Advertisement

BAB.2.32-33: The pulverizer

Part of the Science Networks · Historical Studies book series (SNHS, volume 31)

Keywords

Fractional Part Integer Solution Great Common Divisor Subtractive Term Integral Number 
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.
    [Shukla 1976; p. 48, lines 9–10]Google Scholar
  2. 2.
    [Shukla 1976; p. 48, lines10–16]Google Scholar
  3. 4.
    [Shukla 1976; p. 48, line 10]Google Scholar
  4. 7.
    [Datta& Singh 1980; p.134–135]Google Scholar
  5. 8.
    [Jain 1995; p.57]Google Scholar
  6. 9.
    [Srinivas 1990; p.39]Google Scholar
  7. 13.
    See for instance [Datta& Singh 1938; p.152]Google Scholar
  8. 15.
    [Shukla 1976; p.48, lines 8–10]Google Scholar
  9. 17.
    [Shukla 1976; p.63]Google Scholar
  10. 18.
    [Shukla 1976, p. 63]Google Scholar
  11. 19.
    [Shukla 1976; p.63, lines 2 and 19]Google Scholar
  12. 20.
    [Shukla 1976; Example 1, p.63]Google Scholar
  13. 21.
    [Shukla 1976; p. 63, line 19]. Please refer also to the Glossary for the translations we have adopted of these terms.Google Scholar
  14. 22.
    [Kaye 1908; p. 121]Google Scholar
  15. 23.
    [Sengupta 1927; p.16]Google Scholar
  16. 24.
    [Clark 1930; 27]: “the perpendiculars (from the point where the two diagonals intersect) to the perpendicular sides”.Google Scholar
  17. 25.
    [Sharma& Shukla 1976; p.42]: “the lengths of the perpendiculars on the base and the face (from the point of intersection of the diagonals)”.Google Scholar
  18. 27.
    See [Hayashi 1995; p. 72–75], who also analyses the use of the term in this text, and in non-mathematical texts. One can also see [Shukla 1976; intro p.liv]Google Scholar
  19. 28.
    See [Keller 2000; I p. 104–127]Google Scholar
  20. 29.
    This is exposed by T. Hayashi in [Hayashi 1995]Google Scholar
  21. 30.
    We have discussed the sometimes implicit orientation of fields in [Keller 2000; I. p.228–230]Google Scholar
  22. 32.
    [Datta& Singh 1979; p.168 sqq]Google Scholar
  23. 34.
    For commentaries on approximations of π in India, see [Datta 1926], [Hayashi&Kusuba&Yano 1989], [Hayashi 1997b]Google Scholar
  24. 35.
    [Ahmad 1981]Google Scholar
  25. 36.
    An analysis can be found in [Keller 2000; I p.120–126]Google Scholar
  26. 37.
    See [Datta& Singh 1979; p. 152–154], [Hayashi 1997a; p. 12], [Sarasvati 1979; p. 62sqq]Google Scholar
  27. 38.
    [Sarasvati 1979; p.65]Google Scholar
  28. 41.
    [Sarasvati 1979; p. 63, note 4]Google Scholar
  29. 46.
    See [Hayashi 1997]Google Scholar
  30. 47.
    See [Datta& Singh 1979; p.160sqq] and [Sarasvati 1979; p.63–64]Google Scholar
  31. 48.
    For a more thorough analysis of the types of reasoning involved in the refutation see [Keller 2000; I p.120–126]Google Scholar
  32. 50.
    For remarks of later Sanskrit authors on the links between a chord and a half-chord see [Datta& Singh 1983; p. 40]Google Scholar
  33. 51.
    [Shukla 1976; p. 77, line 15 sqq.]Google Scholar
  34. 53.
    The notations adopted are those used by Takao Hayashi in his article on Ab.2.12, [Hayashi 1997a]Google Scholar
  35. 54.
    [Shukla 1976, p.79, lines 7–8]Google Scholar
  36. 55.
    [Shukla 1976; opcit. lines 8–9]Google Scholar
  37. 56.
    idem, lines 11–13Google Scholar
  38. 57.
    [Shukla 1976; idem lines 9–11]Google Scholar
  39. 58.
    idem, lines 15–17.Google Scholar
  40. 60.
    [Shukla 1976; p.85]Google Scholar
  41. 61.
    [Shukla 1976, p.71]Google Scholar
  42. 62.
    [Shukla 1976; p.79]Google Scholar
  43. 63.
    See [CESS, Volume IV; pp. 187–192]Google Scholar
  44. 64.
    The first edition of the Āryabhaṭīya was published with his commentary: [Kern 1874]Google Scholar
  45. 65.
    [Sastri, 1957]Google Scholar
  46. 66.
    [Kern 1874; p. 32]Google Scholar
  47. 67.
    [Sastri; p.103–104]Google Scholar
  48. 68.
  49. 70.
    [Ōhashi 1994; p.171]Google Scholar
  50. 72.
    [Ōhashi 1994; p.171]Google Scholar
  51. 73.
    Please see the supplement for BAB.2.16. (Volume II, M on the facing page, for an analysis of the use of the rule in this situation)Google Scholar
  52. 78.
    See [Sharma& Shukla 1976; p.118]Google Scholar
  53. 79.
    [Shukla 1976; p. 98, line 13]Google Scholar
  54. 80.
    See BAB.2.17cd. [Shukla 1976; p. 198, line 3–14]Google Scholar
  55. 81.
    See the resolution of example 4 [Shukla 1976; p. 100, line 13–15]Google Scholar
  56. 83.
    See [Sharma& Shukla 1976; p. 62, formula 3]Google Scholar
  57. 85.
    See [Keller 2000; I, 2.1] and [Keller forthcoming]Google Scholar
  58. 86.
    In fact the same word is used in the second half of verse 27 to evoke fractions with a same denominator. See the supplement for BAB.2.27.cd and [Keller 2000; I.2.2]Google Scholar
  59. 87.
    In [Keller 2000; I.2.2] the status of this intermediary form in respect to fractionary quantities and fractions per se is studied.Google Scholar
  60. 89.
    This is presented as such in the edition. We do not know if such a fact was common to all manuscripts. For a discussion, please see [Keller 2000; I.2.2]Google Scholar
  61. 90.
    [Keller 1995]Google Scholar
  62. 91.
    As translated by Sharma& Shukla 1976, p.139-I have added the indentations and the terms in Sanskrit within parentheses.Google Scholar
  63. 92.
    [Shukla 1963; Sanskrit text p.11; English translation p.46]. I have added the subdivisions into different steps of the procedure and the names in Sanskrit of terms which occur also in the paragraph of his commentary on verse 28 of chapter 2.Google Scholar
  64. 93.
    This is the translation adopted by K.S. Shukla of this term (see [Sharma& Shukla 1976; p.26]).Google Scholar
  65. 94.
    [Shukla 1960; Sanskrit text p.15; English translation p. 74–75]. I have added the subdivisions into different steps of the procedure and the names in Sanskrit of terms which occur also in the paragraph of his commentary on verse 28 of the gaṇitapāda of the Āryabhaṭīya.Google Scholar
  66. 97.
    For Āryabhaṭa’s and Bhāskara’s treatment of the pulverizer, see [Jain 1995; p. 422–447]Google Scholar
  67. 99.
    For a brief description of how Bhāskara proceeds to give two different interpretations of the same compound see [Keller 2000; Volume I, I] and in Volume I, Introduction.Google Scholar
  68. 101.
    [Shukla 1976; p.132, lines 15 to 19]Google Scholar
  69. 102.
    [Shukla 1976; p.132 lines 20 to 23]Google Scholar
  70. 104.
    [Shukla 1976; p. 132 lines 23 to 25]Google Scholar
  71. 106.
    [Shukla 1976; p.133, lines 2–3]Google Scholar
  72. 108.
    See for instance, [Gareth& Jones 1998; Proof of Theorem 1.8., p. 10]Google Scholar
  73. 110.
    [Shukla 1976; last paragraph p.149–150]Google Scholar
  74. 111.
    [Shukla 1976; p.135 lines 17 to 21]Google Scholar
  75. 112.
    [Shukla 1976; p.139]Google Scholar
  76. 113.
    We have not translated this versified table. It is summarized, and all values given, in [Shukla 1976; Appendix ii, p.335–339]Google Scholar
  77. 114.
    [Shukla 1976; p.141, line 15–18]Google Scholar
  78. 116.
    The first example given on this topic in Bhāskaraś commentary is explained in the pages 34–35.Google Scholar
  79. 118.
    We have not translated this versified table. This table is summarized in [Shukla 1976; Appendix ii, p.335–339]Google Scholar
  80. 119.
    [Shukla 1960; sk p. 8–9, eng. p. 41]Google Scholar
  81. 120.
    [Shukla 1960; sk p. 8, eng. p.36–37(this is an adaptation — see note 1, p.37)]Google Scholar
  82. 121.
    There seems to be a paradox here, as AΔV is thus defined as a multiple of Ay, therefore AΔV>Ay. This assumption without any comment is also made by K.S. Shukla, when he solves example 12. [Shukla 1976; p.317] (A being what we denote AΔV, 210389 being the reduced number of civil days in a yuga for the sun). We can, nonetheless, remark that Ay is, here, the reduced number of terrestrial days in a yuga and not the total number, so that this is not as absurd as it may seem. However, just why should this be presupposed and whether this is the exact rending of the computation described by Bhāskara, remains to be investigated.Google Scholar
  83. 123.
    [Shukla 1976; p.145, line 16 sqq]Google Scholar
  84. 124.
    [Shukla 1976; p.146, line 13 sqq]Google Scholar
  85. 125.
    [Shukla 1960; p.2–3 skt, p. 7 eng.]Google Scholar
  86. 127.
    [Shukla 1976; p. 147, line 15–17]Google Scholar

Copyright information

© Birkhäuser Verlag 2006

Personalised recommendations