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 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