. W. L. Glaisher: (i) Report of the Comm. on Mathematical Tables, 1873, p. 31-33, 150; (ii) “On circulating decimals with special reference to Henry Goodwyn’s ’Table of circles’ and ’Tabular series of decimal quotients’ (London 1818-1823),” Cambridge Phil. Soc., Proc., v. 3, 1878, p. 185-206; (iii) “On a property of vulgar fractions,” Phil. Mag., s. 5, v. 7, 1879, p. 321-336. We have already quoted DeMorgan’s Statement of 1861 [MTAC, v. 2, p. 87] concerning Goodwyn: “His manuscripts, an enormous mass of similar calculations, came into the possession of Dr. Olinthus Gregory, and were purchased by the Royal Society at the sale of his [Gregory’s] books in 1842.” In the above publications of Glaisher it is stated that no trace of the papers could be found at the Royal Society. Neville’s statement, p. xi, “Goodwyn left a mass of papers, no one knows what became of them,” is therefore slightly misleading. K. G. J. Jacobi, “Über die Kreistheilung und ihre Anwendung auf die Zahlentheorie,” Jn. reine angew. Math., v. 30, 1846, p. 166-182. K. G. Reuschle, Mathematische Abhandlung, enthaltend: Neue Zahlentheoretische Tabellen, Stuttgart, 1856. A. J. C. Cunningham, Quadratic Partitions. London, 1904; Quadratic and Linear Tables. London, 1927. B. van der Pol, Verslagen van de Maatschappij Diligentia, The Hague, 1946. A copy of this diagram woven in red and white squares hangs on the wall of the reviewer’s study, a gift from the author. The curve $t = {x^t}$ has been frequently studied before. For example: (a) J. F. C. Hessel, “Über das merkwürdige Beispiel einer zum Theil punctirt gebildeten Curve das der Gleichung entspricht $y = \sqrt [x]{x}$,” Archiv Math. Phys., v. 14, 1850, p. 169-187; H. Scheffer, “Über die durch die Gleichung $y = \sqrt [x]{x}$ dargestellten Curven,” Archiv Math. Phys., v. 16, 1851, p. 133-137. The role which $\xi = {\omega ^\xi }$ plays in Cantor’s theory of transfinite numbers will be recalled; see, for example, G. Cantor, Math. Annalen, v. 49, 1897, p. 242-246 (also the English translation of P. E. B. Jourdain, Open Court, 1915, p. 195-201).
  • Herbert E. Salzer, Alternative formulas for direct interpolation of a complex function tabulated along equidistant circular arcs, J. Math. Phys. Mass. Inst. Tech. 26 (1947), 56–61. MR 20340, DOI 10.1002/sapm194726156
  • Ronald A. Fisher and Frank Yates, Statistical Tables for Biological, Agricultural and Medical Research, Oliver and Boyd, London, 1948. 3d ed. MR 0030288
  • P. L. Chebyshev, “Sur l’interpolation par la méthode des moindres carrés,” Akad. Nauk, Leningrad, Mémoires, s. 7, v. 1, no. 15, 1859, p. 1-24. Oeuvres, v. 1, p. 471-498.
  • Ronald A. Fisher and Frank Yates, Statistical Tables for Biological, Agricultural and Medical Research, Oliver and Boyd, London, 1948. 3d ed. MR 0030288
  • R. L. Anderson and E. E. Houseman, Tables of orthogonal polynomial values extended to $N=104$, Res. Bull. no. 297, Agricult. Exper. Station, Iowa State Coll. of Agricult. Mech. Arts, Statist. Sect., 1942, pp. 595–672. MR 0009153
  • D. van der Reyden, “Curve fitting by the orthogonal polynomials of least squares,” Onderstepoort Journal of Veterinary Science and Animal Industry, v. 28, 1943, p. 355-404.
  • Raymond T. Birge, Least-squares’ fitting of data by means of polynomials. Mathematical appendix by J. W. Weinberg, Rev. Modern Physics 19 (1947), 298–360. MR 0024683, DOI 10.1103/RevModPhys.19.298
  • J. W. Weinberg, “Mathematical appendix,” ibid., p. 348-360.
  • Lila F. Knudsen, A punched card technique to obtain coefficients of orthogonal polynomials, J. Amer. Statist. Assoc. 37 (1942), 496–506. MR 7132, DOI 10.1080/01621459.1942.10500651
  • M. S. Bartlett, “Properties of sufficiency and statistical tests,” R. Soc. London, Proc., v. 160A, 1937, p. 268-282.
  • E. S. Pearson, The probability integral of the range in samples of $n$ observations from a normal population. I. Foreword and tables, Biometrika 32 (1942), 301–308. MR 6641, DOI 10.1093/biomet/32.3-4.301
  • Karl Pearson, Tables of the Incomplete $\Gamma$-Function. London, 1922. E. C. Molina, Poisson’s Exponential Binomial Limit. New York, 1945. L. R. Salvosa, “Tables of Pearson’s type III function,” Annals Math. Stat., v. 1, 1930, p. 191-198. Appendix, p. 1-187. H. T. Davis & W. F. C. Nelson, Elements of Statistics with Applications to Economic Data. Bloomington, Ind., 1935.
  • H. J. Godwin, Some low moments of order statistics, Ann. Math. Statistics 20 (1949), 279–285. MR 30162, DOI 10.1214/aoms/1177730036
  • J. Neyman & B. Tokarska, “Errors of the second kind in testing ’Student’s’ hypothesis,” Am. Stat. Assn. Jn., v. 31, 1936, p. 318-326. E. Lord, “The use of range in place of standard deviation in the $t$-test,” Biometrika, v. 34, 1947, p. 41-67; “Power of the modified $t$-test ($u$-test) based on the range,” Biometrika, v. 37, 1950, p. 64-77. [RMT 897] E. L. Grant, Statistical Quality Control. New York, 1946.
  • H. B. Mann and A. Wald, On the choice of the number of class intervals in the application of the chi square test, Ann. Math. Statistics 13 (1942), 306–317. MR 7224, DOI 10.1214/aoms/1177731569
  • B. Pal, “On the numerical calculation of the roots of the equations $P_n^m(\mu ) = 0$ and $\tfrac {d}{{d\mu }}P_n^m(\mu ) = 0$ regarded as equations in $n$,” Calcutta Math. Soc., Bull., v. 9, 1918, p. 85-95 and v. 10, 1919, p. 187-194.
  • R. Grammel, Eine Verallgemeinerung der Kreis- und Hyperbelfunktionen, Ing.-Arch. 16 (1948), 188–200 (German). MR 25631, DOI 10.1007/BF00548004
  • A. Walther, Anschauliches zur Gibbsschen Erscheinung und zur Annäherung durch arithmetische Mittel, Math. Z. 42 (1937), no. 1, 355–364 (German). MR 1545681, DOI 10.1007/BF01160084
  • NBSCL, Table of the Bessel Functions ${J_0}(z)$ and ${J_1}(z)$ for Complex Arguments. 2nd ed., New York, 1947 [MTAC, v. 3, p. 25]. J. McDougall & E. C. Stoner, “The computation of Fermi-Dirac functions,” R. Soc. London, Phil. Trans., v. 237A, 1938, p. 67-104. C. Truesdell, “On a function which occurs in the theory of the structure of polymers,” Annals Math., s. 2, v. 46, 1945, p. 150 [MTAC, v. 1, p. 445].
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  • © Copyright 1951 American Mathematical Society
  • Journal: Math. Comp. 5 (1951), 133-160
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