agathe keller - expounding the mathematical seed vol 1

[...]... details in the supplements for BAB.2.9.cd (volume II, p 45), BAB.2 .11 (volume II, p.54), BAB.2 .12 (volume II, p 69), BAB.2 .17 .cd (volume II, p 10 1) and BAB.2 .18 (volume II, p 10 5) B The mathematical matter xxxi Figure 4: An “ancestor” of the trigonometrical circle, as seen in [Shukla 19 76] and in a manuscript KUOML 17 12 Folio 47 recto original whole chord, when considering the half-chord is furthermore... rather than the k¯laksetra of all manuscripts a 81 [Shukla 19 76; p 44; lines 1 4 -1 5]; volume I, p 8 82 See Ab.2.3, volume I, p 1 3 -1 8 83 See BAB.2.24, volume I, p 10 4; volume II, p 10 4 84 See for instance BAB.2 .17 .cd (volume I, p 84; volume II, p 10 1), BAB.2.6.ab (volume I, p 24, volume II, p 22) 80 Reading B The mathematical matter xxxv upon the fact that adding, subtracting, squaring and halving can... Contents of the Chapter on mathematics (ganitap¯da) a Verse 1 Verse 2 Verse 3 Verse 4 Verse 5 Verse 6 Verse 7 Verse 8 Verse 9 Verse 10 Verses 1 1- 1 2 Verse 13 Verses 1 4 -1 6 Verse 17 Verse 18 Verses 1 9-2 2 Verses 2 3-2 4 Verse 25 Verse 26 Verse 27 Verse 28 Verse 29 Verse 30 Verse 31 Verses 3 2-3 3 Prayer Definition of the decimal place value notation Geometrical and arithmetical definition of the square and the cube... Shukla 19 76] 11 [Sharma & Shukla 19 76; xxxv-lviii] 12 Concerning the name siddh¯nta for astronomical treatises, see [Pingree 19 81] a 13 [Shukla 19 76; xxv-xxx], [CESS; volume 4, p 297] 14 These texts have been edited and translated by K S Shukla: [Shukla 19 60], [Shukla 19 63] They had also been previously edited with commentaries, see [Apate 19 46] and [Sastri 19 57] For more details one can refer to the. .. of the verse giving the area of the square (Ab.2.3.ab) A cube is constructed from the surface of a square on which a height is raised The volume of the cube is the product of the area of the square by its height (V = A × H) Similarly, the volume of the sphere is the square root of the area of the circle √ multiplied by the area (V = A × A) The volume of the sphere thus seems to be the product of an area... mathematics Concerning the process presented by Aryabhata and expounded by Bh¯skara see BAB.2.3 2-3 3, volume I, 12 8; volume II, p 14 2 a 62 See BAB.2.2 6-2 7.ab, volume I, p 10 7 explained in volume II, p 11 8 63 See BAB.2.3 2-3 3, references above 58 Summarized B The mathematical matter xxv a specific place (on the left in Shukla’s printed edition) and “divisors” in another (on the right according to the printed text)... Bh¯skara in the [CESS, volume 4, p 29 7-2 99; volume 5] a 15 [Dvivedi 19 02] 16 [Hayashi 19 95] 17 For a discussion of the time when the text would have been written, see [Hayashi 19 95, p 14 8 -1 49] xiv Introduction taries in Sanskrit were produced and preserved At that time, Sanskrit astronomical texts and knowledge spread outside the frontiers of the Indian subcontinent: by the IXth century, there where... involved in This is especially clear in the rules of proportions where “multipliers” are set in 57 See [Keller 2000; volume 1, II.2.] in Table 1 on page xviii 59 See BAB.2.30, volume I, p 12 1; volume II, section V 60 See BAB.2.28, volume I, p .11 8; volume II, p 12 8 61 The “pulverizer” process which solves an indeterminate analysis problem is one of the classi¯ cal problems of medieval Sanskrit mathematics... verses and his personal mathematical input Let us 25 [Kern 18 74] [Sengupta 19 27], [Clark 19 30], and [Sharma & Shukla 19 76] 27 From now on, all verses referred to belong to the mathematical chapter of the Aryabhat¯ ¯ ıya, unless otherwise stated 28 See [Hayashi 19 97a] 29 Let us nevertheless mention [Renou 19 63], [Bronkhorst 19 90], [Bronkhorst 19 91] , [Houben 19 95] and [Filliozat 19 88 b, Appendix], which... numerically by the square root of the area The area of an equilateral trilateral is the product of half of the base and a height (A = 1/ 2 , B × H) In continuity with this computation, the volume of the equilateral pyramid is given with the same consideration: half the area multiplied by the height (V = 1/ 2 , A × H) Bh¯skara furthermore argues for a the “evidence” (pratyaksa) that the volume of the pyramid . Germany Vol. 1/ SN 30: ISBN 10 : 3-7 64 3-7 29 1- 5 e-ISBN: 3-7 64 3-7 59 2-2 ISBN 13 : 97 8-3 -7 64 3-7 29 1- 0 Vol. 2/SN 31: ISBN 10 : 3-7 64 3-7 29 2-3 e-ISBN: 3-7 64 3-7 59 3-0 ISBN 13 : 97 8-3 -7 64 3-7 29 2-7 Set SN 30/ 31: ISBN. vii BAB.2.23 10 3 BAB.2.24 10 4 BAB.2.25 10 5 BAB.2.2 6-2 7.ab 10 7 BAB.2.27cd 11 6 BAB.2.28 11 8 BAB.2.29 11 9 BAB.2.30 12 1 BAB.2. 31 124 BAB.2.3 2-3 3 12 8 Pulverizer 12 8 Pulverizerwithoutremainder 13 2 Planet’spulverizer. Leipzig V.P. Vizgin, Moskva Agathe Keller Expounding the Mathematical Seed Volume 1: The Translation A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya Birkhäuser Verlag Basel