# BAB.2.32-33: The pulverizer

Chapter

## Keywords

Fractional Part Integer Solution Great Common Divisor Subtractive Term Integral Number
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## References

- 1.[Shukla 1976; p. 48, lines 9–10]Google Scholar
- 2.[Shukla 1976; p. 48, lines10–16]Google Scholar
- 4.[Shukla 1976; p. 48, line 10]Google Scholar
- 7.[Datta& Singh 1980; p.134–135]Google Scholar
- 8.[Jain 1995; p.57]Google Scholar
- 9.[Srinivas 1990; p.39]Google Scholar
- 13.See for instance [Datta& Singh 1938; p.152]Google Scholar
- 15.[Shukla 1976; p.48, lines 8–10]Google Scholar
- 17.[Shukla 1976; p.63]Google Scholar
- 18.[Shukla 1976, p. 63]Google Scholar
- 19.[Shukla 1976; p.63, lines 2 and 19]Google Scholar
- 20.[Shukla 1976; Example 1, p.63]Google Scholar
- 21.[Shukla 1976; p. 63, line 19]. Please refer also to the Glossary for the translations we have adopted of these terms.Google Scholar
- 22.[Kaye 1908; p. 121]Google Scholar
- 23.[Sengupta 1927; p.16]Google Scholar
- 24.[Clark 1930; 27]: “the perpendiculars (from the point where the two diagonals intersect) to the perpendicular sides”.Google Scholar
- 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
- 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
- 28.See [Keller 2000; I p. 104–127]Google Scholar
- 29.This is exposed by T. Hayashi in [Hayashi 1995]Google Scholar
- 30.We have discussed the sometimes implicit orientation of fields in [Keller 2000; I. p.228–230]Google Scholar
- 32.[Datta& Singh 1979; p.168 sqq]Google Scholar
- 34.For commentaries on approximations of
*π*in India, see [Datta 1926], [Hayashi&Kusuba&Yano 1989], [Hayashi 1997b]Google Scholar - 35.[Ahmad 1981]Google Scholar
- 36.An analysis can be found in [Keller 2000; I p.120–126]Google Scholar
- 37.See [Datta& Singh 1979; p. 152–154], [Hayashi 1997a; p. 12], [Sarasvati 1979; p. 62sqq]Google Scholar
- 38.[Sarasvati 1979; p.65]Google Scholar
- 41.[Sarasvati 1979; p. 63, note 4]Google Scholar
- 46.See [Hayashi 1997]Google Scholar
- 47.See [Datta& Singh 1979; p.160sqq] and [Sarasvati 1979; p.63–64]Google Scholar
- 48.For a more thorough analysis of the types of reasoning involved in the refutation see [Keller 2000; I p.120–126]Google Scholar
- 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
- 51.[Shukla 1976; p. 77, line 15 sqq.]Google Scholar
- 53.The notations adopted are those used by Takao Hayashi in his article on Ab.2.12, [Hayashi 1997a]Google Scholar
- 54.[Shukla 1976, p.79, lines 7–8]Google Scholar
- 55.[Shukla 1976; opcit. lines 8–9]Google Scholar
- 56.
*idem*, lines 11–13Google Scholar - 57.[Shukla 1976;
*idem*lines 9–11]Google Scholar - 58.
*idem*, lines 15–17.Google Scholar - 60.[Shukla 1976; p.85]Google Scholar
- 61.[Shukla 1976, p.71]Google Scholar
- 62.[Shukla 1976; p.79]Google Scholar
- 63.See [CESS, Volume IV; pp. 187–192]Google Scholar
- 64.The first edition of the
*Āryabhaṭīya*was published with his commentary: [Kern 1874]Google Scholar - 65.[Sastri, 1957]Google Scholar
- 66.[Kern 1874; p. 32]Google Scholar
- 67.[Sastri; p.103–104]Google Scholar
- 68.idem.Google Scholar
- 70.[Ōhashi 1994; p.171]Google Scholar
- 72.[Ōhashi 1994; p.171]Google Scholar
- 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
- 78.See [Sharma& Shukla 1976; p.118]Google Scholar
- 79.[Shukla 1976; p. 98, line 13]Google Scholar
- 80.See BAB.2.17cd. [Shukla 1976; p. 198, line 3–14]Google Scholar
- 81.See the resolution of example 4 [Shukla 1976; p. 100, line 13–15]Google Scholar
- 83.See [Sharma& Shukla 1976; p. 62, formula 3]Google Scholar
- 85.See [Keller 2000; I, 2.1] and [Keller forthcoming]Google Scholar
- 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
- 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
- 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
- 90.[Keller 1995]Google Scholar
- 91.As translated by Sharma& Shukla 1976, p.139-I have added the indentations and the terms in Sanskrit within parentheses.Google Scholar
- 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
- 93.This is the translation adopted by K.S. Shukla of this term (see [Sharma& Shukla 1976; p.26]).Google Scholar
- 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 - 97.For Āryabhaṭa’s and Bhāskara’s treatment of the pulverizer, see [Jain 1995; p. 422–447]Google Scholar
- 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
- 101.[Shukla 1976; p.132, lines 15 to 19]Google Scholar
- 102.[Shukla 1976; p.132 lines 20 to 23]Google Scholar
- 104.[Shukla 1976; p. 132 lines 23 to 25]Google Scholar
- 106.[Shukla 1976; p.133, lines 2–3]Google Scholar
- 108.See for instance, [Gareth& Jones 1998; Proof of Theorem 1.8., p. 10]Google Scholar
- 110.[Shukla 1976; last paragraph p.149–150]Google Scholar
- 111.[Shukla 1976; p.135 lines 17 to 21]Google Scholar
- 112.[Shukla 1976; p.139]Google Scholar
- 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
- 114.[Shukla 1976; p.141, line 15–18]Google Scholar
- 116.The first example given on this topic in Bhāskaraś commentary is explained in the pages 34–35.Google Scholar
- 118.We have not translated this versified table. This table is summarized in [Shukla 1976; Appendix ii, p.335–339]Google Scholar
- 119.[Shukla 1960; sk p. 8–9, eng. p. 41]Google Scholar
- 120.[Shukla 1960; sk p. 8, eng. p.36–37(this is an adaptation — see note 1, p.37)]Google Scholar
- 121.There seems to be a paradox here, as
*A*_{ΔV}is thus defined as a multiple of*A*_{y}, therefore*A*_{ΔV}>*A*_{y}. 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*A*_{ΔV}, 210389 being the reduced number of civil days in a*yuga*for the sun). We can, nonetheless, remark that*A*_{y}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 - 123.[Shukla 1976; p.145, line 16 sqq]Google Scholar
- 124.[Shukla 1976; p.146, line 13 sqq]Google Scholar
- 125.[Shukla 1960; p.2–3 skt, p. 7 eng.]Google Scholar
- 127.[Shukla 1976; p. 147, line 15–17]Google Scholar

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