History of Hindu Mathematics: A Source Book v2.c14: Contents
History of Hindu Mathematics: A Source Book v2.c14
- Language
- English
Contents
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CHAPTER III
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ALGEBRA
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| | | PAGE |
| 1. | GENERAL FEATURES | 1 |
| | Name for Algebra—Algebra defined—Distinction from Arithmetic—Importance of Algebra—Scope of Algebra—Origin of Hindu Algebra. | |
| 2. | TECHNICAL TERMS | 9 |
| | Coefficient—Unknown Quantity—Power—Equation—Absolute Term. | |
| 3. | SYMBOLS | 12 |
| | Symbols of Operation—Origin of Minus Sign—Symbols for Powers and Roots—Symbols for Unknowns. | |
| 4. | LAWS OF SIGNS | 20 |
| | Addition—Subtraction—Multiplication—Division—Evolution and Involution. | |
| 5. | FUNDAMENTAL OPERATIONS | 25 |
| | Number of Operations—Addition and Subtraction—Multiplication—Division—Squaring—Square-root. | |
| 6. | EQUATIONS | 28 |
| | Forming Equations—Plan of writing Equations—Preparation of Equations—Classification of Equations. | |
| 7. | LINEAR EQUATIONS IN ONE UNKNOWN | 36 |
| | Early Solutions—Rule of False Position—Disappearance from later Algebra—Operation with an Optional Number—Solution of Linear Equations. | |
| 8. | LINEAR EQUATIONS WITH TWO UNKNOWNS | 43 |
| | Rule of Concurrence—Linear Equations. | |
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A type of Linear Equations—Solution by False Position—Second Type—Third Type—Brahmagupta's Rule—Mahâvîra's Rule—Bhâskara's Rule.
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10. QUADRATIC EQUATIONS 59
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Early Treatment—Bakhshâlî Treatise—Âryabhaṭa I—Brahmagupta's Rules—Śrîdhara's Rule—Mahâvîra—Âryabhaṭa II—Śrîpati's Rules—Bhâskara II's Rules—Elimination of the Middle Term—Two roots of the Quadratic—Known to Mahâvîra—Brahmagupta.
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11. EQUATIONS OF HIGHER DEGREES 76
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Cubic and Biquadratic—Higher Equations.
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12. SIMULTANEOUS QUADRATIC EQUATIONS 81
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Common Forms—Rule of Dissimilar Operations—Mahâvîra's Rules.
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13. INDETERMINATE EQUATIONS OF THE FIRST DEGREE 87
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General Survey—Its Importance—Three Varieties of Problems—Terminology—Origin of the name—Preliminary Operations—Solution of bx-ay=± c—Âryabhaṭa I's Rule—Bhâskara I's Rules—Brahmagupta's Rules—Mahâvîra's Rules—Âryabhaṭa II—Śrîpati's Rule—Bhâskara II's Rules—Solution of by=ax± 1—Constant Pulveriser—Bhâskara I's Rule—Brahmagupta's Rule—Bhâskara II's Rule—Solution of by+ax=± c—Brahmagupta's Rule—Bhâskara II's Rule—Nârâyaṇa—Illustrative Examples—Particular Cases.
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14. ONE LINEAR EQUATION IN MORE THAN TWO UNKNOWNS 125
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15. SIMULTANEOUS INDETERMINATE EQUATIONS OF THE FIRST DEGREE 127
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Śrîpati's Rule—Bhâskara II's Rule—Special Rules—General Problem of Remainders—Conjunct Pulveriser—Generalised Conjunct Pulveriser—Alternative method.
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16. SOLUTION OF Nx²+1=y² 141
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Square-nature—Origin of the Name—Technical Terms—Brahmagupta's Lemmas—Description by later writers—Principle of Composition—General Solution of the Square-nature—Another Lemma—Rational Solution—Śrîpati's Rational Solution—Illustrative Examples—Solution in Positive Integers.
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17. CYCLIC METHOD 161
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Cyclic Method—Bhâskara's Lemma—Bhâskara's Rule—Nârâyaṇa's Rule—Illustrative Examples.
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18. SOLUTION OF Nx²± c=y² 173
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Form Mn²x²± c=y²—Form a²x²± c=y²—Form c-Nx²=y²—Form Nx²-k²=y².
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19. GENERAL INDETERMINATE EQUATIONS OF THE SECOND DEGREE : SINGLE EQUATIONS 181
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Solution—Solution of ax²+bx+c=y²—Solution of ax²+bx+c=a'y²+b'y+c'—Solution of ax²+by²+c=z²—Solution of a²x²+by²+c=z²—Solution of ax²+bxy+cy²=z².
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20. RATIONAL TRIANGLES 204
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Early Solutions—Later Rational Solutions—Integral Solutions—Mahâvîra's Definitions—Right Triangles having a Given Side—Right Triangles having a Given Hypotenuse—Problems involving Areas and Sides—Problems involving Sides but not Areas—Pairs of Rectangles—Isosceles Triangles with Integral Sides—Juxtaposition of Right Triangles—Isosceles Triangles with a Given Altitude—Pairs of Rational Isosceles Triangles—Rational Scalene Triangles—Triangles having a Given Area.
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21. RATIONAL QUADRILATERALS 228
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Rational Isosceles Trapeziums—Pairs of Isosceles Trapeziums—Rational Trapeziums with Three Equal Sides—Rational Inscribed Quadrilaterals—Inscribed Quadrilaterals having a Given Area—Triangles and Quadrilaterals having a Given Circum-Diameter.
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Mahâvîra's Rule—Bhâskara's Method—Nârâyaṇa's Rule—Form ax2n+2+bx2n=y²—Equation ax⁴+bx²+c=u³.
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23. LINEAR FUNCTIONS MADE SQUARES OR CUBES 250 Square-pulveriser—Cube-pulveriser—Equation bx± c=ay²—Equation bx± c=ayn.
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24. DOUBLE EQUATIONS OF THE FIRST DEGREE 258
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25. DOUBLE EQUATIONS OF THE SECOND DEGREE 266 First Type—Second Type.
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26. DOUBLE EQUATIONS OF HIGHER DEGREES 278 Double Equations in several Unknowns.
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27. MULTIPLE EQUATIONS 283
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28. SOLUTION OF axy=bx+cy+d 297 Bakhshâlî work—An Unknown Author's Rule—Brahmagupta's Rule—Mahâvîra's Rule—Śrîpati's Rule—Bhâskara II's Rule—Bhâskara's Proofs.
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INDEX 309
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*Transliteration* XIII
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*List of abbreviations* XV