Calculation for ‘chain-reduction’ in the Triśatībhāṣya

Abstract

The Triśatībhāṣya is an anonymous commentary on Śrīdhara’s Triśatī. ‘Chain-reduction’ (vallīsavarṇana) is a rule for unifying quantities expressed in several units into the highest one, but the usage of the rule in the Triśatībhāṣya is slightly different. The present paper tries to explain, by comparison with the procedures illustrated in other arithmetic texts, why the commentator applies the ‘chain-reduction’ in an irregular way.

Appendices

Appendix 1: Text and translation of the TrBh

The text edited here of the TrBh is based on A₁. The word(s) cited in the TrBh from the Tr is printed in bold. I remove or add daṇḍa-s (/) in A₁ for the ease of reading. Phonological irregularities have been left in this edition just as they are in the manuscript.

Notations

$^{ac}$ :

Ante correcturam, i.e., the reading before the correction by the scribe

em.:

Emendation (I do not distinguish it from ‘correction’ and ‘conjecture’)

$^{pc}$ :

Post correcturam, i.e., the reading after the correction by the scribe

$\left\langle {\text{A}} \right\rangle$ :

A is supplied by the editor

‘A’:

A is quotation

(A):

A is reference of quoted passage

$^\circ$ :

Truncation (of letters) in long Sanskrit words

1.1 TrBh on Tr E22 (A₁ fols. 9b–10a)

न्यासः।  ‘ ${\mathbf {<}}$विनि ${\mathbf {>}}$ क्षिपेत्’ (PG 37d) इति कृते जातं    एव​। चतुष्केण छेदांशौ गुणये त्। पू र्वांशे एकं प्रक्षिपेत्। प्रक्षिप्ते जातं  ।  ततो विंशत्या छेदांशौ गुणयेत्। अंशेभ्यः एकं पातयेत्। ऊर्द्वस्थानात्पातिते जातं    । ततः पंचकेन​ छेदांशौ गुणयेत्। अंशेभ्यः एककं च पातयेत्। पातिते जातं   । ततः छेदांशौ दलयेत्। दलिते जातं  ।  छेदेन हृते लब्धं पुराणाः पंच ५। ततः शेषे षोडशभिर्हते लभ्यंते पणाः ३। शेषे चतुष्केण हते ततो विंशत्या हते लभ्यंते वराटकाः १८। शेषे षड्भिः शतैश्चत्वारिंशदधिकैरपवर्त्तिते भागाः  ।  एवं वल्लीसवर्णनं समाप्तं॥ छ​॥

nyāsaḥ/ ‘ ${\varvec{<}}$vini${\varvec{>}}$ kṣipet’ (PG 37d)²⁹iti kṛte jātaṃ   eva/³⁰catuṣkeṇa chedāṃśau guṇayet/³¹pūrvāṃśe ekaṃ prakṣipet³²prakṣipte jātaṃ  / tato viṃśatyā chedāṃśau guṇayet/³³aṃśebhyaḥ ekaṃ pātayet/³⁴ūrdvasthānāt pātite jātaṃ / ³⁵ tataḥ paṃcakena chedāṃśau guṇayet/ aṃśebhyaḥ ekakaṃ ca pātayet/³⁶pātite jātaṃ  / tataḥ chedāṃśau dalayet/³⁷ dalite jātaṃ / chedena hṛte labdhaṃ purāṇāḥ paṃca 5/³⁸tataḥ śeṣe ṣoḍaśabhir hate labhyaṃte paṇāḥ 3/ śeṣe catuṣkeṇa hate tato viṃśatyā hate labhyaṃte varāṭakāḥ 18/³⁹śeṣe ṣaḍbhiḥ śataiś catvāriṃśad adhikair apavarttite bhāgāḥ /⁴⁰evaṃ vallīsavarṇanaṃ samāptaṃ// cha//

Setting-down: .

When [the operation] “one should add” (PG 37d) is carried out, the result is exactly . One should multiply [the former (the upper)] denominator and numerator by four. One should add one to the former (the upper) numerator. When [one] is added [to it], the result is . Then, one should multiply [the former (the upper)] denominator and numerator by twenty. One should subtract one from the [upper] numerator. When [one] is subtracted from the above place, the result is . Next, one should multiply [the former (the upper)] denominator and numerator by five and subtract one from the [upper] numerator. When [one is] subtracted [from them], the result is . After that, one should halve the denominator and the numerator. When [they are] halved, the result is . When [the numerator is] divided by the denominator, the quotient is five, i.e., 5 purāṇa-s. Then, when the remainder is multiplied by sixteen, 3 paṇa-s are obtained. When the remainder is multiplied by four and further multiplied by twenty, 18 varāṭaka-s are obtained. When the remainder is reduced by six hundred increased by forty, the parts are . Thus, [the topic of] the ‘chain-reduction’ is completed.

1.2 TrBh on Tr E33 (A₁ fols. 11a–11b)

द्रोणैः कुडवे कृते न्यासः। द्रोणस्य कुडवकरणार्थं युक्तिः प्रस्थे कुडव ४ आढके कुडव १६ द्रोणे कुडव ६४। तदर्द्धभागे ३२। उभयं ९६ त्रिभिर्योगे ९९। अथवा अन्या युक्तिः सवर्ण्णने जातं त्रिकद्विकयोरन्योन्यघाते जातं ६ चतुर्गुणे २४ पुनश्चतुर्गुणे ९६ अधस्थितत्रिकसहिते ९९। तद्वा वल्लीसवण्णने कुडवानयनं। द्रोण १। आढकस्थाने आढकचतुष्टये<न​> द्रोणः स्यादिति चतुष्कदर्शनं ४। प्रस्थस्थाने ० प्रस्थचतुष्टयेनाढकः स्यादिति चतुष्कदर्शनं ४। कुडवचतुष्टयेन प्रस्थं स्यात् ४। एकत्रस्थापनं गुणने एते द्रोणकुडवाः ६४। खार्या १०२४। उभयं जातं १०८८। अंतिमानयनं। अथ स्थापनं। ९९। ८। १०८८। तदंत्यगुणं फलं ८ गुणिते ८७०४ आदिमेन छेदांशविपर्यासेन गुणिता ॥ भागे हृते लब्धानि रूपाणि ८७ रूपभागाः ॥

droṇaiḥ kuḍave kṛte nyāsaḥ/⁴¹droṇasya kuḍavakaraṇārthaṃ yuktiḥ prasthe kuḍava 4 āḍhake kuḍava 16 droṇe kuḍava 64/⁴²tadarddhabhāge 32/ ubhayaṃ 96 tribhir yoge 99/ athavā anyā yuktiḥ savarṇṇane jātaṃtrikadvikayor anyonyaghāte jātaṃ 6 caturguṇe 24 punaś caturguṇe 96 adhasthitatrikasahite 99/⁴³ tadvā vallīsavarṇṇane kuḍavānayanaṃ/⁴⁴droṇa 1/ āḍhakasthāne āḍhakacatuṣṭaye<na> droṇaḥ syād iti catuṣkadarśanaṃ 4/⁴⁵prasthasthāne 0 prasthacatuṣṭayenāḍhakaḥ syād iti catuṣkadarśanaṃ 4/⁴⁶kuḍavacatuṣṭayena prasthaṃ syāt 4/ ekatrasthāpanaṃ guṇaneete droṇakuḍavāḥ 64/⁴⁷khāryā 1024/⁴⁸ubhayaṃ jātaṃ 1088/ aṃtimānayanaṃ/⁴⁹atha sthāpanaṃ/ 99/ 8/ 1088/ tadaṃtyaguṇaṃ phalaṃ 8 guṇite 8704 ādimena chedāṃśaviparyāsena guṇitā /⁵⁰bhāge hṛte labdhāni rūpāṇi 87 rūpabhāgāḥ /.

When the kuḍava is produced by the droṇa-s, setting-down is [as follows]. The principle (yukti) for the sake of converting the droṇa into the kuḍava is: 4 kuḍava-s for one prastha, 16 kuḍava-s for one āḍhaka, 64 kuḍava-s for one droṇa. For a half part of them, there is 32. Both [of 64 and 32 added together] are 96. Increased by three, [the result is] 99. Alternatively, there is the other principle. Reduced to the same color, the result is  . When three and two are multiplied mutually, the result is 6. Multiplied by four, [the result is] 24. Further multiplied by four, [the result is] 96. Increased by three located below (adhasthita),⁵¹ [the result is] 99. Similarly (tadvat),⁵² in the ‘chain-reduction,’ [one] calculates the kuḍava. 1 droṇa is [at the top of the ‘chain’]. Since, in the place of āḍhaka, one droṇa should be made up of four āḍhaka-s, four, i.e., 4 is shown. [The numerator is] 0 in the place of prastha. Because one āḍhaka should be made up of four prastha-s, four, i.e., 4 is shown. One prastha should be made up of four kuḍava-s. 4 [is shown]. Put [the digits] in one place, . [Performing] the multiplication, [the result is] . These are 64 kuḍava-s [contained] in one droṇa. [The kuḍava-s contained] in one khārī are 1024. Both [of 64 and 1024 added together] are 1088. [Thus is] the calculation of the last [term of a Rule of Three]. Now, setting-down is 99, 8, 1088. That (the ‘fruit’ of the ‘standard’) multiplied by the last [quantity]. The ‘fruit’ [of the ‘standard’] is 8. Multiplied [by the last quantity, the result is] 8704. Multiplied by the first [quantity] through interchanging the denominator and the numerator, [the result is] . When [the numerator is] divided [by the denominator], what is obtained is 87 units [and] parts of unity.

Appendix 2: Text and translation of the PGṬ

The text of the PGṬ is based on the following edition and manuscript:

L_ED: Lucknow edition by K. S. Shukla, 1959

J₁: Raghunātha Temple Library, Jammu, 3074

1.1 PGṬ on PG E22 (L_ED p. 35, line 28—p. 36, line 11)

अत्र पुराणानां रूपच्छेदनतास्वातन्त्र्यात्पणादीनां चावयवविशेषत्वे संज्ञाविशेषत्वाच्छेदलाभः। यतः पुराणषोडशभागः पणः, पणचतुर्भागः काकिणी, काकिणीविंशतिभागो वराटकः, अतस्तैरेव रूपच्छेदस्थापनम्— लब्धं पुराणाः ५, पुराणभागाः: ।भागापवाहजातिः समाप्ता वल्ली च​।

atra purāṇānāṃ rūpacchedanatāsvātantryāt paṇādīnāṃ cāvayavaviśeṣatve saṃjñāviśeṣatvāc chedalābhaḥ/ yataḥ purāṇaṣoḍaśabhāgaḥ paṇaḥ, paṇacaturbhāgaḥ kākiṇī, kākiṇīviṃśatibhāgo varāṭakaḥ, atas tair eva rūpacchedasthāpanam– labdhaṃ purāṇāḥ 5, purāṇabhāgāḥ / bhāgāpavāhajātiḥ samāptā vallī ca/

In this [example], the purāṇa-s are independent because they possess unity as the denominator, and the paṇa-s and so on have particular designations as particular parts [of the preceding units]. For that reason, they acquire the denominators. This is because one paṇa is made up of one-sixteenth of one purāṇa, one kākiṇī one-fourth of one paṇa, and one varāṭaka one-twentieth of one kākiṇī. Therefore, by means of those [conversion ratios], the setting-down (sthāpana) of the unity and the denominators is [as follows]: .

What is obtained is 5 purāṇa-s [and] parts of one purāṇa. [The topics of] ‘part-subtraction class’ and ‘chain’ are completed.

1.2 PGṬ on PG E27 (L_ED p. 39, lines 4–21)

धान्यस्य द्रोणः अर्द्धेन प्रस्थाष्टकेन सहितस्तथा कुडवैस्त्रिभिर्युक्तः अष्टाभिर्यैः कैश्चिद्देशनियतैर्व्यावहारिकै रूपैश्चेल्लभ्यते तदा एका खारी द्रोणयुता कियता लभ्यते।

dhānyasya droṇaḥ arddhena prasthāṣṭakena sahitas tathā kuḍavais tribhir yuktaḥ aṣṭābhir yaiḥ kaiścid deśaniyatair vyāvahārikai rūpaiś cel labhyate tadā ekā khārī droṇayutā kiyatā labhyate/

If one droṇa of grain, increased by a half, that is, by eight prastha-s, and increased by three kuḍava-s, are obtained for eight [units of money], that is, for [eight of] certain practical (vyāvahārika) units limited to the region(s), then, for how much one khārī increased by one droṇa [will] be obtained?

अत्र​ धान्यद्रोणो यथोक्तप्रस्थकुडवान्वितो ज्ञातमूल्यत्वात्प्रमाणराशिः। तत्र प्रमाणेच्छाराश्योः सवर्णमुपादीयते। द्रोणानां खारी कार्या, तत्खार्य्यश्च द्रोणाः <वा> कार्याः। द्रोणस्य प्रस्थादिसानुबन्धत्वात्प्रतिपत्तिगौरवं स्यात्। खारी तु षोडशगुणा द्रोणाः तावन्त एव​, एकद्रोणाधिकाः सप्तदश​, द्रोणस्य यदि रूपार्द्धेन योगं कृत्वा कुडवयोगः क्रियते तदा कुडवा चतुःषष्टिच्छेदाः कार्याः, तावत्कुडवैर्द्रोण इति। अथ द्रोणस्यार्द्धेन परभागानुबन्धः कुडवैः वल्ली इति तन्त्रद्वयक्रियायामायासस्तदा द्रोणो रूपच्छिन्न उपरि, तदधः प्रस्थाष्टकं षोडशच्छिन्नं तदधस्त्रयः कुडवा प्रस्थव्यवस्थया चतुर्भक्ताः स्थाप्याः।

उभयत्रापि प्रमाणराशिः सवर्णि<ते> इदं भवति मध्यमराशिः स्वरूपस्थ एव ८, अन्त्यराशिः १७। प्रमाणराशेर्हरत्वाच्छेदांशविपर्यासे ऽनन्तरं प्रभागकर्मणि लब्धं ८७ भागाः: ।

atra dhānyadroṇo yathoktaprasthakuḍavānvito jñātamūlyatvāt pramāṇarāśiḥ/ tatra pramāṇecchārāśyoḥ savarṇam upādīyate/ droṇānāṃ khārī kāryā, tatkhāryyaś ca droṇāḥ <vā> kāryāḥ/ droṇasya prasthādisānubandhatvāt pratipattigauravaṃ syāt/ khārī tu ṣoḍaśaguṇā droṇāḥ tāvanta eva, ekadroṇādhikāḥ saptadaśa, droṇasya yadi rūpārddhena yogaṃ kṛtvā kuḍavayogaḥ kriyate tadā kuḍavā catuḥṣaṣṭicchedāḥ kāryāḥ, tāvatkuḍavair droṇa iti/⁵³atha droṇasyārddhena parabhāgānubandhaḥ kuḍavaiḥ vallī iti tantradvayakriyāyām āyāsas tadā droṇo rūpacchinna upari, tadadhaḥ prasthāṣṭakaṃ ṣoḍaśacchinnaṃ tadadhas trayaḥ kuḍavā prasthavyavasthayā caturbhaktāḥ sthāpyāḥ/⁵⁴

ubhayatrāpi pramāṇarāśiḥ savarṇi<te> idaṃ bhavatimadhyamarāśiḥ svarūpastha eva 8, antyarāśiḥ 17/ pramāṇarāśer haratvāc chedāṃśaviparyāse ’nantaraṃ prabhāgakarmaṇi labdhaṃ 87 bhāgāḥ /

In this [example], one droṇa of grain increased by the aforementioned prastha-s and kuḍava-s is the ‘standard’ quantity, because its price is already known. In that case, the same color for the ‘standard’ and the ‘requisite’ quantities is taken.⁵⁵ The droṇa-s are to be [converted into] khārī, or the khārī-s are to be [converted into] droṇa-s. [However, since] the droṇa [of the ‘standard’ quantity] is accompanied by the prastha and so on, it would be difficult to carry out [the conversion of them into khārī]. On the other hand, one khārī [of the ‘requisite’ quantity] is just as much as one droṇa multiplied by sixteen. Adding one droṇa to it, seventeen [droṇa-s are obtained, and so, let the units of the ‘standard’ quantity be unified into droṇa]. If one droṇa is increased by half of the unit and [three] kuḍava-s are added [to it], then, kuḍava-s possessing sixty-four as the denominator are to be produced. [To be precise], one droṇa is made up of such number of kuḍava-s. Next, ‘other’s part addition’ is by means of a half droṇa, [and] ‘chain’ is by means of kuḍava-s. Thus, [one makes] an effort at calculation in the two principles (tantradvayakriyā). Then, one droṇa divided by unity is at the top. Eight prastha-s divided by sixteen are below that. Three kuḍava-s divided by four are to be placed below them in accordance with the settled rule of prastha (prasthavyavasthā):

There is the ‘standard’ quantity on both sides. Reduced to the same color, this is produced $\begin{array}{*{20}c} {99} \\ {64} \\ \end{array}$. The middle quantity is indeed in its own unit (svarūpastha), i.e., 8. The last quantity is 17. Since the ‘standard’ quantity is the divisor, the denominator and the numerator are interchanged, and after that, the calculation for ‘multi-part’ [is carried out]. What is obtained is 87 [and] $\begin{array}{*{20}c} {91} \\ {99} \\ \end{array}$ parts.