On the Parikarman's of Sanskrit Mathematical Texts Viith-Xiith Century

Elementary Operations, Fundamental operations: On the parikarman's of Sanskrit mathematical texts VIIth-XIIth century.

Agathe Keller

14/11/2011

Monday, November 14, 2011 Bhāskarācārya (b.1114), Siddhānta Śiromāṇi including Līlāvatī and Bījagaṇita

H. T. Colebrooke. Algebra, with Arithmetic and mensuration. J. Murray, London, 1817. F. Patte. Le Siddhāntaśiromāni : l’oeuvre mathématique et astronomique de Bhāskarācarya = Siddhāntaśiromāniḥ : Śri-Bhāskarācarya-viracitaḥ. Droz, 2004. T. Hayashi. Bījagaṇita of Bhāskara. SCIAMVS (10): 2009.

Śrīdharācārya (ca.900), Pāṭīgaṇita and anonymous undated commentary

K. S. Shukla. Pāgaṇita of Śrīdharācarya. Lucknow University, Lucknow, 1959.

Brahmagupta (629) Brahmasphuṭṭasidhānta Pṛthūdakṣvamin (ca.980) Vāsanābhāṣya

H. T. Colebrooke. Algebra, with Arithmetic and mensuration. J. Murray, London, 1817. M. S. Dvivedin. Brāhmasphuṭasiddhānta and Dhyānagrahopadeṣādhyāya by Brahmagupta edited with his own commentary The paṇḍit, XXIV:454, 1902. S. Ikeyama. Brāhmasphuasiddhānta (ch. 21) of Brahmagupta with Commentary of Pṛthudaka, critically ed. with Eng. tr. and Notes, volume 38. Indian Journal of History of Science, New Delhi, India, 2003.

Monday, November 14, 2011 CONTENTS. CONTENTS. Chapter III. Afiscellaneous. Sect. I. Inversion 21 Sect. II. Supposition 23

Page. Sect. III. Concurrence . ; 26 DiSSERTATIOJf • i Sect. IV. Problem concerning Squares ... 27 Sect. V. Assimilation 29 NOTES AND ILLUSTRATIONS. Sect. VI. Rule of Proportion 33 A. Scholiasts of Bha'scara xxv Chapter IV. Mixture.

B. Astronomy of Brahmegupta . . xxviii Sect. I. Interest 39 C. Brahma-sidd'hdnta, Title his of Astronomy xxx Sect. II. Fractions 42 D. Verification of his Text xxxi Sect. III. Purchase and Sale 43 E. Chronology of Astronomical Authorities, according to Astrono- Sect. IV. A Problem 45 mers of Ujjayani xxxiii Sect. V. Alligation 46 F. Age of Brahmegupta, from astronomical data xxxv Sect. VI. Permutation and Combination ... 49 G. Aryabhatta's Doctrine xxxvii Chapter V. Progression.

H. (Reference from p. ix. 1.21.) Scautlness of Additions by later Sect. I. Arithmetical 51 Writers on Aljgfpbra xl Sect. II. Geometrical 55 I. Age of AnrABHATTA xU Chapter VI. Plane FSgure 58 K. Writings and Age of Vara'ha-mihira xlv Chapter VII. Excavations and Content of Solids 97 VIII. Stacks L. Introduction and Progress of Algebra among the Italians ... li Chapter 100 M. Arithmetics of Diophantus Ixi Chapter IX. Saw 101 of Grain N. Progress and Proficiency of the Arabians in Algebra Ixiv Chapter X. Mound i03 of a O. Communication of the Hindus with Western Nations on Astro- Chapter XI. Shadow Gnomon 106 Chapter XII. Pulverizer fCM^/aca) logy and Astronomy Ixxviii 112 Chapter XIII. Combination 123 BHASCARA.

ARITHMETIC (Lildvatl) ALGEBRA (Vija-gatiita.)

I. Chapter Introduction. Axioms. Weights and Measures . . 1 Chapter I. Sect. I. Invocation, &c 129 Chapter II. Sect. I. Invocation. Numeration 4 Sect. II. Algorithm of Negative and Affirmative

Sect. II. Eight Operations of Arithmetic : Addition, &c. 5 Quantities 133 Sect. III. Fractions 13 Sect. III. of Cipher 136

Sect. IV. Cipher 19 Sect. IV. of Unknown Quantity . . ] 39 CONTENTS. Sect. V. of Surds 145 Chapter II. Pulverizer CONTENTS. 156 Chapter III. Afiscellaneous. Square. Sect. I. Inversion 21 Chapter III. Affected Sect. I. 170 '. Sect. II. Supposition 23 Sect. II. Cyclic Method . 175 Sect. III. Miscellaneous 179 Sect. III. Concurrence . ; 26 IV. Simple liquation 185 I Sect. IV. Problem concerning Squares ... 27 Chapter Quadratic, Equations 207 Sect. V. Assimilation 29 Chapter V. &c. Sect. VI. Rule of Proportion 33 Chapter VI. Multiliteral Equations 227 Chapter IV. Mixture. Chapter VII. Varieties of Quadratics 245 VIII. Equation involving a Factum of Unknown Quantities 268 Sect. I. Interest 39 Chapter IX. Conclusion 275 Sect. II. Fractions 42 Chapter Sect. III. Purchase and Sale 43 Sect. IV. A Problem 45 BRAHMEGUPTA. Sect. V. Alligation 46 Sect. VI. Permutation and Combination ... 49 CHAPTER XII. ARITHMETIC (Gariita.) Chapter V. Progression. Sect. I. Algorithm 277 Sect. I. Arithmetical 51 Sect. II. Mixture 287 Sect. II. Geometrical 55 Sect. III. Progression 29() Chapter VI. Plane FSgure 58 Sect. IV. Plane Figure 305 Chapter VII. Excavations and Content of Solids 97 Sect. V. Excavations 312 Chapter VIII. Stacks 100 Sect. VI. Stacks 314 Chapter IX. Saw 101 Sect. VII. Saw 315 Chapter X. Mound of Grain i03 &c^ F7//. Mounds of Grain 316 Chapter XI. Shadow of a Gnomon 106 Sect. IX. Measure by Shadow 317 Chapter XII. Pulverizer fCM^/aca) 112 Sect. X. Supplement 319 Chapter XIII. Combination 123 Monday, November 14, 2011 CHAPTER XVIII. ALGEBRA (CuHaca.)

(Vija-gatiita.) ALGEBRA Sect. I. Pulverizer 325 Sect. II. Algorithm 339 Chapter I. Sect. I. Invocation, &c 129 Sect. III. Simple Equation 344 Sect. II. Algorithm of Negative and Affirmative Sect. IV. Quadratic Equation 346 Quantities 133 Sect. V. Equation of several unknown . . . 348 Sect. III. of Cipher 136 Sect. VI. Equation involving a factum ... 361 Sect. IV. of Unknown Quantity . . ] 39 Sect. VII. Square affected by coefficient . . . 363 Sect. V. of Surds 145 Sect. VIII. Problems 373 Chapter II. Pulverizer 156 Structure de la Lilavati Structure du Bijaganita

8 operations 8 practices 6 methods 4 equations parikarman vyavahāra ! + saṅkalita + x +.. +... 2 - vyavakalita + - x

x pratyutpanna # x x y ÷ bhāgahāra ! ÷ ! 2 a varga △ a2

vargamūla ◎ !a !a 3 a ghana "

3 !a ghanamūla !

Monday, November 14, 2011 ghanamūla BG 2 pūrvaṃ proktaṃ vyaktaṃ avyaktabījaṃ prāyaḥ praśnāḥ no vināvyaktayuktyā/ jñātum śakyāḥ mandadhībhir nitāntam yasmāt tasmāt vacmi bījakriyāṃ ca//

BG.2.1 Since the visible [mathematics] (i.e., mathematics with known numbers) told before [by me in the L] has the invisible [mathematics] as its seed, and since, without reasoning of the invisible [mathematics], problems can hardly be understood (i.e., solved) [even by intelligent persons and] not at all by less- intelligent persons, I speak about the operations with seeds.

Monday, November 14, 2011 3 + x ! ÷ x ÷ + 8 operations 6 rules 2 3 ! - a a a2 - !

Whole Positives rūpa ṛṇa dhana numbers and negatives

bhinna Fractions Zero śunya

śunya Zero Undeter- mined avyakta

surds karaṇī

Monday, November 14, 2011 CONTENTS. CONTENTS. Chapter III. Afiscellaneous. Sect. I. Inversion 21 Sect. II. Supposition 23

ARITHMETIC (Lildvatl) ALGEBRA (Vija-gatiita.)

I. Chapter Introduction. Axioms. Weights and Measures . . 1 Chapter I. Sect. I. Invocation, &c 129 Chapter II. Sect. I. Invocation. Numeration 4 Sect. II. Algorithm of Negative and Affirmative

Sect. II. Eight Operations of Arithmetic : Addition, &c. 5 Quantities 133 Sect. III. Fractions 13 Sect. III. of Cipher 136

Inversion

Suppostion

Concurence saṃkramaṇa

Barter and Exchange

Proportions trairāśikādi

Monday, November 14, 2011 — —

CHAPTER II.

PULVERIZER.'

53—64. 'Rule: In the first place, as preparatory to the investigation of the pulverizer, the dividend, divisor, and atlditive quantity are, if practicable, to be reduced by some number.' If the number, by which the dividend and divisorPulveriserare both measured, do not also measure the additive quantity, the question is an ill put [or impossible] one.* 54 55—56. The last remainder, when the dividend and divisor are mutually divided, is their commonPULVERIZER.measure.* Being divided by that common157

measure, they are termed reduced quantities. Divide mutually the reduced * This is nearly word for word the same with a chapter in the Lildtati on the same subject. dividend and divisor, until unity be the remainder in the dividend. Place (Li7. Ch. 12.) See there, explanations of the terras. the quotients other; the additive quantity beneath them, The method here onetaughtunderis applicablethe chiefly toandthe solution of indeterminate problems that pro- duceandequationscipher involvingat the bottom.more than oneByunknownthe penultquantity.multiplySee ch. 6.the number next above * it, andTen stanzasadd theand lowesttwo halves.term. Then reject the last and repeat the operation * If the dividend and divisor admit a common measure, they must be first reduced by it to their until a pair of numbers be left. The uppermost of these being abraded by least terras; else unity will not be the residue of reciprocal division; but the common measure the reduced dividend, the remainder is the quotient. The other [or lower- will; (or, going a step further, nought.) Ga'n. on Lil. Crishn. on Vy. in like divisor, the is most]* If thebeingdividend and divisormannerhave abradeda common bymeasure,the reducedthe additive also must admitremainderit; and the

threethe multiplier.'terms be correspondently reduced: for the additive, nnkss it be [nought or else] a multiple

of the divisor, must, if negative, equal the residue of a division of the dividend taken into the mul-

tiplier by the divisor; and, if affirmative, must equal the complement of that residue to the divisor. could not measure the remaining portion or residue, if it were greater than it. When therefore the Now, if dividend and divisor be reducible to less terras, the residue of division of the reduced terms, greater number, divided by the lesi, yields a residue, the greatest common measure, in such case, multiplied by the common measure, is equal to the residue of division of the unreduced terms. is equal to ihe remainder, provided this be a measure of the less. If again the less number, divided Therefore the additive, whether equal to the residue, or to its complement, must be divisible by the by the remainder, yield a residue, the common measure cannot exceed this residue ; for the same common measure. Crishn. reason. Therefore no number, though less than the first remainder, can be a common measure, if * two for it The common measure may equal, but cannot exceed, the least of the numbers : it exceed the second remainder: and the greatest common measure is equal to the second remain- Monday, November 14, 2011 must divide it. If it be less, the greater may be considered as consisting of two terras, one the der, provided it measure the first; for then of course it measures the multiple of it, which is the quotient taken into the divisor, the other the residue. The common measure cannot exceed that other portion of the second number. So, if there be a third remainder, the greatest common mea- residue; for, as it measures the divisor, it must of course measure the multiple of the divisor, and sure is either equal to it, if it measure the second ; or is less. Hence the rule, to divide the greater

number by the less, and the less by the remainder, and each residue by the remainder following,

until a residue be found, which exactly measures the preceding one; such last remainder is the

common measure. (§ 54). CRfsHN.

' The substance of Cri'shn'a's demonstration is as follows: When the dividend, taken into the

multiplier, is exactly measured by the divisor, the additive must either be null or a multiple of the

divisor. (See § 63). If the dividend be such, that, being multiplied by the multiplicator and di-

vided by the divisor, it yields a residue, the additive, if negative, must be equal to that remainder;

(and then the subtractive quantity balances the residue;) or, if affirmative, it must be equal to the

difference between the divisor and residue ; (and so the addition of that quantity completes the

amount of the divisor;) or else it must be equal to the residue, or its complement, with the divisor or a multiple of the divisor added. Let the dividend be considered as composed of two portions

or terms: 1st, a multiple of the divisor; 2d, the overplus or residue. The first multiplied by the

multiplier (whatever this be), is of course measured by the divisor. As to the second, or overplus

and remainder, the additive being negative, both disappear when the multiplier is quotient of the

additive divided by the remainder, (the additive being a multiplier of the residue.) But, if the

additive be not a multiple of the remainder, should unity be the residue at the first step of the re-

ciprocal division, the multiplier will be equal to the additive, if this be negative, or to its comple-

ment to the divisor, if it be positive; and the corresponding quotient will be equal to the quotient

of the dividend by the divisor multiplied by the multiplicator, if the additive be negative; or be

equal to the same with addition of unity, if it be affirmative: and, generally, when reciprocal divi-

sion has reached its last step exhibiting a remainder of one, the multiplier, answering to the pre-

ceding residue, become the divisor, as serving for that next before it become dividend, is equal to Datta Singh p. 128: “The eight “fundamental” operations of Hindu gaṇita are (1) addition, (2) subtraction, (3) mutliplication, (4) division, (5) square, (6) square-root, (7) cube, (8) cube-root. Most of these elementary processes have not been mentioned in the Siddhânta works”

Monday, November 14, 2011 Bhāskarācārya (b.1114), Siddhānta Śiromāṇi including Līlāvatī and Bījagaṇita

Śrīdharācārya (ca.900), Pāṭīgaṇita and anonymous undated commentary

K. S. Shukla. Pāgaṇita of Śrīdharācarya. Lucknow University, Lucknow, 1959.

Brahmagupta (629) Brahmasphuṭṭasidhānta Pṛthūdakṣvamin (ca.980) Vāsanābhāṣya

Monday, November 14, 2011 Monday, November 14, 2011 Monday, November 14, 2011 Monday, November 14, 2011 3 + x ! ÷ x ÷ + 8 operations 6 rules 2 3 ! - a a a2 - !

Whole Positives rūpa ṛṇa dhana numbers and negatives

bhinna Fractions Zero śunya

śunya Zero Undeter- mined avyakta

surds karaṇī 4 fold class

Monday, November 14, 2011 bhāgajāti part class

a1 a2 b2a1+a2b1 + b1 b2 b1 b2

LV 30 anyonyahārābhihatau harāṃśau rāśyor samachedavidhānam evaṃ\\ mithas harābhyām apavartitābhyām yadvā harāṃśau sudhiyā atra guṇyau//\\

The numerator and denominator being multiplied reciprocally by the denominators of the two quantities, they are thus reduced to the same denominators. Or both numerator and denominator may be multiplied by the intelligent calculator into the reciprocal denominators abridged by a common measure.

1 1 53 3+ + 1 5 3 15

Monday, November 14, 2011 a1 a2 prabhāgajāti different part class x b1 b2

1 a2 a1a2 a x b1 b2 b1b2

LV032 lavāllavaghnāś ca harāḥ haraghnāḥ bhāgaprabhāgeṣu savarṇanam syāt\\ The numerators multiplied by the numerators, and the denominators by the denominators will be samecoloured when [they are] different parts .

1 1 3 1 1 1 x x 2 x x x 1 2 3 4 16 4 1280

Monday, November 14, 2011 (sva)bhāgānubandhaṃjāti one’s own part additive class

kālasaṃvarṇa samecoloured portion a1 n b1 a1(b2 +a2) b1 n+a1 a1 a2 b1 b2 b1 b1 b2

LV034 chedaghnarūpeṣu lavāḥ dhanaṛṇam ekasya bhāgās adhikaunakāś ced// svāṃśādhikaunas khalu yatra tatra bhāgānubandhe ca lavāpavāhe/ talasthahāreṇa haram nihanyāt svāṃśādhikaunena tu tena bhāgān// The integer being multiplied by the denominator, the numerator is made positive or negative, provided parts of an unit be added or be subtractive. But, if indeed the quantity be increased or diminished by a part of itself, then, in the addition and subtraction of fractions, multiply the denominator by the denominator standing underneath, and the numerator by the same augmented or lessened by it own numerator.

n+a1/b1 a1/b1 + (a1/b1 x a2/b2)

Monday, November 14, 2011 svabhāgāpavahojāti one’s own subtractive part class a1 n b1 a1 (b2 -a2) b1 n-a1 -a1 2 -a b1 b2 b1 b1 b2

n+a1/b1 a1/b1 - (a1/b1 x a2/b2)

bhāgāmātajāti mother part class

Monday, November 14, 2011 bhinnasaṅkalita addition of fractions

LV037. yogas antaram tulyaharāṃśakānām kalpyas haras rūpam ahārarāśeḥ// The sum or the difference of fractions having the a common denominator, is [taken]. Unity is put denominator of a quantity which has no divisor.

Monday, November 14, 2011 Bhāskarācārya (b.1114), Siddhānta Śiromāṇi including Līlāvatī and Bījagaṇita

Śrīdharācārya (ca.900), Pāṭīgaṇita and anonymous undated commentary

K. S. Shukla. Pāgaṇita of Śrīdharācarya. Lucknow University, Lucknow, 1959.

Brahmagupta (629) Brahmasphuṭṭasidhānta Pṛthūdakṣvamin (ca.980) Vāsanābhāṣya

Monday, November 14, 2011