This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
The Sanskrit word for “or”, vā, is non-exclusive [Renou 1984; §382.B], so that all computations in “these mathematics” should be understood as being made of only increase, only decrease, or of both.
Svayambhū (as indicated in the section of the Glossary on the names of gods and planets) is one name of Brahmā. Siddhānta is the name of one genre of astronomical treatise, see [Pingree 1981; p.17sqq].
The Arthaśāstra of Kautilya is one of the classics of Sanskrit literature. It is a treatise on politics. For another discussion on the fact that Āryabhata states his name, see BAB.1.1. in [Shukla 1976, p.5].
This is the name of a city, ancient capital of the Mauryan empire and famous learning center. See [Sharma & Shukla 1976; intro, p.xvii].
These are the name of four of the five siddhāntas summed up in Varāhamihira’s Pañcasiddhāntika. The Paitamahāsiddhānta, the fifth siddhānta of Varahamihira’s treatise, being inspired by the Brahmasiddhānta. See [Neugebauer & Pingree 1971].
All this paragraph is difficult to read. For instance, the last sentence of this paragraph seems to contradict the previous statement that the first part of the verse is concerned with the names of numbers): upayogābhāvānna sankhyāsamjñā lit. Not names of number because of no use. This is however what we understand: the three first quarters of Ab.2.2 fix the names of number up to 109. [Hayashi 1995] has shown that higher numbers didn’t have fixed names. However, this is not a rule explaining how to make the name of numbers. In addition it also gives a name to the first nine places of the decimal-place value notation. These places are defined as representing an increasing set of multiple of tens.
One can understand the verse as meaning: A square is an equi-quadrilateral and the result which is the product of two identical 〈quantities〉 | or A square is an equi-quadrilateral and 〈its〉 area is the product of two identical 〈sides〉 | It is probably ambiguous in order to collect all these significations. Previous translators of this verse have noted this ambiguity. See [Sengupta 1927; p.13], [Clark 1930; p.21], [Shukla 1976; p.34]. Bhāskara expounds the verse in both directions.
Reading tathātra rather than the anyathātra of the printed edition and the yathātra of four manuscripts. The opposition suggested by anyathā does not have much meaning here.
What is exactly called here a vargakarman remains ambiguous. Please refer to volume II, section A.1 on page 2.
For an explanation of the mathematical content of this remark, see the Supplement for this verse commentary, Volume II, section B, p. 15.
Please see the Glossary of the Metaphoric and Peculiar Expressions to name numbers (Volume II, section 4, p. 2).
The former quantity referred to here is the partial cube-root computed, please see the supplement describing the procedure in this verse (Volume II, section B, p. 15).
K. S. Shukla (in the line of Clark’s interpretation [Clark 1930; p.36]) gives the following translation of this verse: “The product of the perpendicular (dropped from the vertex on the base) and half the base gives the measure of the area of a triangle” [Shukla 1976; p.38]. And P. C. Sengupta: “The area of a triangle is its Sarira (body) and is equal to half the product of the base and the altitude (...)” [Sengupta 1927; p.15]. The difference between these two translations lies, first of all, in the interpretation of the compound phala-śarīra. For Sengupta it is a karmadhāraya, meaning “that which is the area which is the body.” Shukla follows both Bhāskara’s and Nīlakantha’s interpretation of it as a genitive tatpurusa, meaning the “body/bulk of the area”. The polysemy of the word śarira may explain these different interpretations: śarira may mean bulk, but also intrinsic nature, this explains Sengupta’s reading of the verse. The second reason for these differences arises from the translation of samadalakotī. According to commentators it is the height in a triangle. The problem is that this is not the compound’s literal meaning (a mediator). Bhāskara comments on this point, below
As noted by J. Bronkhorst, this may be a recalling of the Pāninean sūtra 1.2.58: jātyākhyāyām ekasmin bahuvacanam anyatarasyām Plural optionally can be used for singular when jāti “class” is to be expressed. ([Sharma 1990; II. p. 129])
See [Keilhorn; II p. 246, line 6] This aphorism is also quoted in BAB.4.4, p.248, line 2. In other words, in this particular case, one gives a specific meaning to bhujā in order to give a general rule on the area of triangles.
For an explanation of the computation described here and below, please refer to the supplement for BAB.2.6, Volume II, section C.1, p. 22.
Please refer to the supplement for BAB.2.6.cd (volume II, section C.2, p. 27) for an explanation of the following description and the reasonings exposed below.
This paragraph is translated in [Hayashi 1995; p. 73 and p. 74] and in [Shukla 1976; p.liv.].
Please see the supplement for this commentary (Volume II, section F on page 40), for an explanation of this statement and the following. This sentence can also be translated as: “The sixth part of the circumference is a two-rā’si 〈field〉.”
As stated in AB.2.2., an ayuta is the name of ten thousand.
This anonymous collective voice is used from time to time in this commentary, as in BAB.2.3, and must be referring to scholars who had commented on this point.
Concerning cardinal directions, one can refer to Volume II, Supplement to verse 11 H.1.2.
Reading apavartitāu instead of apartitāu of the printed edition.
This way of expressing numbers is defined by Āryabhata in Ab.1.2 and subsequently used when producing versified tables, such as the one referred to here which is given in Ab.1.12. As this way of naming numbers is not used in the ganitapāda we have not described them here. There has been a relatively abundant literature on the subject. For a final word, one can refer to [Shukla & Sharma 1976; p. 3–5]. [Wish 1827; pp. 55–56] is the first to point out it’s existence. For controversies one can look up [Kaye 1908; p. 116–119], [Fleet 1912; p. 459–461], [Ganguly 1927] [Chattopādhyāya 1927; pp. 110–115], [Sen 1963; 298–302].
In his commentary to the two following verses, Bhāskara attempts to reconstruct the table of half-chords given by Āryabhata in the verse 12 of the first part of the Āryabhatīya (Ab.1.12). From the way he proceeds, that is described in the Supplements for BAB.2.11 and BAB.2.12, evidently approximations are used. Takao Hayashi in [Hayashi 1997] has also examined closely the different approximations used by Bhāskara and others, and also by Āryabhata himself in the establishment of this table of Rsinuses.
According to the setting down this should be a name for 21. It actually occurs in the Āryabhatīya, in Ab.2.2 as 109.
This whole paragraph is translated in the introduction of [Shukla 1976; 7.43.3, p. lxiv.]. And also in [Hayashi 1997; p.212–213].
This is a translation of the verse according to Bhāskara’s understanding of it. Historians of mathematics consider it a misinterpretation of Āryabhata’s verse. See [Hayashi 1997] and the Annex to this verse.
“They” may be a reference to the followers of Prabhākara, according to the statement below, [Shukla 1976; p. 84 line 13].
This is the enumeration of sine differences given in Ab.1.12, see [Sharma-Shukla 1976; p.29].
Reading: ādigrahanam madhyaparijñām ca’ instead of what is in the printed edition.
The word dalitam is used in all manuscripts as it is indicated in [Shukla 1976; note 3, p. p.105]; even though Shukla omits it from the main text of his edition.
The result could be stated as four fifths. We have discussed this in [Keller 2000; I, 2.2.4.b].
For a mathematical presentation of the contents of this verse, and a discussion please see [Keller 2000; I. 2.2].
We have adopted the translation of [Sharma 1990; II p. 129].
We have adopted Shukla’s translation of these verses in [Shukla 1960, p. 201].
Ab.1.4d states [Sharma-Shukla 1976; p.6]: budhāhnyajākordayācca lankāyām∥ (These revolutions commenced) at the begining of the sign Aries on Wednesday at sunrise at Lankā (when it was the commencement of the current yuga). As 1001 is divisible by seven, when a 1000 days have past, the new day is once again a Wednesday.
For a definition of the ucca, please see the Appendix d. The ucca of Jupiter, according to K.S. Shukla quoting the Brhajjātaka [Shukla 1976; p. 322], is in the fifth degree of Cancer (which is the fourth sign).
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag
About this chapter
Cite this chapter
(2006). Chapter on Mathematics. In: Expounding the Mathematical Seed, Volume 1: The Translation. Science Networks · Historical Studies, vol 30. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7592-2_2
Download citation
DOI: https://doi.org/10.1007/3-7643-7592-2_2
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7291-0
Online ISBN: 978-3-7643-7592-8
eBook Packages: Humanities, Social Sciences and LawHistory (R0)