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Mathematics in the Pacific Basin

  • Garry J. Tee (a1)


The development of systematic mathematics requires writing, and hence a non-literate culture cannot be expected to advance mathematics beyond the stage of numeral words and counting. The hundreds of languages of the Australian aborigines do not seem to have included any extensive numeral systems. However, the common assertions to the effect that ‘Aborigines have only one, two, many’ derive mostly from reports by nineteenth century Christian missionaries, who commonly understood less mathematics than did the people on whom they were reporting. Of course, in recent decades almost all Aborigines have been involved with the dominant European-style culture of Australia, and even those who are not literate have mostly learned to use English-style numerals and to handle money. Similar qualifications should be understood when speaking of any recent primitive culture.



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1 Koestler, Arthur, The Act of Creation, Hutchinson, London, 1964, p. 622.

2 Crossley, John N., The Emergence of Number, Upside Down A Book Company, Steel's Creek, Australia, 1980, pp. 2933.

3 Pospisil, Leopold and de Solla Price, Derek J., ‘A survival of Babylonian arithmetic in New Guinea?’, Indian Journal of the History of Science, (1966), 1, pp. 3033; and ‘Kapauku numeration’, Journal of the Polynesian Society, (1977), 86, 271272.

4 Bowers, Nancy and Lepi, Punda, ‘Kaugel Valley systems of reckoning’, Journal of the Polynesian Society, (1977), 86, (1), pp. 105116.

5 Mariner, William, An Account of the Natives of the Tonga Islands in the South Pacific Ocean. With an original Grammar and Vocabulary of their language. Compiled and arranged from the extensive communications of Mr. William Mariner, several years resident in those islands. By John Martin, M.D. (2 volumes), John Murray, London, 1817, Volume 2, pp. 388391. Dr John Martin complained pedantically (Volume 2, p. 382) that the Tongans, when speaking their own language, flagrantly ignored the rules (of Latin grammar!) for distinguishing nouns from verbs.

6 Leonard, George, He Huinahelu oia ka Helunaau me ka Helukakau, i Huiia, Ka na Misionari mea Pai, Honolulu, 1852, Arithmetic textbook by a Bishop of Hawaii, using dollars and cents on p. 98; Colburn, Warren, He Helunaau ke mea e Maa'i ke Kanaka, Ellsworth, Boston, 1868. Textbook of arithmetic in Hawaiian, by a Bishop of Honolulu; Thomson, James B., Ka Huinahelu Hou; oia hoi ka Arimatika Kulanui, Papa Hoonaauao, Honolulu, 1870. Textbook of arithmetic, in Hawaiian, with dollars and cents used on p. 227. Copies of these three texts are held in the George Grey Collection, in Auckland Public Library.

7 (William Colenso) Ko Nga Tepara, He mea ta i te perehi i Paihia, 1835: single sheet with tables for addition (up to 12 + 15) and multiplication (up to 12 x 12), and tables for pounds, shillings and pence. The only known copy was acquired by the Alexander Turnbull Library in 1984.

8 Anonymous, Koe Fika Nomiba. Ko hono tolu oe tohi. Koe falakiseni moe Tesimale. Nae Buluji i he Kollij ko Tubou, 1871, p. 64. The George Grey Collection in Auckland Public Library has a copy of that textbook of arithmetic in Rarotongan.

9 Anonymous, Ai Vola Ni Fika, Printed by T.D. Hartwell, next the Wesleyan Church, Newton, 1871, p. 29. The George Grey Collection in Auckland Public Library has a copy of that textbook of arithmetic in Fijian. The author explained in the Preface that ‘As this book has been prepared solely for the use of Fijians, the wants and the mental capacity of those tribes alone have been taken into consideration.… It has not been thought advisable to extend this work beyond Vulgar Fractions; for though natives might perhaps be found capable of understanding the more advanced rules, yet neither the national character nor the national prospects for the future give sufficient promise of such knowledge being turned to any practical use; and it is to be feared that the possession of knowledge which cannot be utilised is not only useless, but positively hurtful to savages.’

10 Warner, Sylvia Townsend, Mr. Fortune's Maggot, Viking Press, New York, 1927.

11 Williams, William, A Dictionary of the New Zealand Language, (1st edn), Press of the Church Missionary Society, Paihia, 1844.

12 Conant, Levi Leonard, The Number Concept: Its Origin and Development, MacMillan, New York, 1896, pp. 122123.

13 Henrici, Peter, Elements of Numerical Analysis, John Wiley, New York, 1964, p. 291;

Э. И. Березкина ‘О математических методах древних (к историм систем сислений)’, Ист. и Метод. Естест Наук, (1982), вьІп. 29, 31–40.

(Berezkina, È. I., ‘On mathematical methods of the ancients (on the history of number systems)’, (in Russian), History and methodology of the Natural Sciences, (1982), No. 29, pp. 3140.)

14 Scholefield, Guy H. (ed.), A Dictionary of New Zealand Biography, (2 vols), Government Printer, Wellington, 1940, article on Frederick William Frankland; an obituary note on Frankland was published in the Bulletin of the American Mathematical Society, (1916), 23, p. 54. I am indebted to the referee for informing me of Frederick William Frankland. (I am also grateful to the referee for the very prompt acceptance of this paper for publication.)

15 Frankland, Frederick William, ‘On the simplest continuous manifoldness of two dimensions and of finite extent’, Transactions of the New Zealand Institute, (1876), 9, pp. 272279 (reprinted in Proceedings of the London Mathematical Society, (1877), 8, pp. 5764; and in Nature, (1877), 15, pp. 515517 and Nature, (1880), 22, 170171); ‘The non-Euclidean geometry vindicated: a reply to Mr. Skey’, Transactions of the New Zealand Institute, (1885), 18, pp. 5869.

16 Gardner, W.J., Colonial Cap and Gown: Studies in the mid-Victorian Universities of Australasia, University of Canterbury, Christchurch, 1979.

17 Grace A. Lockhart had graduated as Bachelor of Science from Mount Allison University, New Brunswick, in 1875 (Gardner16, p. 8 1).

18 Lamb, Horace, A Treatise on the Mathematical Theory of the Motion of Fluids, Cambridge University Press, Cambridge, 1879. The many later editions were entitled Hydrodynamics.

19 ‘University Plucking Match’, cartoon in Adelaide Punch, 11 January 1879.

20 Potts, R.B., ‘Mathematics at the University of Adelaide 1874–1944’, The Australian Mathematical Society Gazette, (1977), 4, pp. 19 and 3744; Radok, R., A Portrait of Horace Lamb, Mathematics Department, James Cook University, Townsville, 1980; Home, Rod W., ‘The problem of intellectual isolation in scientific life: W.H. Bragg and the Australian scientific community, 1886–1909’, Historical Records of Australian Science, (1984), 6, pp. 1930; Moyal, Ann, A Bright and Savage Land: Scientists in Colonial Australia, Collins, Sydney, 1986, pp. 165166.

21 Crowe, Michael J., A History of Vector Analysis, University of Notre Dame Press, Notre Dame, 1967, Ch. 6.

22 McAulay, Alexander, Utility of Quaternions in Physics, MacMillan, London, 1893, and Octonions: a Development of Clifford's Biquaternions, Cambridge University Press, Cambridge, 1898.

23 Lancaster, Harold Oliver, ‘The departments of mathematics in the University of Sydney’, The Australian Mathematical Society Gazette, (1986), 13, pp. 2938.

24 Tarnish Bell, Robert John, An Elementary Treatise on Co-ordinate Geometry of Three Dimensions, Bell, London, 1910 (3rd edn 1950).

25 Young Sommerville, Duncan M'Laren, Bibliography of non-Euclidean Geometry, (1st edn) University of St. Andrews Press, 1911 (2nd edn, Chelsea, New York, 1970); The Elements of non-Euclidean Geometry, Bell, London, 1914 (Dover edn, New York, 1958); Analytical Conies, Bell, London, 1924 (3rd edn 1933); An Introduction to the Geometry of N Dimensions, Methuen, London, 1925 (Dover edn, New York, 1958); Analytical Geometry of Three Dimensions, Cambridge University Press, Cambridge, 1934.

26 Forder, Henry George, The Foundations of Euclidean Geometry, Cambridge University Press, Cambridge, 1927 (reprinted by Dover, New York, 1958; Rumanian translation fundamentele geometriei euclidiene, Editura Ştiinţifică, Bucureşti, 1970); A School Geometry, Cambridge University Press, Cambridge, 1930 (2nd edn 1938); Higher Course Geometry, Cambridge University Press, Cambridge, 1931 (reprinted 1949 and 1955); The Calculus of Extension, Cambridge University Press, Cambridge, 1941 (reprinted by Chelsea Press, New York, 1960); Geometry, Hutchinson, London and Longmans Green, New York, 1950 (2nd edn, Hutchinson, London, 1960 and Harper, New York, 1962; Turkish translation Geometri, Millî Eğitim Basimeví, Istanbul, 1968).

27 Butcher, John Charles (ed.), A Spectrum of Mathematics: Essays Presented to H.G. Forder, Auckland University Press and Oxford University Press, Auckland, 1971.

28 Loxton, John H., ‘Celebration of the 80th birthday of Kurt Mahler FRS, FAA’, The Australian Mathematical Society Gazette, (1984), 11, pp. 12. In the month following the delivery of this lecture at the Royal Institution, Kurt Mahler died at the Australian National University, on 25 February 1988: cf. Neumann, Bernhard H. and van der Poorten, Alf, ‘Kurt Mahler 1903–1988’, The Australian Mathematical Society Gazette, (1988), 15, pp. 2527. The Australian National University has set up a Mahler Memorial Fund, whose aims will include the promotion of the theory of numbers among senior high school students and undergraduates.

29 Potts, R.B., ‘Mathematics at the University of Adelaide 1944–1958’, The Australian Mathematical Society Gazette, (1985), 12, pp. 2530.

30 Newman, Michael F., ‘Hanna Neumann (1914–1971)’, In: Grinstein, Louise S. and Campbell, Paul J. (eds), Women of Mathematics: A Biobibliographic Sourcebook, Greenwood Press, New York, 1987, pp. 156160; The Selected Papers of Bernhard and Hanna Neumann (6 vols), with commentaries by Bernhard Neumann, have been published in May 1988 by the Charles Babbage Research Centre (University of Manitoba), as a contribution to the Australian Bicentennial celebrations.

31 Nield, Donald A., ‘University mathematics in Auckland: a historical essay’, Mathematical Chronicle, (1983), 12, pp. 133.

32 Rutherford's first name is often given as Ernest, but his name is spelt Earnest Rutherford on his birth certificate, cf. Hoare, M.E. and Bell, L.G. (eds), In Search of New Zealand's Scientific Heritage (1984), Bulletin 21, The Royal Society of New Zealand, p. 119.

33 Eve, E.S., Rutherford, Cambridge University Press, Cambridge, 1939.

34 Rutherford, Earnest, ‘The succession of changes in radioactive bodies’, Phil. Trans. Roy. Soc; Series A, (1904), 204, pp. 169219. (The Bakerian Lecture for 1904. Reprinted in The Collected Papers of Lord Rutherford (ed. Chadwick, James), George Allen and Unwin, London, Vol. 1, 1962, pp. 671722).

35 Rutherford, Earnest, ‘The scattering of a and β particles by matter and the structure of the atom’, Phil. Mag; Series 6, (1911), 21, pp. 669688. (Reprinted in The Collected Papers of Lord Rutherford 34, vol. 2, 1963, pp. 238254.)

36 Weatherburn, Charles Ernest, Elementary Vector Analysis with Applications to Geometry and Physics, Bell, London, 1921 (revised edition 1956); Advanced Vector Analysis with Applications to Mathematics and Physics, Bell, London, 1924 (and 1957); Differential Geometry of Three Dimensions, Cambridge University Press, Cambridge, 1927; An Introduction to Riemannian Geometry and the Tensor Calculus, Cambridge University Press, Cambridge, 1938 (and 1942); A First Course in Mathematical Statistics, Cambridge University Press, 1946 (and 1947).

37 cf. Lancaster.23

38 John Tee, Garry, ‘Two New Zealand mathematicians’, In: Crossley, John N. (eel.), History of Mathematics: Proceedings of the First Australian Conference, Department of Mathematics, Monash University, Clayton, Victoria, Australia, 1981, pp. 180199. (L.J. Comrie and A.C. Aitken).

39 Comrie, Leslie John, Modern Babbage Machines, The Office Machinery Users' Association, London, 1933.

40 John Comrie, Leslie, ‘Babbage's dream comes true’, Nature, (1946 October 26), 158, pp. 567569.

41 Wilkes, Maurice V., ‘How Babbage's dream came true’, Nature, (1975 October 16), 257, pp. 541544.

42 Randell, Brian (ed.), The Origins of Digital Computers: Selected Papers (3rd edn), Springer-Verlag, Berlin, 1982. (Randell's annotated Bibliography describes c. 850 publications, of which twenty-two are publications by Comrie—more than for any other author (pp. 450–452).)

43 Aitken, Alexander Craig, ‘On Bernoulli's numerical solution of algebraic equations’, Proc. Roy. Soc. Edinb; (1925), 46, pp. 289305; ‘On interpolation by iteration of proportional parts, without the use of differences’, Proc. Edinb. Math. Soc; (1931), 3, pp. 5676, and many other papers.

44 Aitken, Alexander Craig, Determinants and Matrices, Oliver and Boyd, Edinburgh, 1939; and Statistical Mathematics, Oliver and Boyd, Edinburgh, 1939. (Both texts have been reprinted in many editions.)

45 Smith, Steven B., The Great Mental Calculators, Columbia University Press, New York, 1983, Ch. 31.

46 Craig Aitken, Alexander, Gallipoli to the Somme: Recollections of a Neiu Zealand Infantryman, Oxford University Press, Oxford, 1963.

47 cf.Tee.38

48 cf. Nield.31

49 cf. Lancaster.23

50 The Proceedings (ed. John N. Crossley38) publish thirteen papers from that conference.

51 John (sic! for Jock) Hoe, Les systèmes d'équations polynômes dans le Siyuán yùjiàn (1303), Mémoires de l'Institut des Hautcs Ètudes Chinoises, (1977), tome 6, Paris; and Jock Hoe, ‘Zhu Shijie and his Jade Mirror of the Four Unknowns’, In: Crossley,38 pp. 103–134.

52 John Tee, Garry, ‘Mathematical science in New Zealand’, Canita Bhāratī, (1987), 9, pp. 19.

53 Hawkins, William Francis, The Mathematical Work of John Napier (1550–1617), Ph.D. thesis, University of Auckland, 1982 (to be published by University Microfilms International). (Abstract published in Bulletin of the Australian Mathematical Society, (1982), 26, pp. 455468.)

54 Hawkins, William Francis, ‘The first calculating machine (John Napier, 1617)’, The New Zealand Mathematical Society Newsletter, (12 1979), No. 16, Supplement, pp. 123. (Reprint in Annals of the History of Computing, (1988), 10, 3751).

55 cf.Tee.52

56 Several publications date the award to 13 July 1823—but the medal is inscribed with the date 1824, and the presentation was made by Henry Thomas Colebrooke, who was elected as President of the Society in February 1824.

57 John Tee, Garry, ‘The heritage of Charles Babbage in Australasia’, Annals of the History of Computing, (1983), 5, pp. 4559 (reprinted in The World—Te Reo, (08 1983), pp. 519); ‘Charles Babbage (1791–1871) and his New Zealand connections’, In: Hoare, M.E. and Bell, L.G. (eds),32 pp. 8190 (reprinted in The New Zealand Mathematics Magazine, (01 1986), 22, pp. 112123).

58 Hankins, Thomas L., Sir William Rowan Hamilton, The Johns Hopkins University Press, Baltimore and London, 1980, p. 321.

59 cf. Tee.52 The tombstone on Sydney Margaret Hamilton's grave, in Rosebank Road cemetery in Auckland, tells more about her brother than about herself: cf. Segedin, Marin G., ‘Sir William Rowan Hamilton’, The New Zealand Mathematics Magazine, (19671968), 5, pp. 128131. Hamilton's biographer, Archdeacon Robert Perceval Graves, had arranged for that tombstone to be placed on Sydney's grave (letter from Graves to Sir George Grey, 14 July 1892, Grey Collection, Auckland Public Library, GL-26(2)).

60 Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time, printed by Philip Dixon Hardy, Dublin 1835; and Researches Respecting Quaternions, First Series, read 13 11, 1843, printed by M.H. Gill, Dublin, 1847.

61 John Tee, Garry, ‘Mathematics and ANZAAS’, The New Zealand Mathematical Society Newsletter, (1986), No. 37, p. 34.

62 ‘At an ANZAAS Meeting in Adelaide, Schwerdtfeger gave a talk on “The Pfaffian invariant of a skew-symmetric matrix”. The audience consisted of his wife, his two honours students (Wall and Potts), Mrs. Marta Sved, and one unknown, a non-mathematical newspaper reporter, who was intrigued by the mystery of the title of the talk. As one can imagine, the subsequent newspaper article was strange-reading publicity for mathematics at Adelaide!’ [Potts,29 p. 28].

63 Blakers, A.L., ‘The Australian Mathematical Society: foundation and early years’, The Australian Mathematical Society Gazette, (1976), 3, pp. 3352 and 6586.

64 Sved, Marta, ‘Paul Erdös—portrait of our new Academician’, The Australian Mathematical Society Gazette, (1987), 14, pp. 5962.

65 Kerr, Roy Patrick, ‘Gravitational field of a spinning mass as an example of algebraically special metrics’, Physical Review Letters, (1963 September 1), 11, pp. 237238.

66 Sheffield, Charles, ‘Killing vector’, In: Vectors, Ace Books, New York, 1979.

67 Chandrasekhar, Subrahmanyan, Shakespeare, Newton and Beethoven, or, Patterns of Creativity, The Norma and Edward Ryerson Lecture at the University of Chicago for 1975.

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Mathematics in the Pacific Basin

  • Garry J. Tee (a1)


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