Little is known at present to historians of mathematics regarding the achievements of the early Hindu mathematicians and our indebtedness to them. Though it is now generally admitted that the decimal place-value system of numeral notation was invented and first used by the Hindus, it is not yet fully realized to what extent we are indebted to them for our elementary mathematics. This is due to the lack of a reliable and authentic history of Hindu mathematics. Our object in writing the present book has been to make up for this deficiency by giving a comprehensive account of the growth and development of the science of mathematics in India from the earliest known times down to the seventeenth century of the Christian era.

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The subject is treated by topics. Under each topic are collected together and set forth in chronological order translations of relevant Sanskrit texts as found in the Hindu works. The texts have been elucidated, wherever necessary, by adding explanatory notes and comments, and also by illustrative examples culled from original sources. We have tried to avoid repetition as far as has been consistent with our aim. However, on several occasions it has been considered desirable to repeat the same rule in the words of different authors in order to emphasize the continuity or rather the gradual evolution of mathematical thought and terminology in India. Comparative study of this kind has helped us to throw light on certain obscure Sanskrit passages and technical terms whose full significance had not been understood before. In translating the texts we have tried to be as literal and faithful as possible without sacrificing the spirit of the original. Sometimes it has not been possible to find exact parallels to Sanskrit words and technical terms in English. In all such cases we have tried to maintain the spirit of the original in the English version.

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The above plan of the book has been adopted in pursuance of our intention to place before those who have no access to the Sanskrit sources all evidence, unfavourable as well as favourable, so that they can judge for themselves the claims of Hindu mathematics, without depending solely on our statements. In order to facilitate comparison with the development of mathematics in other countries the various topics have been arranged generally in accordance with the sequence in Professor D. E. Smith’s *History of Mathematics*, Vol. II. This has sometimes necessitated divergence from the arrangement of topics as found in the Hindu works on mathematics.

v1.c3.4

In search of material for the book we had to examine the literature of the Hindus, non-mathematical as well as mathematical, whether in Sanskrit or in Prâkrit (Pâlî and Ardha Mâgadhî). Very few of the Hindu treatises on mathematics have been printed so far, and even these are not generally known. The manuscript works that exist in the various Sanskrit libraries in India and Europe are still less known. We have not spared labour in collecting as many of these as we could. Sanskrit mathematical works mentioned in the bibliography given at the end of this volume have been specially consulted by us. We are thankful to the authorities of the libraries at Madras, Bangalore, Trivandrum, Trippunithura and Benares, and those of the India Office (London) and the Asiatic Society of Bengal (Calcutta) for supplying us transcripts of the manuscripts required or sending us manuscripts 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, London.

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It would not have been possible to carry our study as far as has been done without the spade work of previous writers. Foremost among these must be mentioned the late Pandit Sudhakar Dvivedi of Benares, whose editions of the *Lîlâvatî*, *Brâmasphuṭa-siddhânta*, *Triśatikâ*, *Mahâsiddhânta*, *Siddhânta-tattva-viveka*, etc., have been of immense help. Colebrooke's translations of the arithmetic and algebra of Brahmagupta and Bhâskara II, Kern's edition of the *Âryabhaṭîya* and Rangacarya's edition (with English translation) of the *Gaṇita-sâra-saṁgraha* of Mahâvîra have also been of much use. The recent work of G. R. Kaye, however, has been found to be extremely unreliable. His translation of the *Gaṇitapâda* of the *Âryabhaṭîya* and his edition of the Bakhshâlî Manuscript are full of mistakes and are misleading.

v1.c3.6

It has been decided to publish the book in three parts. The first part deals with the history of the numeral notation and of arithmetic. The second is devoted to algebra, a science in which the ancient Hindus made remarkable progress. The third part contains the history of geometry, trigonometry, calculus and various other topics such as magic squares, theory of series and permutations and combinations. Each part is complete in itself, so that one interested in any particular branch of mathematics need not consult all of them.

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Part I which is now being published contains two chapters. Chapter I gives an account of the various devices employed by the Hindus for denoting numbers. The gradual evolution of the decimal place-value system of notation has been traced and all evidence relating to its use in India collected together for the first time. This evidence shows that the system was in use in India during the earliest centuries of the Christian era, if not earlier. We hope that the facts set forth in this chapter will finally set at rest the controversy about its place of origin. Considerations of space have prevented us from giving details regarding the introduction of the Hindu numerals into Arabia, Northern Africa and Europe. A brief account has, however, been included.

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Chapter II deals with arithmetic in general. We have become so familiar with our methods of performing the fundamental arithmetical operations of addition, subtraction, multiplication, division and the extraction of roots that we seldom pause to think how and when these methods were invented. The problem, however, has deep interest for the teacher and historian of mathematics. And an account of the evolution of these methods in the land of their birth should be welcome. We have given details and illustrations of different methods of performing these operations on a *pāṭī*, ("board"), as followed in India from the fifth century onwards. It has been shown that our present methods are simple variations of those of the ancient Hindus. The rule of three, the rules of supposition and false position, and rules relating to calculations involving interest, exchange of commodities, fineness of gold, etc., are all due to the Hindus. In fact, practically the whole of elementary arithmetic can be traced back to them. Thus the importance of chapter II cannot be over-emphasised.

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The scheme of transliteration of Sanskrit words and proper names has been indicated in the beginning. Arabic words and proper names that occur in the book have been taken down as found in various secondary sources consulted.

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Acknowledgment has been made in footnotes to the texts of various sources from which we have derived assistance. Of the books which have been found specially helpful, we would mention Bühler's Indian Palæography and Ojha's Prâcîna Lipi Mâlâ (in Hindi). We are indebted to Mr. Ojha for permitting us to reproduce from his book the tables of Brâhmî and Kharoṣṭhî numerals. We have pleasure in expressing our obligation to Mr. T. N. Singh for his help in the revision of proof-sheets. He has caused the removal of many obscurities and has made many valuable suggestions of which we have availed ourselves. Our thanks are also due to Mr. R. D. Misra for preparing the index to this volume.

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In conclusion we express our thanks to the Law Journal Press for their unfailing courtesy and for expediting the work of printing.

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LUCKNOW *September, 1935* BIBHUTIBHUSAN DATTA AVADHESH NARAYAN SINGH

v1.c4.1

CHAPTER I

v1.c4.2

NUMERAL NOTATION

v1.c4.3

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| | | PAGE |
| 10. | WORD NUMERALS | 53 |
| | Explanation of the System—List of Words—Word Numerals without Place-value—Word Numerals with Place-value—Word Numerals in Inscriptions—Origin and Early History—Date of Invention. | |
| 11. | ALPHABETIC NOTATIONS | 63 |
| | Alphabetic System of Âryabhaṭa I—Explanation—*Kaṭapayâdi* Systems—*Akṣarpallî*—Other Letter Systems. | |
| 12. | THE ZERO SYMBOL | 75 |
| | Earliest use—Form of the Symbol—Other uses of the Symbol. | |
| 13. | THE PLACE-VALUE NOTATION IN HINDU LITERATURE | 83 |
| | Jaina Canonical Works—Purâṇas—Works on Philosophy—Literary Works. | |
| 14. | DATE OF INVENTION OF THE PLACE-VALUE NOTATION | 86 |
| 15. | HINDU NUMERALS IN ARABIA | 88 |
| 16. | HINDU NUMERALS IN EUROPE | 92 |
| | Boethius Question—Definite Evidence. | |
| 17. | MISCELLANEOUS REFERENCES TO THE HINDU NUMERALS | 95 |
| | Syrian Reference—Arabic References—The Terms *Hindasi*, etc.—European References. | |
| 18. | TABLES | 105 |

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CHAPTER II

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ARITHMETIC

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| 1. | GENERAL SURVEY | 123 |
| | Terminology and Scope—Sources—Exposition and Teaching—Decay of Mathematics—The Fundamental Operations. | |

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2. ADDITION 130 Terminology—The Operation—Direct Process—Inverse Process.

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3. SUBTRACTION 132 Terminology—The Operation—Direct Process—Inverse Process.

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4. MULTIPLICATION 134 Terminology—Methods of Multiplication—Doorjunction Method—Transmission to the West—Gelosia Method—Cross Multiplication Method—Multiplication by Separation of Places—Zigzag Method—Parts Multiplication Method—Algebraic Method.

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5. DIVISION 150 Terminology—The Operation—The Method of Long Division.

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6. SQUARE 155 Terminology—The Operation—Minor Methods of Squaring.

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7. CUBE 162 Terminology—The Operation—Minor Methods of Cubing.

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8. SQUARE-ROOT 169 Terminology—The Operation.

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9. CUBE-ROOT 175 Terminology—The Operation.

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10. CHECKS ON OPERATIONS 180

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11. FRACTIONS 185 Early Use—Weights and Measures—Terminology—Writing of Fractions—Reduction to Lowest Terms—Reduction to Common Denominator—Fraction in Combination—Lowest Common Multiple—The Eight Operations—Addition and Subtraction—Multiplication—Division—Square and Square-root—Cube and Cube-root—Unit Fractions.

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| 12. | THE RULE OF THREE | 203 |
| | Terminology—The Method—Inverse Rule of Three—Appreciation of the Rule of Three—Proportion in the West—Compound Proportion—The Method—Illustration—The Rule of Three as a Particular Case. | |
| 13. | COMMERCIAL PROBLEMS | 218 |
| | Interest in Ancient India—Interest in Hindu *Gaṇita* Problem involving a Quadratic Equation—Other Problems—Miscellaneous Problems on Interest—Barter and Exchange—Other types of Commercial Problems. | |
| 14. | MISCELLANEOUS PROBLEMS | 230 |
| | *Regula Falsi*—The Method of Inversion—Problems on Mixture—Problems involving Solution of Quadratic Equations. | |
| 15. | THE MATHEMATICS OF ZERO | 238 |
| | Zero in Arithmetic—Zero in Algebra—Zero as an Infinitesimal—Infinity—Indeterminate Forms. | |
| | BIBLIOGRAPHY | 247 |
| | INDEX | 251 |
| | *Transliteration* | xvii |
| | *List of abbreviations* | xix |

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* |

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| प् | फ् | ब् | भ् | म् |
| :-: | :-: | :-: | :-: | :-: |
| *p* | *ph* | *b* | *bh* | *m*, *ṃ* = final म् |

v1.c5.8

Un-classified :

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| य् | र् | ल् | व् | श् | ष् | स् | ह् |
| :-: | :-: | :-: | :-: | :-: | :-: | :-: | :-: |
| *y* | *r* | *l* | *v* | *ś* | *ṣ* | *s* | *h* |

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Compound :

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| क्ष् | त्र् | ज्ञ् |
| :-: | :-: | :-: |
| *kṣ* | *tr* | *jñ* |

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Pâlî : ळ = *l'*

List of Abbreviations

v1.c6.1

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*LMâ* Laghu-mânasa *MaiS* Maitrâyaṇî Saṁhitâ *MâŚl* Mânava Śulba *MBh* Mahâ-Bhâskarîya *MSi* Mahâ-siddhânta *NBi* Bîjagaṇita of Nârâyaṇa *PâSâ* Pâṭî-sâra of Munîśvara *PLM* Prâcîna-lipi-mâlâ *PSi* Pañca-siddhântikâ *RV* Ṛgveda *ŚBr* Śatapatha Brâhmaṇa *ŚiDVr* Śiṣya-dhî-vṛddhida *SiŚe* Siddhânta-śekhara *SiŚi* Siddhânta-śiromaṇi *SiTVi* Siddhânta-tattva-viveka *SûSi* Sûrya-siddhânta *TBr* Taittirîya Brâhmaṇa *Triś* Triśatikâ *TS* Taittirîya Saṁhitâ *YJ* Yâjuṣa-Jyotiṣa *ZDMG* Zeitschrift der deutschen morgenländischen Gesselschaft

Chapter I. Numeral Notation

1. A Glimpse of Ancient India

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The student of ancient Indian History is struck by the marvellous attainments of the Hindus, both in the Arts and the Sciences, at a very early period. The discoveries at Mohenjo-daro reveal that as early as 3,000 B.C. the inhabitants of the land of the Sindhu—the Hindus—built brick houses, planned cities, used metals such as gold, silver, copper and bronze, and lived a highly organised life. In fact, they were far in advance of any other people of that period. The earliest works available, the *Vedas* (*c.* 3,000 B.C. or probably much earlier), although consisting mainly of hymns of praise and poems of worship, show a high state of civilisation. The *Brâhmaṇa* literature (*c.* 2,000 B.C.) which follows the *Vedas*, is partly ritualistic and partly philosophical. In these works are to be found well-developed systems of metaphysical, social and religious philosophy, as well as the germs of most of the sciences and arts which have helped to make up the modern civilisation. It is here that we find the beginnings of the science of mathematics (arithmetic, geometry, algebra, etc.) and astronomy. This *Brâhmaṇa* period was followed by more than two thousand years of continuous progress and brilliant achievements. Although during this period there were several foreign invasions as well as internal wars and many great kingdoms rose and fell, yet the continuity of intellectual progress was maintained. The constitution of Hindu society was mainly responsible for this. The foreign invaders, instead of being a hindrance, contributed to progress and the strengthening of Hindu society by bringing in new blood. They settled in the land, adopted the religion and customs of the conquered and were completely absorbed into Hindu society. There were a class of people—the Brâhmaṇas—who took the vow of poverty, and devoted themselves, from one generation to another, to the cultivation of the sciences and arts, religion and philosophy. The Brâhmaṇas, thanks to their selflessness and intellectual attainments, were highly respected by the kings and the people alike. They were the law-givers and advisers of the kings. In fact, this body of selfless thinkers and learned men were the real rulers of the land.

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The great Epic, the *Râmâyaṇa*, was composed by Vâlmîki, the father of Sanskrit poetry, about 1,000 B.C., Pâṇini, the grammarian, perfected Sanskrit grammar about 700 B.C. and Suśruta wrote on the sciences of medicine and surgery about 600 B.C.¹ A century later, Mahâvîra and Buddha taught their unique systems of religious and moral philosophy, and the doctrine of *Nirvâṇa*. With the spread of these religions evolved the Jaina and Buddhist literatures. Some of the earlier *Purâṇas* and *Dharma-śâstras* were written about this time. The period 400 B.C. to 400 A.D., however, seems to have been a period of great activity and progress. During this period flourished the great Jaina metaphysician Umâsvâti, Patañjali, the grammarian and philosopher, Kauṭilya, the celebrated politician, Nâgârjuna, the chemist, Caraka, the physician, and the immortal poets Aśvaghoṣa, Bhâsa and Kâlidâsa. The great astronomical *Siddhântas*, the *Sûrya*, the *Pitâmaha*, the *Vaśiṣṭha*, the *Parâśara* and others were written during this period and the decimal place-value notation was perfected.

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¹ There is considerable divergence of opinion regarding the dates of the pre-historic works and personalities mentioned in this section. We have given those dates that appear most plausible.

2. Hindus and Mathematics

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**Appreciation of Mathematics.** It is said that in ancient India no science did ever attain an independent existence and was cultivated for its own sake. Whatever of any science is found in Vedic India is supposed to have originated and grown as the handmaid of one or the other of the six "members of the Veda," and consequently with the primary object of helping the Vedic rituals. It is also supposed, sometimes, that any further culture of the science was somewhat discouraged by the Vedic Hindus in suspicion that it might prove a hindrance to their great quest of the knowledge of the Supreme by diverting the mind to other external channels. That is not, indeed, a correct view on the whole. It is perhaps true that in the earlier Vedic Age, sciences grew as help to religion. But it is generally found that the interest of people in a particular branch of knowledge, in all climes and times, has always been aroused and guided by specific reasons. Religion being the prime avocation of the earlier Hindus, it is not unnatural that the culture of other branches of knowledge grew as help to it and was kept subsidiary. But there is enough evidence to show that in course of time all the sciences outgrew their original purposes and were cultivated for their own sake. A new orientation had indeed set in in the latter part of the Vedic Age.

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There is a story in the *Chândogya Upaniṣad*¹ whose value in support of our view cannot be over-estimated. It is said that once upon a time Nârada approached the sage Sanatkumâra and begged of him the *Brahma-vidyâ* or the supreme knowledge. Sanatkumâra asked Nârada to state what sciences and arts he had already studied so that he (Sanatkumâra) might judge what still remained to be learnt by him. Thereupon Nârada enumerated the various sciences and arts studied by him. This list included astronomy (*nakṣatra-vidyâ*) and arithmetic (*râśi-vidyâ*). Thus the culture of the science of mathematics or of any other branch of secular knowledge, was not considered to be a hindrance to spiritual knowledge. In fact, *Aparâ-vidyâ* ("secular knowledge") was then considered to be a helpful adjunct to *Parâ-vidyâ* ("spiritual knowledge").¹

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¹ *Chândogya Upaniṣad*, vii. 1, 2, 4.

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¹ *Muṇḍakopaniṣad*, i. 1, 3-5.

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Importance to the culture of *Gaṇita* (mathematics) is also given by the Jainas. Their religious literature is generally classified into four branches, called *anuyoga* ("exposition of principles"). One of them is *gaṇitânuyoga* ("the exposition of the principles of mathematics"). The knowledge of *Saṁkhyâna* (literally, "the science of numbers," meaning arithmetic and astronomy) is stated to be one of the principal accomplishments of the Jaina priest.² In Buddhist literature too, arithmetic (*gaṇanâ*, *saṁkhyâna*) is regarded as the first and the noblest of the arts.³ All these will give a fair idea of the importance and value set upon the culture of *gaṇita* in ancient India.

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² *Bhagavatî-sûtra*, Sûtra 90; *Uttarâdhyayana-sûtra*, xxv. 7, 8, 38.

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³ *Vinaya Piṭaka*, ed. Oldenberg, Vol. IV, p. 7; *Majjhima Nikâya*, Vol. I, p. 85; *Cullaniddesa*, p. 199.

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The following appreciation of mathematics, although belonging to a much later date, will be found to be interesting, especially, as it comes from the pen of Mahāvîra (850 A.D.), one of the best mathematicians of his time:

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“In all transactions which relate to worldly, Vedic or other similar religious affairs calculation is of use. In the science of love, in the science of wealth, in music and in drama, in the art of cooking, in medicine, in architecture, in prosody, in poetics and poetry, in logic and grammar and such other things, and in relation to all that constitutes the peculiar value of the arts, the science of calculation (*gaṇita*) is held in high esteem. In relation to the movements of the sun and other heavenly bodies, in connection with eclipses and conjunctions of planets, and in connection with the *tripraśna* (direction, position and time) and the course of the moon—indeed in all these it is utilised. The number, the diameter and the perimeter of islands, oceans and mountains; the extensive dimensions of the rows of habitations and halls belonging to the inhabitants of the world, of the interspace between the worlds, of the world of light, of the world of the gods and of the dwellers in hell, and other miscellaneous measurements of all sorts—all these are made out by the help of *gaṇita*. The configuration of living beings therein, the length of their lives, their eight attributes, and other similar things; their progress and other such things, their staying together, etc.—all these are dependent upon *gaṇita* (for their due comprehension). What is the good of saying much? Whatever there is in all the three worlds, which are possessed of moving and non-moving beings, cannot exist as apart from *gaṇita* (measurement and calculation).

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“With the help of the accomplished holy sages, who are worthy to be worshipped by the lords of the world, and of their disciples and disciples’ disciples, who constitute the well-known series of preceptors,

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I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are picked from the sea, gold is from the stony rock and pearl from the oyster shell; and give out according to the power of my intelligence, the *Sâra-saṁgraha*, a small work on *gaṇita*, which is (however) not small in value."¹

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¹ *GSS*, i. 9-19.

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**Mathematics in Hindu Education.** The elementary stage in Hindu education lasted from the age of five till the age of twelve. This period slightly differed in the case of sons of kings and noblemen. The main subjects of study were *lipi* or *lekhâ* (alphabets, reading and writing), *rûpa* (drawing and geometry) and *gaṇanâ* (arithmetic). It is said in the *Arthaśâstra* of Kauṭilya (400 B.C.) that having undergone the ceremony of tonsure, the student shall learn the alphabets (*lipi*) and arithmetic (*saṁkhyâna*).² We find in the Hâthîgumphâ Inscription³ that king Khâravela (163 B.C.) of Kaliṅga spent nine years (from the age of sixteen to the age of 25) in learning *lekhâ*, *rûpa* and *gaṇanâ*. Prince Gautama began his education when he was eight years of age "firstly (with) writing and then arithmetic as the most important of the 72 sciences and arts."⁴ Mention of *lekhâ*, *rûpa* and *gaṇanâ* is also found in the Jaina canonical works.⁵

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² *Arthaśâstra*, ed. by R. Shamasastri, i. 5, 2; Eng. trans., p. 10.

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³ *Hathigumpha and three other inscriptions*, ed. by Bhagavanlal Indraji, p. 22.

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⁴ *Antagaḍa-dasâo and Anuttarovavâiya-dasâo*, Eng. trans. by L. D. Barnett, 1907, p. 30; cf. *Kalpasûtra* of Bhadrabâhu, Sûtra 211.

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⁵ E.g., *Samavâyâṅga-sûtra*, Sûtra 72.

3. Scope and Development of Hindu Mathematics

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*Gaṇita* literally means "the science of calculation" and is the Hindu name for mathematics. The term is a very ancient one and occurs copiously in Vedic literature. The *Vedâṅga Jyotiṣa* (*c.* 1,200 B.C.) gives it the highest place of honour among the sciences which form the *Vedâṅga*: "As the crests on the heads of peacocks, as the gems on the hoods of snakes, so is *gaṇita* at the top of the sciences known as the *Vedâṅga*."¹ In ancient Buddhist literature we find mention of three classes of *gaṇita*: (1) *mudrâ* ("finger arithmetic"), (2) *gaṇanâ* ("mental arithmetic") and (3) *saṁkhyâna* ("higher arithmetic in general"). One of the earliest enumerations of these three classes occurs in the *Dîgha Nikâya*,² and it is also found in the *Vinaya Piṭaka*,³ *Divyâvadâna*⁴ and *Milindapañho*.⁵ The word *saṁkhyâna* has been used for *gaṇita* in several old works.⁶ At this remote period *gaṇita* included astronomy, but geometry (*kṣetra-gaṇita*) belonged to a different group of sciences known as *Kalpasûtra*.

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¹ "Yathâ śikhâ mayurâṇâṁ nâgânâṁ maṇayo yathâ Tadvadvedâṅgaśâstrâṇâṁ gaṇitaṁ mûrdhani sthitam." *Vedâṅga Jyotiṣa*, 4.

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² I, p. 51.

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³ IV, p. 7.

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⁴ *Divyâvadâna*, ed. by E. B. Cowell and R. A. Neil, Cambridge, 1886, pp. 3, 26 and 88.

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⁵ *Milindapañho*, Eng. trans. by Rhys Davids, Oxford, 1890, p. 91.

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⁶ E.g., *Kalpasûtra* of Bhadrabâhu, ed. by H. Jacobi, Leipzig, 1897; *Bhagavatî-sûtra*, Bombay, 1918, p. 112; *Arthaśâstra*, i. 5. 2.

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It is believed that some time before the beginning of the Christian era, there was a renaissance of Hindu *Gaṇita*.⁷ The effect of this revival on the scope of *gaṇita* was great. Astronomy (*jyotiṣa*) became a separate subject and geometry (*kṣetra-gaṇita*) came to be included within its scope. The subjects treated in the Hindu *Gaṇita* of the early renaissance period consisted of the following:¹ *Parikarma* ("fundamental operations"), *Vyavahâra* ("determinations"), *Rajju* ("rope," meaning geometry), *Râśi* ("rule of three"), *Kalâsavarṇa* ("operations with fractions"), *Yâvat tâvat* ("as many as," meaning simple equations), *Varga* ("Square," meaning quadratic equations), *Ghana* ("Cube", meaning cubic equations), *Varga-varga* (biquadratic equations) and *Vikalpa* ("permutations and combinations").

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⁷ Bibhutibhusan Datta, "The scope and development of Hindu Gaṇita," *Indian Historical Quarterly*, V, 1929, pp. 479-512.

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¹ "Parikammaṁ vavahâro rajju râsî kalâsavanne ya । Jâvantâvati vaggo ghano tataha vaggavaggo vikappo ta ॥" *Sthânâṅgasûtra*, Sûtra 747.

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Thus *gaṇita* came to mean mathematics in general, while 'finger arithmetic' as well as 'mental arithmetic' were excluded from the scope of its meaning For the calculations involved in *gaṇita*, the use of some writing material was essential. The calculations were performed on a board (*pâṭî*) with a piece of chalk or on sand (*dhûlî*) spread on the ground or on the *pâṭî*. Thus the terms *pâṭî-gaṇita* ("science of calculation on the board") or *dhûlî-karma* ("dust-work"), came to be used for higher mathematics. Later on the section of *gaṇita* dealing with algebra was given the name *Bîja-gaṇita*. The first to effect this separation was Brahmagupta (628), but he did not use the term *Bîja-gaṇita*. The chapter dealing with algebra in his *Brâhma-sphuṭa-siddhânta* is called *Kuṭṭaka*. Śrîdharâcârya (750) regarded *Pâṭî-gaṇita* and *Bîja-gaṇita* as separate and wrote separate treatises on each. This distinction between *Pâṭîgaṇita* and *Bîjagaṇita* has been preserved by later writers.

v1.c7.s3.12

Having given a brief survey of the position and scope of mathematics in Ancient India, we turn to the purpose in hand—that of giving a connected account of the development and growth of the different branches of mathematics. The numeration system of the Hindus will engage our attention first.

4. Numeral Terminology

v1.c7.s4.1

**Scale of Notation.** We can definitely say that from the very earliest known times, ten has formed the basis of numeration in India.¹ In fact, there is absolutely no trace of the extensive use of any other base of numeration in the whole of Sanskrit literature. It is also characteristic of India that there should be found at a very early date long series of number names for very high numerals. While the Greeks had no terminology for denominations above the *myriad* (10⁴), and the Romans above the *mille* (10³), the ancient Hindus dealt freely with no less than eighteen denominations. In modern times also, the numeral language of no other nation is as scientific and perfect as that of the Hindus.

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¹ Various instances are to be found in the *Ṛgveda*; noted by Macdonell and Keith, *Vedic Index*, Vol. I, p. 343.

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In the *Yajurveda Saṃhitā* (*Vājasaneyī*)² the following list of numeral denominations is given: *Eka* (1), *daśa* (10), *śata* (100), *sahasra* (1000), *ayuta* (10,000), *niyuta* (100,000), *prayuta* (1,000,000), *arbuda* (10,000,000), *nyarbuda* (100,000,000), *samudra* (1,000,000,000), *madhya* (10,000,000,000), *anta* (100,000,000,000), *parārdha* (1,000,000,000,000). The same list occurs at two places in the *Taittirīya Saṃhitā*.³ The *Maitrāyaṇī*⁴ and *Kâṭhaka*¹ *Saṁhitâs* contain the same list with slight alterations. The *Pañcaviṁśa Brâhmaṇa* has the *Yajurveda* list upto *nyarbuda* inclusive, and then follow *nikharva*, *vâḍava*, *akṣiti*, etc. The *Śâṅkhyâyana Śrauta Sûtra* continues the series after *nyarbuda* with *nikharva*, *samudra*, *salila*, *antya*, *ananta* (= 10 billions). Each of these denominations is 10 times the preceding, so that they were aptly called *daśaguṇottara saṁjñâ*² ("decuple terms").

v1.c7.s4.4

² *Yajurveda Saṃhitā*, xvii. 2.

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³ iv. 40. 11. 4; and vii. 2. 20. 1.

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⁴ ii. 8. 14; the list has *ayuta*, *prayuta*, then again *ayuta*, then *nyarbuda*, *samudra*, *madhya*, *anta*, *parārdha*.

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¹ xvii. 10; the list is the same with the exception that *niyuta* and *prayuta* change places. In xxxix. 6, after *nyarbuda* a new term *vâḍava* intervenes.

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² Cf. Bhâskara II, *L*, p. 2.

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Coming to later times, *i.e.*, about the 5th century B.C., we find successful attempts made to continue the series of number names based on the centesimal scale.³ We quote below from the *Lalitavistara*,⁴ a well-known Buddhist work of the first century B.C., the dialogue between Arjuna, the mathematician, and Prince Gautama (Bodhisattva):

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³ *Śatottara gaṇanâ* or *Śatottara saṁjñâ* (names on the centesimal scale).

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⁴ *Lalitavistara*, ed. by Rajendra Lal Mitra, Calcutta, 1877, p. 168.

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"The mathematician Arjuna asked the Bodhisattva, 'O young man, do you know the counting which goes beyond the *koṭi* on the centesimal scale?

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Bodhisattva: I know.

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Arjuna: How does the counting proceed beyond the *koṭi* on the centesimal scale?

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Bodhisattva: Hundred *koṭis* are called *ayuta*, hundred *ayutas* *niyuta*, hundred *niyutas* *kaṅkara*, hundred *kaṅkaras* *vivara*, hundred *vivaras* *kṣobhya*, hundred *kṣobhyas* *vivâha*, hundred *vivâhas* *utsaṅga*, hundred *utsaṅgas* *bahula*, hundred *bahulas* *nâgabala*, hundred *nâgabalas* *tiṭi-* *lambha*, hundred *titilambhas* *vyavasthâna-prajñapti*, hundred *vyavasthâna-prajñaptis* *hetuhila*, hundred *hetuhilas* *karahu*, hundred *karahus* *hetvindriya*, hundred *hetvindriyas* *samâpta-lambha*, hundred *samâpta-lambhas* *gaṇanâgati*, hundred *gaṇanâgatis* *niravadya*, hundred *niravadyas* *mudrâ-bala*, hundred *mudrâ-balas* *sarva-bala*, hundred *sarva-balas* *visamjñâ-gati*, hundred *visamjñâ-gatis* *sarvajñâ*, hundred *sarvajñâs* *vibhutamgamâ*, hundred *vibhutamgamâs* *tallakṣaṇa*.¹”

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¹ Thus *tallakṣaṇa*=10⁵³. This and the following show that the Hindus anticipated Archimedes by several centuries in the matter of evolving a series of number names which “are sufficient to exceed not only the number of a sand-heap as large as the whole earth, but one as large as the universe.” *Cf.* ‘*De harenae numero*’ in the 1544 edition of the *Opera* of Archimedes; quoted by Smith and Karpinski, *Hindu Arabic Numerals*, Boston, 1911, p. 16.

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Another interesting series of number names increasing by multiples of 10 millions is found in Kâccâyana's Pali Grammar.² “For example: *dasa* (10) multiplied by *dasa* (10) becomes *sata* (100), *sata* (100) multiplied by ten becomes *sahassa* (1,000), *sahassa* multiplied by ten becomes *dasa* *sahassa* (10,000), *dasa* *sahassa* multiplied by ten becomes *sata* *sahassa*³ (100,000), *sata* *sahassa* multiplied by ten becomes *dasa* *sata* *sahassa* (1,000,000), *dasa* *sata* *sahassa* multiplied by ten becomes *koṭi* (10,000,000). Hundred-hundred-thousand *koṭis* give *pakoṭi*.⁴ In this manner the further terms are formed. What are their names? .......... hundred hundred-thousands is *koṭi*, hundred-hundredthousand *koṭis* is *pakoṭi*, hundred-hundred-thousand *pakoṭis* is *koṭippakoṭi*, hundred-hundred-thousand *koṭippakoṭis* is *nahuta*, hundred-hundred-thousand *nahutas* is *ninnahuta*, hundred-hundred-thousand *ninnahutas* is *akkhobhiṇi*; similarly we have *bindu*, *abbuda*, *nirabbuda*, *ahaha*, *ababa*, *atata*, *sogandhika*, *uppala*, *kumuda*, *puṇḍarîka* *paduma*, *kathâna*, *mahâkathâna*, *asaṅkhyeya*.”

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² “Grammaire Pâlie de Kâccâyana,” *Journ. Asiatique*, Sixieme Serie, XVII, 1871, p. 411. The explanations to sūtras 51 and 52 are quoted here.

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³ Also called *lakkha* (*lakṣa*).

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⁴ Also called *koṭi-koṭi*, *i.e.*, (10,000,000)²=10¹⁴. The following numbers are in the denomination *koṭi-koṭi*. Compare the *Anuyogadvâra-sūtra*, Sūtra 142.

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In the *Anuyogadvâra-sûtra*² (*c.* 100 B.C.), a Jaina canonical work written before the commencement of the Christian era, the total number of human beings in the world is given thus: “a number which when expressed in terms of the denominations, *koṭi-koṭi*, etc., occupies twenty-nine places (*sthâna*), or it is beyond the 24th place and within the 32nd place, or it is a number obtained by multiplying sixth square (of two) by (its) fifth square, (*i.e.*, 2⁹⁶), or it is a number which can be divided (by two) ninety-six times.” Another big number that occurs in the Jaina works is the number representing the period of time known as *Śîrṣaprahelikâ*. According to the commentator Hema Candra (b. 1089)³, this number is so large as to occupy 194 notational places (*aṅka-sthânehi*). It is also stated to be (8,400,000)²⁸.

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² *Sûtra* 142.

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³ The figures within brackets after the names of authors or works denote dates after Christ.

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**Notational Places.** Later on, when the idea of place-value was developed, the denominations (number names) were used to denote the places which unity would occupy in order to represent them (denominations) in writing a number on the decimal scale. For instance, according to Âryabhaṭa I (499) the denominations are the names of ‘places’. He says: “*Eka* (unit) *daśa* (ten), *śata* (hundred), *sahasra* (thousand), *ayuta* (ten thousand), *niyuta* (hundred thousand), *prayuta* (million), *koṭi* (ten million), *arbuda* (hundred million), and *vṛnda* (thousand million) are respectively from *place to place* each ten times the preceding.”¹ The first use of the word ‘place’ for the denomination is met with in the Jaina work quoted above.

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¹ Thus *asaṅkhyeya* is (10)¹⁴⁰ = (10,000,000)²⁰.

v1.c7.s4.26

¹ *Ā*, ii. 2.

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In most of the mathematical works, the denominations are called “names of places,” and eighteen of these are generally enumerated. Śrīdhara (750) gives the following names:² *eka*, *daśa*, *śata*, *sahasra*, *ayuta*, *lakṣa*, *prayuta*, *koṭi*, *arbuda*, *abja*, *kharva*, *nikharva*, *mahāsaroja*, *śaṅku*, *saritā-pati*, *antya*, *madhya*, *parārdha*, and adds that the decuple names proceed even beyond this. Mahāvīra (850) gives twenty-four notational places:³ *eka*, *daśa*, *śata*, *sahasra*, *daśa-sahasra*, *lakṣa*, *daśa-lakṣa*, *koṭi*, *daśa-koṭi*, *śata-koṭi*, *arbuda*, *nyarbuda*, *kharva*, *mahākharva*, *padma*, *mahā-padma*, *kṣoṇi*, *mahā-kṣoṇi*, *śaṅkha*, *mahā-śaṅkha*, *kṣiti*, *mahā-kṣiti*, *kṣobha*, *mahākṣobha*.

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² *Triś*, R. 2-3; the term used is *daśaguṇāḥ saṁjñāḥ*, i.e., “decuple names.”

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³ *GSS*, i. 63-68; “The first place is what is known as *eka*; the second is *daśa*” etc.

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Bhāskara II's (1150) list agrees with that of Śrīdhara except for *mahāsaroja* and *saritāpati* which are replaced by their synonyms *mahāpadma* and *jaladhi* respectively. He remarks that the names of places have been assigned for practical use by ancient writers.⁴

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⁴ *L*, p. 2.

v1.c7.s4.32

Nārāyaṇa (1356) gives a similar list in which *abja*, *mahāsaroja* and *saritāpati* are replaced by their synonyms *saroja*, *mahābja* and *pārāvāra* respectively.

v1.c7.s4.33

**Numerals in Spoken Language.** The Sanskrit names for the numbers from one to nine are: *eka*, *dvi*, *tri*, *catur*, *pañca* *ṣaṭ*, *sapta*, *aṣṭa*, *nava*. These with the numerical denominations already mentioned suffice to express any required number. In an additive system it is immaterial how the elements of different denominations, of which a number is composed, are spoken. Thus one-ten or ten-one would mean the same. But it has become the usual custom from times immemorial to adhere to a definite mode of arrangement, instead of speaking in a haphazard manner.

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In the Sanskrit language the arrangement is that when a number expression is composed of the first two denominations only, the smaller element is spoken first, but when it is composed also of higher denominations, the bigger elements precede the smaller ones, the order of the first two denominations remaining as before. Thus, if a number expression contains the first four denominations, the normal mode of expression would be to say the thousands first, then hundreds, then units and then tens. It will be observed that there is a sudden change of order in the process of formation of the number expression when we go beyond hundred. The change of order, however, is common to most of the important languages of the world.¹ Nothing definite appears to be known as to the cause of this sudden change.

v1.c7.s4.35

¹ Only in very few languages is the order continuously descending. In English the smaller elements are spoken first in the case of numbers upto twenty only.

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The numbers 19, 29, 39, 49, etc., offer us instances of the use of the *subtractive* principle in the spoken language. In Vedic times we find the use² of the terms *ekânna-viṁśati* (one-less-twenty) and *ekânna-catvâriṁśat* (one-less-forty) for nineteen and thirty-nine respectively. In later times (Sûtra period) the *ekânna* was changed to *ekona*, and occasionally even the prefix *eka* was deleted and we have *ûna-vimśati*, *ûna-trimśat*, etc.—forms which are used upto the present day. The alternative expressions *nava-daśa* (nine-ten), *nava-vimśati* (nine-twenty), etc., were also sometimes used.¹

v1.c7.s4.37

² *Taittirîya Saṁhitâ*, vii. 2. 11.

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¹ 19 = nava-daśa (*Vâjasaneyî Saṁhitâ*, xiv. 23; *Taittirîya Saṁhitâ*, xiv. 23. 30). 29 = nava-viṁśati (*Vâjasaneyî Saṁhitâ*, xiv. 31). 99 = nava-navati (*Ṛgveda*, i. 84. 13).

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Practically the whole of Sanskrit literature is in verse, so that for the sake of metrical convenience, various devices were resorted to in the formation of number expressions, the most common being the use of the additive² method. The following are a few examples of common occurrence taken from mathematical works :

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² 3339 = trîṇi śatâni trîsahasrâṇi triṁśa ca nava ca, i.e., “three hundreds and three thousands and thirty and nine.” (*Ṛgveda*, iii. 9. 9; also x. 52. 6.)

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*Subtractive:*
(1) the number 139 is expressed as 40+100-1;³
(2) 297 is expressed as 300-3.⁴

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³ *GSS*, i. 4: *cattvâriṁśascaikona śatâdhika* (“forty increased by one-less-hundred”).

v1.c7.s4.43

⁴ *L*, p. 4, Ex. 1: *Trihînasya śata-trayasya* (“three less three hundred”).

v1.c7.s4.44

*Multiplicative:*
(1) the number eighteen is expressed as 2 × 9;⁵
(2) twenty-seven is expressed as 3 × 9 and 12 as 2 × 6;⁶
(3) 28,483 is expressed as 83+400+(4000 × 7).⁷

v1.c7.s4.45

⁵ *Â*, ii. 3: *dvi-navaka*.

v1.c7.s4.46

⁶ *Triś*, Ex. 43: *tri-navaka* (“three nines”), *dvi-ṣaṭ* (“two sixes”).

v1.c7.s4.47

⁷ *GSS*, i. 28: *tryaśîtimiśrâṇi catuśśatâni catussahasraghna nagânvitâni* (“eighty-three combined with four hundred and four thousand multiplied by seven”).

v1.c7.s4.48

The expression of the number 12345654321 in the form "beginning with 1 upto 6 and then diminishing in order" is rather interesting.¹

v1.c7.s4.49

¹ *GSS*, i. 27: *ekâdiṣaḍantâni krameṇa hînâni*.

v1.c7.s4.50

What are known as alphabetic and word numerals were generally employed for the expression of large numbers. A detailed account of these numerals will be given later on.

5. The Development of Numerical Symbolism

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**Writing in Ancient India.** It is generally held that numerical symbols were invented after writing had been in use for some time, and that in the early stages the numbers were written out in full in words. This seems to be true for the bigger units, but the signs for the smaller units are as old as writing itself.

v1.c7.s5.2

Until quite recently historians were divided as to the date when writing was in use in India. There were some who stated that writing was known even in the Vedic age, but the majority following Weber, Taylor, Bühler and others were of opinion that writing was introduced into India from the West about the eighth century B.C. These writers built up theories deriving the ancient Indian script, as found in the inscriptions of Aśoka, from the more ancient writing discovered in Egypt and Mesopotamia. The Semitic origin was first suggested by Sir W. Jones, in the year 1806, and later on supported by Kopp (1821), Lespius (1834), and many others. The supporters of this theory, however, do not completely agree amongst themselves. For, whilst W. Deccke and I. Taylor derive the Indian script from a South-Semitic script, Weber and Bühler derive it from the Phoenician or a North-Semitic script.¹ Bühler rejects the derivation from a South-Semitic script, stating that the theory requires too many assumptions, and makes too many changes in the letter forms to be quite convincing. He, however, supports Weber's derivation from a North-Semitic script and has given details of the theory.² Ojha³ has examined Bühler's theory in detail and rejects it stating that it is fanciful and that the facts are against it. He states that only one out of the twenty-two letters of the Phoenician (North-Semitic) script resembles a phonetically similar Brâhmî letter. He supports his argument in a most convincing manner by a table of the two alphabets, with phonetically similar letters arranged in a line. He further shows that following Bühler's method of derivation almost any script could be proved to be the parent of another.⁴

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¹ For minor differences in the theories set up by different writers and also for several other theories, see Bühler, *Palaeography*, p. 9; the notes give the references.

v1.c7.s5.4

² Bühler, *l.c.*, pp. 9f.

v1.c7.s5.5

³ *PLM*, pp. 18-31.

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⁴ Recently several other eminent historians have expressed their disagreement with Bühler's derivation. See Bhandarkar, "Origin of the Indian Alphabet," *Sir Asutosh Mukerji Jubilee Volumes*, Vol. III, 1922, p. 493; H. C. Ray, "The Indian Alphabet," *IA*, III, 1924, p. 233; also *Mohenjo-daro and the Indus Valley Civilisation*, 1931, p. 424, where the following remark occurs: "I am convinced that all attempts to derive the Brâhmî alphabet from Semitic alphabets were complete failures."

v1.c7.s5.7

Other scholars, who held that writing was known in India as early as the Vedic age, based their conclusion upon literary evidence. The *Vasiṣṭha Dharmasûtra*, which originally belonged to a school of the *Ṛgveda* offers clear evidence of the use of writing in the Vedic period. Vasiṣṭha (xvi. 10, 14-15) mentions written documents as legal evidence, and the first of these sûtras is a quotation from an older work or from traditional lore. Another quotation from the *Ṛgveda* itself (x. 62. 7), which refers to the writing of the number eight is: *Sahasraṁ me dadato aṣṭakarṇyaḥ*, meaning "gave me a thousand cows on whose ears the number eight was written." The above interpretation, although doubted by some scholars, seems to be correct, as it is supported by Pâṇini.¹ Moreover, the practice of making marks on the ears of cows to denote their relation to their owners, seems to have been prevalent in ancient India.²

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¹ *Karṇo varṇa lakṣaṇât* (vi. 2. 112) and also (vi. 3. 115) support the interpretation.

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² *Atharvaveda* (vi. 141) mentions the method of making *mithuna* marks on the ears. In (xii. 4. 6) the practice is denounced. The *Maitrâyaṇî Saṁhitâ* has a chapter dealing with this topic. The method of making such marks is dealt with in iv. 2. 9.

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At another place in the *Ṛgveda* (x. 34), we find mention of a gambler lamenting his lot and saying that "having staked on one,³ he lost his faithful wife...." Again, in the *Atharvaveda* (vii. 50, (52), 5) we find the mention of the word "written amount".⁴ Pâṇini's grammar (*c.* 700 B.C.) contains the terms *yavanânî* ("Semitic writing") and the compounds *lipikâra* and *libikâra* (iii. 2. 21) (writer), which show that writing was known in his time. In addition to these passages, the Vedic works contain some technical terms, such as *akṣara* (a letter of the alphabet), *kâṇḍa* (chapter), *paṭala*, *grantha* (book), etc., which have been quoted as evidence of writing. These specific references to written documents when considered with the advanced state of Vedic civilisation, especially the high development of trade and complicated monetary transactions, the use of prose in the *Brâhmaṇas*, the collection, the methodical arrangement, the numeration, the analysis of the Vedic texts and the phonetic and lexicographic researches found in the *Vedâṅgas*, form sufficient grounds for assigning a very early date to the use of writing in India.¹ Although these arguments possess considerable weight, they were not generally recognised, as will always happen if an *argumentum ex impossibili* is used. R. Shamasastry (1906) has published a derivation based upon ancient Indian hieroglyphic pictures which he believes to be preserved in the *tântric* figures. His learned article has not attracted the attention it deserves.

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³ Here 'one' refers to the number stamped on the dice.

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⁴ *Ajaiṣaṁ tvâ saṁlikhitamajaiṣamuta saṁrudham*.

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¹ Cf. Bühler, *l.c.*, p. 3.

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Recent discoveries have, however, sounded the death knell of all theories deriving the Indian script from foreign sources. Pottery belonging to the Megalithic (*c.* 1,500 B.C.) and Neolithic (6,000 B.C.—3,000 B.C.) ages, preserved in the Madras Museum, has been found to be inscribed with writing. And according to Bhandarkar² five of these marks are identical with the Brâhmî characters of the time of Aśoka. The excavations at Mohenjo-daro and Harappa have also brought to light written documents, seals and inscriptions, dating from before 3,000 B.C. Thus it would be now absurd to trace the Brâhmî to any Semitic alphabet of the eighth or ninth century B.C.

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² *l.c.*

Earliest Numerals

v1.c7.s5.t1.1

The numerical figures contained in the seals and inscriptions of Mohenjo-daro, have not been completely deciphered as yet. The vertical stroke and combinations of vertical strokes arranged side by side, or one group below another, have been found. The numbers one to thirteen seem to have been written by means of vertical strokes, probably, as in the figures given below:³

v1.c7.s5.t1.2

³ Marshall, *l.c.*, pp. 450-52. See also "Mohenjo-daro—Indus Epigraphy" by G. R. Hunter (*JRAS*, April, 1932, pp. 470, 478ff.) who is more pronounced about the numerical values of some of the signs.

v1.c7.s5.t1.4

It is not yet quite certain whether there were special signs for greater numbers such as 20, 30, the hundreds and higher numbers. There are numerous other signs which are believed to represent such numbers, but there seems to be no means of finding out the true values of these signs at present.

v1.c7.s5.t1.5

Between the finds of Mohenjo-daro and the inscriptions of Aśoka, which contain numerals, there is a gap of 2,700 years or more. No written documents containing numerals and belonging to this intervening period have been so far discovered. The literary evidence, however, points to the use of numerical symbols at a very early date. The reference to the writing of the number eight in the *Ṛgveda* and the use of numerical denominations as big as 10¹² in the *Yajurveda Saṁhitā* and in several other Vedic works, quoted before, offer sufficient grounds for concluding that, even at that remote period, the Hindus must have possessed a well developed system of numerical symbols. The conclusion is supported by the fact that the Greek and the Roman numerical terminologies did not go beyond 10⁴, even after writing and a satisfactory numerical symbolism had been in use for several centuries.

v1.c7.s5.t1.6

The writings on the inscriptions of Aśoka show that in his time the use of numerical symbols in India was quite common.¹ The variations in the forms of the numerical signs suggest that the symbols had been in use for a long time.

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¹ Megasthenes speaks of mile-stones indicating the distances and the halting places on the roads. The distances must have been written in numerical figures (Bühler, *l.c.*, p. 6; also *Indika of Megasthenes*, pp. 125-26). The complicated system of keeping accounts mentioned in the *Arthaśāstra* of Kauṭilya confirms the conclusion.

v1.c7.s5.t1.8

Most of the inscriptions of Aśoka and the following period are written in a script which has been called *Brāhmī*, whilst some are in a different script known as *Kharoṣṭhī*. The forms of the numerical symbols in the two scripts are different. We consider them separately.

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Linked texts (85)

Ṛgvedasaṃhitā — Sanskrit
Aṣṭādhyāyī — Pāṇini, Sanskrit
Atharvavedasaṃhitā — Atharvan & Aṅgiras, Sanskrit
Kauṭilīya Arthaśāstra — Kauṭilya, Sanskrit
Taittirīyasaṃhitā — Sanskrit
Maitrāyaṇīsaṃhitā — Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Lāṭyāyanaśrautasūtra — Lāṭyāyana, Sanskrit
Taittirīyabrāhmaṇa — Sanskrit
Vasiṣṭhadharmasūtra — Vasiṣṭha, Sanskrit
Kāṭhakasaṃhitā — Sanskrit
Chāndogyopaniṣad — Sanskrit
Vājasaneyisaṃhitā — Sanskrit
Śatapathabrāhmaṇa — Yājñavalkya & Śāṇḍilya (Vedic teacher), Sanskrit
Agnipurāṇa — Vyāsa, Sanskrit
Cūḷaniddesa — Pāli
Milindapañha — Pāli
Kātyāyanaśrautasūtra — Kātyāyana (Vājasaneyi sūtrakāra), Sanskrit
Śāṅkhāyanaśrautasūtra — Śāṅkhāyana, Sanskrit
Gaṇitasārasaṅgraha — Mahāvīra (mathematician), Sanskrit
Kaccāyanavyākaraṇa — Kaccāyana (grammarian), Pāli
Bṛhatsaṃhitā — Varāhamihira, Sanskrit
Divyāvadāna — Sanskrit
Uttarajjhayaṇa — Ardhamāgadhī Prakrit
Ṭhāṇaṃga — Sudharman, Ardhamāgadhī Prakrit
Līlāvaī — Kouhala, Māhārāṣṭrī Prakrit
Nāradasmṛti — Nārada, Sanskrit
Aṇuogaddārāiṃ — Āryarakṣita, Ardhamāgadhī Prakrit
Sūryasiddhānta — Maya, Sanskrit
Ἰνδική — Λούκιος Φλάουιος Ἀρριανός, Ancient Greek
Chandaḥsūtra — Piṅgala, Sanskrit
Pañcaviṃśabrāhmaṇa — Sanskrit
Aitareyabrāhmaṇa — Mahidāsa Aitareya, Sanskrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
تحقيق ما للهند من مقولة مقبولة في العقل أو مرذولة — Arabic
Nandamapūṇḍi grant of Rājarāja I Narendra — Sanskrit
Muṇḍakopaniṣad — Sanskrit
Ārcajyotiṣa — Lagadha, Sanskrit
Cipḷūṇ plates of Pulakeśin II — Sanskrit
Takua Pa, Tank inscription — Tamil
Dīghanikāya — Pāli
Epigraphia Indica
Harappā
Kalpasūtra — Bhadrabāhu, Ardhamāgadhī Prakrit
Mahābhārata — Vyāsa, Sanskrit
Vinayapiṭaka — Pāli
Majjhimanikāya — Pāli
Mohenjo-daro
Vāsiṣṭhasiddhānta — Sanskrit
Corpus Inscriptionum Indicarum
Pauliśasiddhānta — Sanskrit
Indian Antiquary
Pannavaṇā — Śyāmārya, Ardhamāgadhī Prakrit
Samavāyaṃga — Sudharman, Ardhamāgadhī Prakrit
Viyāhapannatti — Sudharman, Ardhamāgadhī Prakrit
Vālmīkirāmāyaṇa — Vālmīki, Sanskrit
Aṃtagaḍadasāo — Sudharman, Ardhamāgadhī Prakrit
Aṇuttarovavāiyadasāo — Sudharman, Ardhamāgadhī Prakrit
Śāstra
Kharoṣṭhī inscriptions
Nāsik
Nānāghāṭ
Pañcasiddhāntikā — Varāhamihira, Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit
Mahābhāskarīya — Bhāskara I, Sanskrit
Bakhshālī Manuscript — Sanskrit
Laghubhāskarīya — Bhāskara I, Sanskrit
Triśatikā (Pāṭīgaṇitasāra) — Śrīdhara (mathematician), Sanskrit
Koṇḍyāta grant of Veṅkaṭa II of Śaka 1558 — Sanskrit
Ahar stone inscription — Sanskrit
Sānglī plates of Govinda IV — Sanskrit
Praśasti of Nānāka (II) — Sanskrit
Two grants of Indraraja III.; Saka-samvat 836: No. II (grant of Tenna) (ed. Bhandarkar 1908) — Sanskrit
Kapadvanj plates of Kṛṣṇa II, Śaka 832 — Sanskrit
Second inscription of the Vāïllabhaṭṭasvāmin temple at Gwalior, year 933 — Sanskrit
The First Plate of a Valabhī Grant of Unknown Date — Sanskrit
Orissa plates of Vidyādharabhañjadēva — Sanskrit
Soraṭūr stone-tablet inscription of the time of Akālavarṣadēva (Kannaradēva), Śaka 873 — Kannada
Peheva inscription from the temple of Garībnāth — Sanskrit
British Museum inscription No. 1, Samvat 783 — Sanskrit
Asni Inscription of Mahipala.—(Vikrama)-Samvat 974 (ed. Fleet 1887) — Sanskrit
Khajuraho fragment of a stone inscription of Harshadeva — Sanskrit
Kanker copper plate of Pamparājadeva, [Kalachuri] Samvat 966 — Sanskrit
Tewar stone inscription of the reign of Jayasiṁhadeva. The [Chedi] year 928 (ed. Kielhorn 1894) — Sanskrit
The Banawasi Inscription of Haritiputa-Satakamni — Prakrit

History of Hindu Mathematics: A Source Book v1: Volume 1

Contents

Volume 1

Front matter

Part I: Numeral Notation and Arithmetic

Preface