Abstract
In Fig. 1, PNQP is segment of a circle (i.e. circular disc) whose centre is at O and whose radius is \( OP=OQ=r \). Due to the figure’s resemblance to an archer’s bow, the arc PNQ (= s in length) was called cāpa (‘bow’), the chord \( PQ (=c) \) was called jyā or jīvā (‘bow-string’) and the segment’s height \( MN (=h) \) was called bāṅa or śara (‘arrow’) in ancient India. The cāpakṣetra (‘bow-figure’) or segment of a circle had great importance in Indian cosmography and geography, especially in the Jaina school. The Bharata-kṣetra (=Bhārata-varṣa or ‘land of India’) of those times was in the shape of a bow-figure which formed the southernmost part of the central continent or Jambūdvīpa (‘Jambū Island’) which is stated to be circular and of diameter one lac (100,000) yojanas. This cartographic description may be taken to represent the oldest map of India as part of Asia. The maximum north–south breadth of the country was 526 \( \frac{6}{19} \) yojanas.
Studies in the History of Exact Sciences (2004), Leiden, pp. 517–532.
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Notes
- 1.
See the Bhāṣya under sūtra 11 of chapter III in [Umāsvāti 1932, 170]. He is placed in the first century ad by [Pingree 1970, 59]. But the date (and even authorship) is controversial.
- 2.
See [Umāsvāti, 170], and [Gupta 1979, 91–2].
- 3.
[Gupta 1972], [Gupta 1972–73]; also [Gupta 1979, 93].
- 4.
[Shukla 1976, LVI, 74] (under II, 10).
- 5.
For an example see [Katz 1993, 20].
- 6.
[Van der Waerden 1983, 177–9].
- 7.
[Gupta 2001] contains details.
- 8.
For circular areas the rule is found in Umāsvāti’s Bhāṣya, 170 and in the Jiu Zhang Suanshu (I.32) (c. ad 100). See [Lam 1994, 13]; also see [H\(\phi \)yrup 1996, 21–3], and [Hayashi 1990, 5].
- 9.
[van der Waerden 1983, 39–40, 172–7].
- 10.
[Lam 1994, 13] and [van der Waerden 1983, 36–40].
- 11.
[Heath 1981, 330].
- 12.
When h is small we can neglect \( {h}^2 \) in (1) to get \( \left( \frac{c}{2}\right) ^2 = dh \) which becomes the parabola \( {y}^2 =dx \) with a proper choice of coordinate axes.
- 13.
[Heath 1981, 303], and [H\(\phi \)yrup 1996, 13, 16].
- 14.
[Midonick 1968, 197]. For controversy about date and authorship of the Hebrew work, see [Katz 1993, 152] and [H\(\phi \)yrup 1996, 25].
- 15.
[Martzloff 1997, 327].
- 16.
[L. C. Jain 1963, 190], and [Viśuddhamati 1975, 597].
- 17.
[Hayashi 1990].
- 18.
[Dvivedi 1899, 35].
- 19.
[Nahata & Nahata 1961, part II, 56].
- 20.
[Dvivedi 1910, 171] where the reading in the footnote is correct and accepted here.
- 21.
[Dvivedi 1910, 172] [Billard 1971, 157–62] shifts the date of the Mahā-siddhānta to early sixteenth century. Also see [Mercier 1993].
- 22.
The use of \( \pi =\frac{19}{6}\) is found earlier in the Tiloya-paṇṇatī, I, 118 (see Viśuddhamatī 1984, 26]) and elsewhere (see [Hayashi 1991, 333–5]).
- 23.
[Gupta 1991]. Alberuni credits Brahmagupta (fl.628) with a knowledge of \(\pi =\frac{22}{7}\) (see [Sachau 1964, Vol. I, 168]). It was also known to Śrīdhara [Hayashi 1985, 755].
- 24.
[Datta & Singh 1980, 167].
- 25.
[Apte 1937, part II 218].
- 26.
[Pingree 1979, 903]; and [Hayashi 1985, 751–6].
- 27.
For Śrīdhara’s rule \(\sqrt{N}=\frac{\sqrt{Na^2}}{a}\), see [Shukla 1959, 175] and Triśatikā rule 46.
- 28.
[Viśuddhamatī 1986, 636]. Yativṛṣabha is placed between ad 473 and 609.
- 29.
[Anupam Jain 1990, 163].
- 30.
[Shukla 1976, 73].
- 31.
[L. C. Jain 1963, 198].
- 32.
[Viśuddhamatī 1975, 597]; see Sect. 3.
- 33.
[Gupta 1987, 52–3].
- 34.
[Gupta 1989, 21–2].
- 35.
[Gupta 1981] contains a survey on averaging.
- 36.
Ancient mathematicians may have easily noted that the area of a semicircle (\( \frac{3r^2}{2}\) with \(\pi =3\)) is in fact the mean of the areas of the triangle \((=r^2)\) and outer rectangle \((=2r^2)\) on the same base.
- 37.
[Srinivasachar & Narasimhachar 1931, 124].
- 38.
[Datta 1991, 17]. Datta’s interpretation is good when the bow-string PMQ (Fig. 1) is assumed to take the position POQ in the pulled state.
- 39.
Ibid., and [Srinivasachar & Narasimhachar 1931, 56]. [Chakravarti 1934, 28] says that the formula \( A= \frac{ch}{2} \) is found in the Śulbasūtras but gives no details.
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Ramasubramanian, K. (2019). Area of a Bow-Figure in India. In: Ramasubramanian, K. (eds) Gaṇitānanda. Springer, Singapore. https://doi.org/10.1007/978-981-13-1229-8_25
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