History of Hindu Mathematics: A Source Book v2: Volume 2

History of Hindu Mathematics: A Source Book v2

Language: English

Volume 2

Front matter

Part II: Algebra

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BY BIBHUTIBHUSAN DATTA AND AVADHESH NARAYAN SINGH

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इदं नम ऋषिभ्यः पूर्वजेभ्यः पूर्वेभ्यः पथिकृद्भ्यः
(*RV*, x. 14. 25)

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*To the Seers, our Ancestors, the first Path-makers*

Preface

The present work forms Part II of our History of Hindu Mathematics and is devoted to the history of Algebra in India. It is intended to be a source book, and the subject is treated topicwise. Under each topic are collected together and set forth in chronological order translations of relevant Sanskrit texts as found in the Hindu mathematical works. This plan necessitates a certain amount of repetition. But it shows to the reader at a glance the improvements made from century to century

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To gather materials for the book we have examined all the published mathematical treatises of the Hindus as well as most of the important manuscripts available in Indian libraries, a list of the most important of which has already been included in Part I. We have great pleasure in once more expressing our thanks to the authorities of the libraries at Madras, Bangalore, Trivandrum, Tripunithura, Baroda, Jammu, and Benares, and those of the India Office (London) and the Asiatic Society of Bengal for supplying transcripts of manuscripts or sending them to us for consultation. We are indebted also to Dr. R. P. Paranjpye, Vice-Chancellor of the Lucknow University, for help in securing for our use several manuscripts or their transcripts from the State libraries in India and the India Office.

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In translating Sanskrit texts we have tried to be as literal and faithful as possible without sacrificing the spirit of the original, in order to preserve which we have at a few places used literal translations of Sanskrit technical terms instead of modern terminology. For instance, we have used the term 'pulveriser' for the equation *ax*+*by*=1, and the term 'Square-nature' for the equation *Nx*²+*c*=*y*².

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The use of symbols—letters of the alphabet to denote unknowns—and equations are the foundations of the science of algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra.

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A portion of the subject matter of this book has been available to scholars through papers by various authors and through Colebrooke's *Algebra with Arithmetic and Mensuration from the Sanscrit of Brahmegupta and Bhascara*, but about half of it is being presented here for the first time. For want of space it has not been possible to give a detailed comparison of the Hindu achievements in Algebra with those of other nations. For this the reader is referred to the general works on the history of mathematics by Cantor, Smith, and Tropfke, to Dixon's *History of the Theory of Numbers* and to Neugebauer's *Mathematische Keilschrift-Texte*. A study of this book along with the above standard works will reveal to the reader the remarkable progress in algebra made by the Hindus at an early date. It will also show that we are indebted to the Hindus for the technique and the fundamental results of algebra just as we owe to them the place-value notation and the elements of our arithmetic.

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We have pleasure in expressing our thanks to Mr. T. N. Singh and Mr. Ahmad Ali for help in correcting proofs and to Mr. R. D. Misra for preparing the index to this volume.

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<pre> LUCKNOW BIBHUTIBHUSAN DATTA *March, 1938* AVADHESH NARAYAN SINGH </pre>

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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

Transliteration

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VOWELS

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| Short : | अ | इ | उ | ऋ | ऌ | | | | |
| | *a* | *i* | *u* | *ṛ* | *ḷ* | | | | |
| | | | | | | | | | |
| Long : | आ | ई | ऊ | ॠ | ॡ | ए | ऐ | ओ | औ |
| | *â* | *î* | *û* | *ṝ* | *ḹ* | *e* | *ai* | *o* | *au* |

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| *Anusvâra* : | ं = *ṁ* |
| *Visarga* : | ः = *ḥ* |
| Non-aspirant : | ऽ = ’ |

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CONSONANTS

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| Classified : | क् | ख् | ग् | घ् | ङ् |
| | *k* | *kh* | *g* | *gh* | *ṅ* |
| | | | | | |
| | च् | छ् | ज् | झ् | ञ् |
| | *c* | *ch* | *j* | *jh* | *ñ* |
| | | | | | |
| | ट् | ठ् | ड् | ढ् | ण् |
| | *ṭ* | *ṭh* | *ḍ* | *ḍh* | *ṇ* |

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| | त् | थ्. | द् | ध् | न् | |
| | *t* | *th* | *d* | *dh* | *n* | |
| | | | | | | |
| | प् | फ् | ब् | भ् | म् | |
| | *p* | *ph* | *b* | *bh* | *m*, *ṃ* — final म् | |
| | | | | | | |
| Un-classified : | य् | र् | ल् | व् | श् ष् स् | ह् |
| | *y* | *r* | *l* | *v* | *ś* *ṣ* *s* | *h* |
| | | | | | | |
| Compound : | क्ष् | त्र् | ज्ञ् | | | |
| | *kṣ* | *tr* | *jñ* | | | |
| | | | | | | |
| Pâlî : | ∞ = l' | | | | | |

List of Abbreviations

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Chapter III. Algebra

1. General Features

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**Name for Algebra.** The Hindu name for the science of algebra is *bîjagaṇita*. *Bîja* means "element" or "analysis" and *gaṇita* "the science of calculation." Thus *bîjagaṇita* literally means "the science of calculation with elements" or "the science of analytical calculation." The epithet dates at least as far back as the time of Pṛthûdakasvâmî (860) who used it. Brahmagupta (628) calls algebra *kuṭṭaka-gaṇita*, or simply *kuṭṭaka*.¹ The term *kuṭṭaka*, meaning "pulveriser", refers to a branch of the science of algebra dealing particularly with the subject of indeterminate equations of the first degree. It is interesting to find that this subject was considered so important by the Hindus that the whole science of algebra was named after it in the beginning of the seventh century. Algebra is also called *avyakta-gaṇita* or "the science of calculation with unknowns" (*avyakta*=unknown) in contradistinction to the name *vyakta-gaṇita* or "the science of calculation with knowns" (*vyakta*=known) for arithmetic including geometry and mensuration.

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¹ See Bibhutibhusan Datta, "The scope and development of the Hindu Gaṇita," *IHQ*, V, 1929, pp. 479-512; particularly pp. 489f.

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**Algebra Defined.** Bhâskara II (1150) has defined algebra thus:

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"Analysis (*bîja*) is certainly the innate intellect assisted by the various symbols (*varṇa*), which, for the

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instruction of duller intellects, has been expounded by the ancient sages who enlighten mathematicians as the sun irradiates the lotus; that has now taken the name algebra (*bîjagaṇita*).”¹

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¹ *BBi*, p. 99.

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That algebraic analysis requires keen intelligence and sagacity has been observed by him on more than one occasion.

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“Neither does analysis consist in symbols, nor are there different kinds of analyses; sagacity alone is analysis, for wide is imagination.”²

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² *BBi*, p. 49; *SiŚi*, *Gola*, xiii. 5.

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“Analysis is certainly clear intelligence.”³

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³ *L*, p. 15; *SiŚi*, *Gola*, xiii. 3.

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“Or intelligence alone is analysis.”⁴

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⁴ *BBi*, p. 49.

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In answer to the question, “if (unknown quantities) are to be discovered by intelligence alone what then is the need of analysis ?” he says:

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“Because intelligence is certainly the real analysis; symbols are its helps. The innate intelligence which has been expressed for the duller intellects by the ancient sages, who enlighten mathematicians as the sun irradiates the lotus, with the help of various symbols, has now obtained the name of algebra.”⁵

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⁵ *BBi*, p. 100.

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Thus, according to Bhâskara II, algebra may be defined as the science which treats of numbers expressed by means of symbols, and in which there is scope and primary need for intelligent artifices and ingenious devices.

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**Distinction from Arithmetic.** What distinguishes algebra from arithmetic, according to the Hindus, will be found to some extent in their special names. Both deal with symbols. But in arithmetic the values of the symbols are *vyakta*, that is, known and definitely determinate, while in algebra they are *avyakta*, that is, unknown, indefinite. The relation between these two branches of *gaṇita* is considered by Bhâskara II to be this:

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“The science of calculation with unknowns is the source of the science of calculation with knowns.”¹

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He has put it more explicitly and clearly thus:

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“Algebra is similar to arithmetic in respect of rules (of fundamental operations) but appears as if it were indeterminate. It is not indeterminate to the intelligent; it is certainly not sixfold,² but manifold.”³

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The true distinction between arithmetic and algebra, besides that of symbols employed, lies, in the opinion of Bhâskara II, in the demonstration of the rules. He remarks:

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the other symbolical (*râśigata*).”¹ We do not know who was the first in India to use geometrical methods for demonstrating algebraical rules. Bhâskara II (1150) ascribes it to “ancient teachers.”²

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¹ *BBi*, p. 125.

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² *BBi*, p. 127.

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**Importance of Algebra.** The early Hindus regarded algebra as a science of great importance and utility. In the opening verses of his treatise³ on algebra Brahmagupta (628) observes:

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³ Forming chapter xviii of his *Brâhma-sphuṭa-siddhânta*.

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“Since questions can scarcely be known (*i.e.*, solved) without algebra, therefore, I shall speak of algebra with examples.

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“By knowing the pulveriser, zero, negative and positive quantities, unknowns, elimination of the middle term, equations with one unknown, factum and the Square-nature, one becomes the learned professor (*âcârya*) amongst the learned.”⁴

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⁴ *BrSpSi*, xviii. 1-2.

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Similarly Bhâskara II writes:

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“What the learned calculators (*sâmkhyâḥ*) describe as the originator of intelligence, being directed by a wise being (*satpuruṣa*) and which alone is the primal cause (*bîja*) of all knowns (*vyakta*), I venerate that Invisible God as well as that Science of Calculation with Unknowns... Since questions can scarcely be solved without the reasoning of algebra—not at all by those of dull perceptions—I shall speak, therefore; of the operations of analysis.”⁵

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⁵ In the first part of this passage every principal term has been used with a double significance. The term *sâmkhyâḥ* (literally, “expert calculators”) signifies the “Sâmkhya philosophers” in one sense, “mathematicians” in the other; *satpuruṣa* “the self-existent being of the Sâmkhya philosophy” or “a wise mathematician”; *vyakta* “manifested universe” or “the science of calculation with knowns.”

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Nârâyaṇa (1350) remarks:

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"I adore that Brahma, also that science of calculation with the unknown, which is the one invisible root-cause of the visible and multiple-qualitied universe, also of multitudes of rules of the science of calculation with the known."

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"As out of Him is derived this entire universe, visible and endless, so out of algebra follows the whole of arithmetic with its endless varieties (of rules). Therefore, I always make obeisance to Śiva and also to (*avyakta-*) *gaṇita* (algebra)."

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He adds:

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"People ask questions whose solutions are not to be found by arithmetic; but their solutions can generally be found by algebra. Since less intelligent men do not succeed in solving questions by the rules of arithmetic, I shall speak of the lucid and easily intelligible rules of algebra."

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**Scope of Algebra.** The science of algebra is broadly divided by the Hindus into two principal parts. Of these the most important one deals with analysis (*bîja*). The other part treats of the subjects which are essential for analysis. They are: the laws of signs, the arithmetic of zero (and infinity), operations with unknowns, surds, the pulveriser (or the indeterminate equation of the first degree), and the Square-nature (or the so-called Pellian equation). To these some writers add concurrence and dissimilar operations, while others include them in arithmetic. At the end of the first section of his treatise on algebra Bhâskara II is found to have observed as follows:

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¹ ^1 *NBi*, I, R. 1.

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² ^2 *NBi*, II, R. 1.

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³ ^3 *NBi*, I, R. 5-6.

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⁴ ^4 All writers, except Brahmagupta and Śrîpati, are of the latter opinion.

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"(The section of) this science of calculation which is essential for analysis has been briefly set forth. Next I shall propound analysis, which is the source of pleasure to the mathematician."¹

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¹ *BBi*, p. 43.

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Analysis is stated by all to be of four kinds, for equations are classified into four varieties (*vide infra*). Thus each class of equations has its own method of analysis.

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**Origin of Hindu Algebra.** The origin of Hindu algebra can be definitely traced back to the period of the *Sulba* (800-500 B.C.) and the *Brâhmaṇa* (*c.* 2000 B.C.). But it was then mostly geometrical.² The geometrical method of the transformation of a square into a rectangle having a given side, which is described in the important *Sulba* is obviously equivalent to the solution of a linear equation in one unknown, *viz.*, $ax = c².$ The quadratic equation has its counterpart in the construction of a figure (an altar) similar to a given one but differing in area from it by a specified amount. The usual method of solving that problem was to increase the unit of measure of the linear dimensions of the figure. One of the most important altars of the obligatory Vedic sacrifices was called the *Mahâvedi* (the Great Altar). It has been described to be of the form of an isosceles trapezium whose face is 24 units long, base 30 and altitude 36. If x be the enlarged unit of measure taken in increasing the size of the altar by m units of area, we must have $36x × ((24x + 30x))/2 = 36 × ((24 + 30))/2 + m,$ or 972x² = 972 + m.

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² Bibhutibhusan Datta, *The Science of the Śulba*, Calcutta, 1932.

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Therefore x = √(1 + m/972).

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If m be put equal to 972 (n—1), so that the area of the enlarged altar is n times its original area, we get x = √n, some particular cases of which are described in the *Sulba*. The particular cases, when n = 14 or 14 3/7, are found as early as the *Śatapatha Brâhmaṇa*¹ (*c.* 2000 B.C.).

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¹ *ŚBr*, X. 2. 3. 7ff.

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The most ancient and primitive form of the "Fire-altar for the sacrifices to achieve special objects" was the *Śyenacit* (or "the altar of the form of the falcon").

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Fig. 1.

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Its body (*ABCD*) consists of four squares of one square *puruṣa* each; each of its wings (*EFGH*, *E'F'G'H'*) is a rectangle of one *puruṣa* by one *puruṣa* and a *prâdeśa* (= 1/10 of a *puruṣa*). This Fire-altar was enlarged in two ways: *first*, in which all the constituent parts were affected in the same proportion; *second*, in which the breadth of the portions *LFGM* and *L'F'G'M'* of the wings were left unaffected. If *x* be the enlarged unit for enlargement in the first case we shall have to solve the quadratic equation

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$2x × 2x + 2x × (x + x/5) + x × (x + x/10) = 7½ + m,$

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where *m* denotes the increment of the Fire-altar in size.

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$Therefore x² = 1 + 2m/15.$

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In particular, when m = 94, we shall have

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$x² = 13 8/15 = 14 (approximately),$

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which occurs in the *Śatapatha Brāhmaṇa*.

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In the second case of enlargement the equation for *x* will be

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$2x × 2x + 2x × (x + ⅕) + x × (x + 1/10) = 7½ + m,$ $or 7x² + ½x = 7½ + m,$

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which is a complete quadratic equation.

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The problem of altar construction gave rise also to certain indeterminate equations of the second degree such as,

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(1) & x² + y² = z², / (2) & x² + a² = z²;

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and simultaneous indeterminate equations of the type

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ax + by + cz + dw &= p, / x + y + z + w &= q.

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Further particulars about these equations will be given later on.

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**Coefficient.** In Hindu algebra there is no systematic use of any special term for the coefficient. Ordinarily, the power of the unknown is mentioned when the reference is to the coefficient of that power. In explanation of similar use by Brahmagupta his commentator Pṛthûdakasvâmî writes “the number (*aṅka*) which is the coefficient of the square of the unknown is called the ‘square’ and the number which forms the coefficient of the (simple) unknown is called ‘the unknown quantity.’ ”¹ However, occasional use of a technical term is also met with. Brahmagupta once calls the coefficient *saṁkhyâ*² (number) and on several other occasions *guṇaka*, or *guṇakâra* (multiplier).³ Pṛthûdakasvâmî (860) calls it *aṅka*⁴ (number) or *prakṛti* (multiplier). These terms reappear in the works of Śrîpati (1039)⁵ and Bhâskara II (1150).⁶ The former also used *rûpa* for the same purpose.⁷

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¹ *BrSpSi*, xviii. 44 (*Com*).

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² *BrSpSi*, xviii. 63.

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³ *BrSpSi*, xviii. 64, 69-71.

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⁴ *BrSpSi*, xviii. 44 (*Com*).

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⁵ *SiŚe*, xiv. 33-5.

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⁶ *BBi*, pp. 33-4.

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⁷ *SiŚe*, xiv. 19.

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**Unknown Quantity.** The unknown quantity was called in the *Sthânâṅga-sûtra*⁸ (before 300 B.C.) *yâvat-tâvat* (as many as or so much as, meaning an arbitrary quantity). In the so-called Bakhshâlî treatise, it was called *yadṛcchâ*, *vâñchâ* or *kâmika* (any desired quantity).⁹ This term was originally connected with the Rule of False Position.¹⁰ Âryabhaṭa I (499)

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⁸ *Sûtra* 747; *cf.* Bibhutibhusan Datta, “The Jaina School of Mathematics,” *BCMS*, XXI, 1929, pp. 115-145; particularly pp. 122-3.

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⁹ *BMs*, Folios 22, verso; 23, recto & verso.

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¹⁰ Bibhutibhusan Datta, “The Bakshshâlî Mathematics,” *BCMS*, XXI, pp. 1-60; particularly pp. 26-8, 66.

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calls the unknown quantity *gulikâ* (shot). This term strongly leads one to suspect that the shot was probably then used to represent the unknown. From the beginning of the seventh century the Hindu algebraists are found to have more commonly employed the term *avyakta* (unknown).¹

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¹ *BrSpSi*, xviii. 2, 41; *SiŚe*, xiv. 1-2; *BBi*, pp. 7ff.

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**Power.** The oldest Hindu terms for the power of a quantity, known or unknown, are found in the *Uttarâdhyayana-sûtra* (c. 300 B.C. or earlier).² In it the second power is called *varga* (square), the third power *ghana* (cube), the fourth power *varga-varga* (square-square), the sixth power *ghana-varga* (cube-square), and the twelfth power *ghana-varga-varga* (cube-square-square), using the multiplicative instead of the additive principle. In this work we do not find any method for indicating odd powers higher than the third. In later times, the fifth power is called *varga-ghana-ghâta* (product of cube and square, *ghâta* = product), the seventh power *varga-varga-ghana-ghâta* (product of square-square and cube) and so on. Brahmagupta's system of expressing powers higher than the fourth is scientifically better. He calls the fifth power *pañca-gata* (literally, raised to the fifth), the sixth power *ṣaḍ-gata* (raised to the sixth); similarly the term for any power is coined by adding the suffix *gata* to the name of the number indicating that power.³ Bhâskara II has sometimes followed it consistently for the powers one and upwards.⁴ In the *Anuyogadvâra-sûtra*⁵, a work written before the commencement of the Christian Era, we find certain interesting terms for higher powers, integral as well as fractional, particularly successive squares (*varga*) and square-roots (*varga-mûla*). According to it the term *prathama-varga* (first square) of a quantity, say *a*, means a²; *dvitîyavarga* (second square) = (a²)² = a⁴; *tṛtîya-varga* (third square) = ((a²)²)² = a⁸; and so on. In general,

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² Chapter xxx, 10, 11.

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³ *BrSpSi*, xviii. 41, 42.

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⁴ *BBi*, p. 56.

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⁵ *Sûtra* 142.

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*n*th *varga* of a = a² × 2 × 2 × to n terms = a2n.

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Similarly, *prathama-varga-mûla* (first square-root) means √a; *dvitîya-varga-mûla* (second square-root) = √()(√a) = a1/4; and, in general,

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*n*th *varga-mûla* of a = a1/2n.

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Again we find the term *tṛtîya-varga-mûla-ghana* (cube of the third square-root) for (a1/2³)³ = a3/8.

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The term *varga* for "square" has an interesting origin in a purely concrete concept. The Sanskrit word *varga* literally means "rows," or "troops" (of similar things). Its application as a mathematical term originated in the graphical representation of a square, which was divided into as many *varga* or troops of small squares, as the side contained units of some measure.¹

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¹ G. Thibaut, *Śulba-sûtras*, p. 48. Compare also Bibhutibhusan Datta, "On the origin of the Hindu terms for root," *Amer. Math. Mon.*, XXXVIII, 1931, pp. 371-6.

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**Equation.** The equation is called by Brahmagupta (628) *sama-karaṇa*² or *samî-karaṇa*³ (making equal) or more simply *sama*⁴ (equation). Pṛthûdakasvâmî (860) employs also the term *sâmya*⁵ (equality or equation); and Śrîpati (1039) *sadṛśî-karaṇa*⁶ (making similar). Nârâyaṇa (1350) uses the terms *samî-karaṇa*, *sâmya* and *samatva* (equality).⁷ An equation has always two *pakṣa* (side). This term occurs in the works of Śrīdhara (*c.* 750), Padmanābha¹ and others.²

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² *BrSpSi*, xviii. 63.

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³ *BrSpSi*, xviii, subheading for the section on equations.

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⁴ *BrSpSi*, xviii. 43.

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⁵ *BrSpSi*, xii. 66 (*Com*).

v2.c17.s1.101

⁶ *SiŚe*, xiv. 1.

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⁷ *NBi*, II, R. 2-3.

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¹ The algebras of Śrīdhara and Padmanâbha are not available now. But the term occurs in quotations from them by Bhâskara II (*BBi*, pp. 61, 67).

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² *BrSpSi*, xviii. 43 (*Com*); *SiŚe*, xiv. 14, 20; *BBi*, pp. 43-4.

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**Absolute Term.** In the Bakhshālî treatise³ the absolute term is called *dṛśya* (visible). In later Hindu algebras it has been replaced by a closely allied term *rūpa*⁴ (appearance), though it continued to be employed in treatises on arithmetic.⁵ Thus the true significance of the Hindu name for the absolute term in an algebraic equation is obvious. It represents the visible or known portion of the equation while its other part is practically invisible or unknown.

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³ *BMs*, Folio 23, verso; Folio 70, recto and verso (c).

v2.c17.s1.107

⁴ *BrSpSi*, xviii. 43-4; *SiŚe*. xiv, 14, 19; etc.

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⁵ *Triś*, pp. 11, 12.

3. Symbols

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**Symbols of Operation.** There are no special symbols for the fundamental operations in the Bakhshālî work. Any particular operation intended is ordinarily indicated by placing the tachygraphic abbreviation, the initial syllable of a Sanskrit word of that import, after, occasionally before, the quantity affected. Thus the operation of addition is indicated by *yu* (an abbreviation from *yuta*, meaning added), subtraction by + which is very probably from *kṣa* (abbreviated from *kṣaya*, diminished), multiplication by *gu* (from *guṇa* or *guṇita*, multiplied) and division by *bhâ* (from *bhāga* or *bhājita*, divided). Of these again, the most systematically employed abbreviation is that for the operation of subtraction and next comes that of division. In the case of the other two operations the indicatory words are often written in full, or altogether omitted. In the latter case, the particular operations intended to be carried out are understood from the context. We take the following instances from the Bakhshâlî Manuscript:

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(i)¹ 0 & 5 / 1 & 1 yu means x/1 + 5/1, and 11 / 1 yu 5 / 1 means 11/1 + 5/1.

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¹ Folio 59, recto.

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(ii)² 3 & 3 & 3 & 3 & 3 & 3 & 3 & 10 gu / 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 / means 3 × 3 × 3 × 3 × 3 × 3 × 3 × 10.

v2.c17.s2.5

² Folio 47, recto.

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(iii)³ 0 & 1 & 3 & 2 & 5 & 3 & 7 & 4 & 9 / 1 & 1 & 2 & 1 & 2+ & 1 gu & 2+ & 1 gu & 2+ means x(1+3/2) + 2x(1+3/2) - 5x/2 + 3x(1+3/2) - 7x/2 + 4x(1+3/2) - 9x/2.

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³ Folio 25, verso. The beginning and end of this illustration are mutilated but the restoration is certain.

v2.c17.s2.8

(iv)⁴ 1 & 1 & 1 & 1 bhâ & 36 / 1+ & 1 & 1 & 1 & / 2 & 3 & 4+ & 6 & 1 / means 36/((1-½)(1+⅓)(1-¼)(1+⅙)).

v2.c17.s2.9

⁴ Folio 13, verso.

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(v)⁵ 40 bhâ & 160 & 13 / 1 & 1 & 1 / & & 2 means 160/40 × 13½.

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⁵ Folio 42, recto.

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In later Hindu mathematics the symbol for subtraction is a dot, occasionally a small circle, which is placed above the quantity, so that 7 or 7 (°) means -7; other operations are represented by simple juxtaposition.

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Bhâskara II (1150) says, "Those (known and unknown numbers) which are negative should be written with a dot (*bindu*) over them."¹ A similar remark occurs in the algebra of Nârâyaṇa (1350).² Their silence about symbols of other fundamental operations proves their non-existence.

v2.c17.s2.14

¹ *BBi*, p. 2.

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² *NBi*, I, R. 7.

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**Origin of Minus Sign.** The origin of · or ° as the minus sign seems to be connected with the Hindu symbol for the zero, 0. It is found to have been used as the sign of emptiness or omission in the early Bakhshâlî treatise as well as in the later treatises on arithmetic (*vide infra*).³ It is placed over the number affected in order to distinguish it from its use in a purely numerical significance when it is placed beside the number. The origin of the Bakhshâlî minus sign (+) has been the subject of much conjecture. Thibaut suggested its possible connection with the supposed Diophantine negative sign ⋔ (reversed ψ, tachygraphic abbreviation for λειψις meaning wanting). Kaye believes it. The Greek sign for minus, however, is not ⋔, but ↑. It is even doubtful if Diophantus did actually use it; or whether it is as old as the Bakhshâlî cross.⁴ Hoernle⁵ presumed the Bakhshâlî minus sign to be the abbreviation *ka* of the Sanskrit word *kanita*, or *nu* (or *nû*) of *nyûna*, both of which mean diminished and both of which abbreviations in the Brâhmî characters would be denoted by a cross. Hoernle was right, thinks Datta,⁶ so far as he sought for the origin of + in a tachygraphic abbreviation of some Sanskrit word. But as neither the word *kanita* nor *nyûna* is found to have been used in the Bakhshâlî work in connection with the subtractive operation, Datta finally rejects the theory of Hoernle and believes it to be the abbreviation *kṣa*, from *kṣaya* (decrease) which occurs several times, indeed, more than any other word indicative of subtraction. The sign for *kṣa*, whether in the Brâhmî characters or in Bakhshâlî characters, differs from the simple cross (+) only in having a little flourish at the lower end of the vertical line. The flourish seems to have been dropped subsequently for convenient simplification.

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³ p. 16.

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⁴ *Cf.* Smith, *History*, II, p. 396.

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⁵ *IA*, XVII, p. 34.

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⁶ Datta, *Bakh. Math.*, (BCMS, XXI), pp. 17-8.

Symbols for Powers and Roots

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The symbols for powers and roots are abbreviations of Sanskrit words of those imports and are placed after the number affected. Thus the square is represented by *va* (from *varga*), cube by *gha* (from *ghana*), the fourth power by *va-va* (from *varga-varga*), the fifth power by *va-gha-ghâ* (from *varga-ghana-ghâta*), the sixth power by *gha-va* (from *ghana-varga*), the seventh power by *va-va-gha-ghâ* (from *varga-varga-ghana-ghâta*) and so on. The product of two or more unknown quantities is indicated by writing *bhâ* (from *bhâvita*, product) after the unknowns with or without interposed dots; e.g., *yâva-kâgha-bhâ* or *yâvakâghabhâ* means (ya)² (ka)³. In the Bakhshâlî treatise the square-root of a quantity is indicated by writing after it *mû*, which is an abbreviation for *mûla* (root). For instance,¹

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¹ Folio 59, recto; compare also folio 67, verso.

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| 11 | *yu* | 5 | *mû* | 4 |
| 1 | | 1 | | 1 |

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means √(11+5)=4

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and

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| 11 | 7+ | | *mû* | 2 |
| 1 | 1 | | | 1 |

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means √(11-7)=2.

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In other treatises the symbol of the square-root is *ka* (from *karaṇî*, root or surd) which is usually placed before the quantity affected. For example,² *ka* 9 *ka* 450

v2.c17.s2.t1.9

² *BBi*, p. 15.

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*ka* 75 *ka* 54 means √9+√450+√75+√54.

v2.c17.s2.t1.11

**Symbols for Unknowns.** In the Bakhshâlî treatise there is no specific symbol for the unknown. Consequently its place in an equation is left vacant and to indicate it vividly the sign of emptiness is put there. For instance,¹

v2.c17.s2.t1.12

¹ *BMs*, Folio 22, verso.

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$0 & 2 & 3 & 4 & dṛśya & 200 / 1 & 1 & 1 & 1 & & 1 means x+2x+3x+4x=200.$

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The use of the zero sign to mark a vacant place is found in the arithmetical treatises of later times when the Hindus had a well-developed system of symbols for the unknowns. Thus we find in the *Triśatikâ*² of Śrîdhara (*c.* 750) the following statement of an arithmetical progression whose first term (*âdiḥ*) is 20, number of terms (*gacchaḥ*) 7, sum (*gaṇitaṁ*) 245 and whose common difference (*uttaraḥ*) is unknown:

v2.c17.s2.t1.15

² *Triś*, p. 29.

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Linked texts (16)

Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Ṛgvedasaṃhitā — Sanskrit
Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Uttarajjhayaṇa — Ardhamāgadhī Prakrit
Ṭhāṇaṃga — Sudharman, Ardhamāgadhī Prakrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Aṇuogaddārāiṃ — Āryarakṣita, Ardhamāgadhī Prakrit
Āpastambaśulbasūtra — Āpastamba, Sanskrit
Bījagaṇita — Bhāskara II, Sanskrit
Brāhmaṇa — Sanskrit
Indian Antiquary
Siddhāntaśekhara — Śrīpati, Sanskrit
Bakhshālī Manuscript — Sanskrit
Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit