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  1. Lakshmikantham,v and S.Leela, The Origin of Mathematics, University Press of America Inc., Latham, New York,2000


    ECIT Sourcebook on Indic Contributions in Math and Science

    This sourcebook will consist primarily of reprinted articles on Indic contributions in math and science, as well as several new essays to contextualize these works. It will bring together the works of top scholars which are currently scattered thoughout disparate journals, and will thus make them far more accessible to the average reader.

    There are two main reasons why this sourcebook is being assembled. First, it is our hope that by highlighting the work of ancient and medieval Indian scientists we might challenge the stereotype that Indian thought is "mystical" and "irrational". Secondly, by pointing out the numerous achievements of Indian scientists, we hope to show that India had a scientific "renaissance" that was at least as important as the European renaissance which followed it, and which, indeed, is deeply indebted to it.

    Currently, the following table of contents is proposed for this volume:

    1. Editors' Introduction (Subhash Kak)

    Section 1: Mathematics

    2. D. Gray, 2000. Indic Mathematics etc.

    3. Joseph, George Ghevarughese. 1987. "Foundations of Eurocentrism in Mathematics". In Race & Class 28.3, pp. 13-28.

    4. A. Seidenberg, 1978. The origin of Mathematics. Archive for History of Exact Sciences 18.4, pp. 301-42.

    5. Frits Staal, 1965. Euclid and Panini. Philosophy East and West 15.2, pp. 99-116.

    6. Subhash Kak, 2000. Indian binary numbers and the Katapayadi notation. ABORI, 81.

    7. Subhash Kak, 1990. The sign for zero. Mankind Quarterly, 30, pp. 199-204.

    8. C.-O. Selenius, 1975. Rationale of the chakravala process of Jayadeva and Bhaskara II. Historia Mathematica, 2, pp. 167-184.

    9. K.V. Sarma, 1972. Anticipation of modern mathematical discoveries by Kerala astronomers. In A History of the Kerala School of Hindu Astronomy. Hoshiarpur: Vishveshvaranand Institute.

    Section 2: Science, General

    10. Staal, Frits. 1995. "The Sanskrit of Science". In Journal of Indian Philosophy 23, pp. 73-127.

    11. Subbarayappa, B. V. 1970. "India's Contributions to the History of Science". In Lokesh Chandra, et al., eds. India's Contribution to World Thought and Culture. Madras: Vivekananda Rock Memorial Committee, pp. 47-66.

    12. Saroja Bhate and Subhash Kak, 1993. Panini's grammar and computer science. ABORI, 72, pp. 79-94.

    Section 3: Astronomy

    13. Subhash Kak, 1992. The astronomy of the Vedic altars and the Rgveda. Mankind Quarterly, 33, pp. 43-55.

    14. Subhash Kak, 1995. The astronomy of the age of geometric altars. Quarterly Journal of the Royal Astronomical Society, 36, pp. 385-395.

    15. Subhash Kak, 1996. Knowledge of planets in the third millennium BC. Quarterly Journal of the Royal Astronomical Society, 37, pp. 709-715.

    16. Subhash Kak, 1998. Early theories on the distance to the sun. Indian Journal of History of Science, 33, pp. 93-100.

    17. B.N. Narahari Achar, 1998. Enigma of the five-year yuga of Vedanga Jyotisa, Indian Journal of History of Science, 33, pp. 101-109.

    18. B.N. Narahari Achar, 2000. On the astronomical basis of the date of Satapatha Brahmana, Indian Journal of History of Science, 35, pp. 1-19.

    19. B.L. van der Waerden, 1980. Two treatises on Indian astronomy, Journal for History of Astronomy 11, pp.50-58.

    20. K. Ramasubramanian, M.D. Srinivas, M.S. Sriram, 1994. Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion. Current Science, 66, pp. 784-790.

  3. What Eleventh-Century Spain Knew About Indian Science and Math
    By Alok Kumar, PhD

To their credit, the Indians have made great strides in the study of numbers (3) and of geometry. They have acquired immense information and reached the zenith in their knowledge of the movements of the stars (astronomy) and the secrets of the skies (astrology) as well as other mathematical studies. After all that, they have surpassed all the other peoples in their knowledge of medical science and the strengths of various drugs, the characteristics of compounds and the peculiarities of substances.

  1. Science and Technology in Ancient India, ed. by Dr.P.R.K. Rao, Vijnan Bharati,Mumbai , India, January 2002

  2. Gupta, R.C. (1980), "Indian Mathematics and Astronomy in Eleventh Century Spain," gaNita-bhAratI 2:53-7
    ---, (1982), "Indian Mathematics Abroad upto the Tenth Century AD," gaNita-bhAratI 4:10-6
    ---, (1989), "Sino-Indian Interaction and the Great Chinese Buddhist Astronomer-Mathematician I-Hsing (AD 683-727)," gaNita-bhAratI  11:39-49

  3. Hayashi, Takao (1994), "Indian Mathematics," in: Grattan-Guinness: 118-30
    ---, (1995), The Bakhshali Manuscript. An ancient Indian mathematical treatise, Groningen: Egbert Forsten
    ---, (forthcoming a), "Geometria per la costruzione di altari," Storia della Scienza: La Scienza in India, Roma: Istituto della Enciclopedia Italiana
    ---, (forthcoming b), "Indian Mathematics" in: Flood (forthcoming).

  4. Pingree, David. 1970-. Census of the Exact Sciences in Sanskrit. Philadelphia: American Philosophical Society. Four volumes.
  5. _____________. 1981. JyotiAA.stra : Astral and Mathematical Literature . Wiesbaden: Harrassowitz. History of Indian Literature vol. 6 fasc. 4.
  6. Joseph, George Ghevarughese, "Foundations of Eurocentrism in Mathematics," Race and Class, XXVII, 3(1987), p.13-28.
  7. This is an interesting paper by David Pingree with lot of original source material. the subject is the explosion of Sanskrit knowledge Materials in the two centuries prior to the arrival of the British. I was not aware that it was a period of noteworthy intellectual ferment. The last contribution to this explosion was in 1750 ce. Coincidentally, the date is in congruence with the start of British colonial domination in India . Such studies will reveal why and how the traditional learning systems based on Sanskrit disappeared from the map of India to be replaced by new learning and knowledge systems based on European notions.



References on Bhaskara


K Shankar Shukla, Bhaskara I, Bhaskara I and his works II. Maha-Bhaskariya (Sanskrit) (Lucknow, 1960).
K Shankar Shukla, Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya (Sanskrit) (Lucknow, 1963).

R C Gupta, Bhaskara I's approximation to sine, Indian J. History Sci. 2 (1967), 121-136.
R C Gupta, On derivation of Bhaskara I's formula for the sine, Ganita Bharati 8 (1-4) (1986), 39-41.
T Hayashi, A note on Bhaskara I's rational approximation to sine, Historia Sci. No. 42 (1991), 45-48.
P K Majumdar, A rationale of Bhaskara I's method for solving ax ± c = by, Indian J. Hist. Sci. 13 (1) (1978), 11-17.
P K Majumdar, A rationale of Bhatta Govinda's method for solving the equation ax - c = by and a comparative study of the determination of "Mati" as given by Bhaskara I and Bhatta Govinda, Indian J. Hist. Sci. 18 (2) (1983), 200-205.
A Mukhopadhyay and M R Adhikari, A step towards incommensurability of and Bhaskara I : An episode of the sixth century AD, Indian J. Hist. Sci. 33 (2) (1998), 119-129.
A Mukhopadhyay and M R Adhikari, The concept of cyclic quadrilaterals: its origin and development in India (from the age of Sulba Sutras to Bhaskara I, Indian J. Hist. Sci. 32 (1) (1997), 53-68.
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya II, Ganita 22 (2) (1971), 61-78.
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya III, Ganita 23 (1) (1972), 57-79
K S Shukla, Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya IV, Ganita 23 (2) (1972), 41-50.
I I Zaidullina, Bhaskara I and his work (Russian), Istor. Metodol. Estestv. Nauk No. 36 (1989), 45-49.

References for The Indian Sulvasutras


B Datta, The science of the Sulba (Calcutta, 1932).
G G Joseph, The crest of the peacock (London, 1991).

R C Gupta, New Indian values of from the Manava sulba sutra, Centaurus 31 (2) (1988), 114-125.
R C Gupta, Baudhayana's value of square root of 2, Math. Education 6 (1972), B77-B79.
S C Kak, Three old Indian values of , Indian J. Hist. Sci. 32 (4) (1997), 307-314.
R P Kulkarni, The value of known to Sulbasutrakaras, Indian J. Hist. Sci. 13 (1) (1978), 32-41.
G Kumari, Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
A Mukhopadhyay and M R Adhikari, The concept of cyclic quadrilaterals : its origin and development in India (from the age of Sulba Sutras to Bhaskara I, Indian J. Hist. Sci. 32 (1) (1997), 53-68.
A E Raik and V N Ilin, A reconstruction of the solution of certain problems from the Apastamba Sulbasutra of Apastamba (Russian), in A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin, Studies in the history of mathematics 19 "Nauka" (Moscow, 1974), 220-222; 302.



References for The Indian Contributions to the development of Place Value systems

Description: Now in paperback, here is Georges Ifrah's landmark international bestseller the first complete, universal study of the invention and evolution of numbers the world over. A riveting history of counting and calculating, from the time of the cave dwellers to the twentieth century, this fascinating volume brings numbers to thrilling life, explaining their development in human terms, the intriguing situations that made them necessary, and the brilliant achievements in human thought that they made possible. It takes us through the numbers story from Europe to China, via ancient Greece and Rome, Mesopotamia, Latin American, India, and the Arabic countries. Exploring the many ways civilizations developed and changed their mathematical systems, Ifrah imparts a unique insight into the nature of human thought and into how our understanding of numbers and the ways they shape our lives have changed and grown over thousands of years

Original Sources (Georges Ifrah, Universal History of Numbers)


1. Trishatiká by ShrIdharachârya (date unknown) [B. Datta and A. N. Singh (1938) p. 591
2. Karanapaddhati by Putumanasomayâjin (eighteenth century CE) [K. S. Sastri (1937)]
3. Siddhântatattvaviveka by Kamâlakara (seventeenth century CE) [S. Dvivedi (1935)]
4. Siddhántadarpana by Nilakanthasomayájin (1500 CE) 1K. V. Sarma (undated)]
5. Drigganita by Parameshvara (1431 CE) I Sarma (1963)1
6. VdAyapañchddhydvi (Anon., fourteenth century CE) [Sarma and Sastri (1962)]
7. Siddhantashiromani by Bháskaráchârya (1150 CE) [B. D. Sastri (1929)1
8. Rajamriganka by Bhoja (1042 CE) [Billard (1971), p. 101
9. Siddhántashekhara by Shripati (1039 CE) [Billard (1971), p. 101
10. ShishyadhIivrddhidatantra by Lalla (tenth century CE) [Billard (1971), p. 10]
11. Laghubháskariyaivivarana by Shankaranâràyana (869 CE) [Billard (1971), p.8]
12. Ganitasarasamgraha by MahãvIrâcháryà (850 CE) FM. Rangacarya (1912)]
13. Grahacharanibandhana by Haridatta (c. 850 CE) [Sarma (1954)]
14. Bhaskarivabhasya by Govindasvàmin (c. 830 CE) IBillard (1971),
15. Commentary on the Aryabhatiya by Bhâskara (629 CE) [K. S. Shukia and K. V. Sarma (1976)1
16. Brahmasphutasiddhánta by Brahmagupta (628 CE) [S. Dvivedi (1902)]
17. Pañchasiddhantiká by Varâhamihira (575 CE) [0. Neugebauer and D. Pingree (1970)1
Examples taken from the work of Bhaskara I




Vedic Mathematics

By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj (1884-1960)

Book ref: ISBN 0 8426 0967 9
Published by Motilal Banarasidas

From the Introduction by Smti Manjula Trivedi  16-03-1965.
An extract:

Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae from the Atharvaveda after assiduous research and ‘Tapas’ (austerity) for about eight years in the forests surrounding Sringeri. Obviously these formulae are not to be found in the present recensions of Atharvaveda. They were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda.

From the Preface by the author Jagadguru

Swami Sri Bharati Krsna Tirthaji Maharaj

We may however, at this point draw the earnest attention of every one concerned to the following salient items thereof:


  1. The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including Arithmetic, Algebra, Geometry – plane and solid, Trigonometry – plane and spherical, Conics – geometrical and analytical, Astronomy, Calculus – differential and integral etc.) In fact, there is no part of mathematics, pure or applied, that is beyond their jurisdiction.

  2. The Sutras are easy to understand, easy to apply and easy to remember, and the whole work can be truthfully summarised in one word ‘Mental’!

  3. Even as regards complex problems involving a good number of mathematical operations (consecutively or even simultaneously to be performed), the time taken by the Vedic method will be a third, a fourth, a tenth, or even a much smaller fraction of the time required according to modern (i.e. current) Western methods.

  4. And in some very important and striking cases, sums requiring 30, 50, 100 or even more numerous and cumbrous ‘steps’ of working (according to the current Western methods) can be answered in a single and simple step of work by the Vedic method!  And little children (of only 10 or 12 years of age) merely look at the sums written on the blackboard and immediately shout out and dictate the answers. And this is because, as a matter of fact, each digit automatically yields its predecessor and its successor! And the children have merely to go on tossing off (or reeling off) the digits one after another (forwards or backwards) by mere mental arithmetic (without needing pen or pencil, paper, slate etc.).

  5. On seeing this kind of work actually being performed by the little children, the doctors, professors and other ‘big-guns’ of mathematics are wonder-struck and exclaim: ‘Is this mathematics or magic’? And we invariably answer and say: ‘It is both. It is magic until you understand it; and it is mathematics thereafter’. And then we proceed to substantiate and prove the correctness of this reply of ours!

  6. As regards the time required by the students for mastering the whole course of Vedic Mathematics as applied to all its branches, we need merely state from our actual experience that 8 months (or 12 months) at an average rate of 2 or 3 hours per day should suffice for completing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also of foreign universities.

  7. And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge amount of time, energy, and money on and which even now it solves with the utmost difficulty and that also after vast labour involving large numbers of difficult, tedious and cumbersome ‘steps’ of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista (the appendix portion) of the Atharvaveda in a few simple steps and by methods that can be conscientiously described as mere ‘mental arithmetic’.

  8. It is thus in the fitness of things that the Vedas include 1. Ayurveda (anatomy, physiology, hygiene, sanitary science, medical science, surgery etc.), not for the purpose of achieving perfect health and strength in the after-death future but in order to attain them here and now in our present physical bodies. 2.Dhanurveda (archery and other military sciences), not for fighting with one another after our transportation to heaven but in order to quell and subdue all invaders from abroad and all insurgents from within. 3. Gandharva Veda (the science of art and music) and  4. Sthapatya Veda (engineering, architecture etc. and all branches of mathematics in general). All these subjects, be it noted, are inherent parts of the Vedas i.e., are reckoned as ‘spiritual’ studies and catered for as such therein.

  9. Similar is the case with Vedangas (i.e., grammar, prosody, astronomy, lexicography etc.) which according to the Indian cultural conceptions, are also inherent parts and subjects of Vedic (i.e. religious) study.

From the Foreward by Swami Pratyagatmananda Saraswati
Varanasi, 22-03-1965

An extract:

Vedic Mathematics by the late Shankaracharya (Bharati Krsna Tirtha) of Govardhan Pitha is a monumental work. In his deep-layer explorations of cryptic Vedic mysteries relating especially to their calculus of shorthand formulae and their neat and ready application to practical problems, the late Shankaracharya shows the rare combination of the probing insight of revealing intuition of a Yogi with the analytic acumen and synthetic talent of a mathematician.

With the late Shankaracharya we belong to a race, now fast becoming extinct, of diehard believers who think that the Vedas represent an inexhaustible mine of profoundest wisdom in matters of both spiritual and temporal; and that this store of wisdom was not, as regards its assets of fundamental validity and value at least, gathered by the laborious inductive and deductive methods of ordinary systemic enquiry, but was direct gift of revelation to seers and sages who in their higher reaches of Yogic realisation were competent to receive it from a source, perfect and immaculate.

Whether or not the Vedas are believed as repositories of perfect wisdom, it is unquestionable that the Vedics lived not as merely pastoral folk possessing a half or a quarter developed culture and civilisation. The Vedic seers were, again, not mere ‘navel-gazers’ or ‘nose-tip gazers’. They proved themselves adepts in all levels and branches of knowledge, theoretical and practical. For example, they had their varied objective science both pure and applied.

Let us take a concrete illustration. Suppose in a time of drought we require rains by artificial means. The modern scientist has his own theory and art (technique) for producing the result. The old seer scientist had his both also, but different from these now availing. He had his science and technique, called Yajna, in which Mantra, Yantra, and other factors must co-operate with mathematical determinateness and precision. For this purpose, he had developed the six auxiliaries of the Vedas in each of which mathematical skill and adroitness, occult or otherwise, play the decisive role. The Sutras lay down the shortest and surest lines. The correct intonation of the Mantra, the correct configuration of the Yantra (in the making of the Vedi etc., e.g. the quadrate of a circle), the correct time or astral conjunction factor, the correct rhythams etc. All had to be perfected so as to produce the desired results effectively and adequately. Each of these required the calculus of mathematics. The modern technician has his logarithmic tables and mechanic’s manuals. The old Yajnik had his Sutras.  

References (from Ian Pearce's article

Boyer, C. B. (1968). A History of Mathematics. USA: John Wiley and Sons, INC.
Al-Daffa, A. A. (1977). The Muslim Contribution to Mathematics. USA: Humanities Press. [All references denoted by AA'D]
Datta, B. and Singh, A. N. (1962). History of Hindu Mathematics, a source book, Parts 1 and 2, (single volume). Bombay: Asia Publishing House. [Denoted by AS/BD]
Duncan, D. E. (1998). The Calendar. London: Fourth Estate. [Denoted by DD]
Gurjar, L. V. (1947). Ancient Indian Mathematics and Vedha. Poona: [Denoted by LG]
Joseph, G. G. (2000). The Crest of the Peacock, non-European roots of Mathematics. Princeton and Oxford: Princeton University Press. [Denoted by GJ]
Katz, V. J. (1998). A History of Mathematics (an introduction). USA: Addison-Wesley.
Rashed, R. (1994). The Development of Arabic Mathematics: between Arithmetic and Algebra, especially "Appendix 1: The Notion of Western Science: Science as a Western Phenomenon. Netherlands: Kluwer Academic Publishers. [Denoted by RR]
Srinivasiengar, C. N. (1967). The History of Ancient Indian Mathematics. Calcutta: The World Press Private LTD. [Denoted by CS]
Struik, D. J. (1948). A Concise History of Mathematics. New York: Dover Publications, INC.

Journal articles:
Almeida, D. F., John, J. K. and Zadorozhnyy, A. (2001). Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications. Journal of Natural Geometry 20, 77-104. [All references denoted by DA/JJ/AZ1]
Channabasappa, M. N. (1984). Mathematical Terminology Peculiar to the Bakhshali Manuscript. Ganita-Bharati (Bulletin of the Indian Society of the History of Mathematics) 6, 13-18.
Dwary, N. N. (1991). Mathematics in Ancient and Medieval India. The Mathematics Education (Historical) 8, 39-41. [Denoted by ND1]
Gupta, R. C. (1990). The Chronic Problem of Ancient Indian Chronology. Ganita-Bharati 12, 17-26. [Denoted by RG1]
Gupta, R. C. (1986). Highlights of Mathematical Developments in India. The Mathematics Education 20, 131-138. [Denoted by RG2]
Gupta, R. C. (1976). Aryabhata, Ancient India's Greatest Astronomer and Mathematician. The Mathematics Education 10, B69-B73. [Denoted by RG3]
Gupta, R. C. (1982). Indian Mathematics Abroad upto the tenth Century A.D. Ganita-Bharati 4, 10-16. [Denoted by RG4]
Gupta, R. C. (1974). An Indian Approximation of Third Order Taylor Series Approximation of the Sine. Historica Mathematica 1, 287-289. [Denoted by RG5]
Gupta, R. C. (1983). Spread and Triumph of Indian Numerals. Indian Journal of History of Science 18, 23-38
Gupta, R. C. (1987). South Indian Achievements in Medieval Mathematics. Ganita-Bharati 9, 15-40.
Gupta, R. C. (1976). Development of Algebra. The Mathematics Education 10, B53-B61.
Gupta, R. C. (1980). Indian Mathematics and Astronomy in the Eleventh Century Spain. Ganita-Bharati 2, 53-57.
Gupta, R. C. (1977). On Some Mathematical Rules from the Aryabhatiya. Indian Journal of History of Science 12, 200-206.
Kak, S. C. (2000). Birth and Early Development of Indian Astronomy. The History of Non-Western Astronomy, 303-340. [Denoted by SK1]
Kak, S. C. (1993). Astronomy of the Vedic Altars. Vistas in Astronomy 36, 117-140.
Kak, S. C. (1988). A Frequency Analysis of the Indus Script. Cryptologia 12, 129-143.
Kak, S. C. (1999). The Solar Numbers In Angkor Wat. Indian Journal of history of Science 34, 117-126.
Kak, S. C. (1987). The Paninian Approach to Natural Language Processing. International Journal of Approximate Reasoning 1, 117-130.
Kak, S. C. (2000). Indian Binary Numbers and the Katapayadi Notation. Annals (B.O.R. Institute) 131, 269-272.
Kak, S. C. (2000). An Interesting Combinatoric Sutra. Indian Journal of History of Science 35, 123-127.
Kak, S. C. (1997). Three Old Indian Values of . Indian Journal of History of Science 32, 307-314.
Majumdar, P. K. (1981). A Rationale of Brahmagupta's Method of Solving
ax + c = by. Indian Journal of History of Science 16, 111-117.
Rajagopal, C. T. and Rangachari, M. S. (1977-78). On an Untapped Source of Medieval Keralese Mathematics. Archive for History of Exact Sciences 18, 89-102. [Denoted by CR/MR1]
Rajagopal, C. T. and Rangachari, M. S. (1986). On Medieval Keralese Mathematics. Archive for History of Exact Sciences 35, 91-99.
Rajagopalan, K. R. (1982). State of Indian Mathematics in the Southern Region upto 10th Century A.D. Ganita-Bharati 4, 78-82. [Denoted by KR1]
Sinha, R. S. (1981). Contributions of Ancient Indian Mathematicians. The Mathematics Edducation 15, 69-82. [Denoted by SS1]
Sinha, R. S. (1951). Bhaskara's Lilavati. Bulletin of Allahbad University Mathematical Association 15, 9-16.
Srinivasan, S. (1982). Evolution of Weights and Measures in Ancient India. Ganita-Bharati 4, 17-25. [Denoted by SSr1]

Internet Sources:
MacTutor History of Mathematics: History Topics and Mathematical biographies
(Written and compiled by Professor Edmund F Robertson and Dr John J O'Connor, University of St Andrews. I used 38 articles form this website.)

Kalyanaraman, S. The Sarasvati-Sindhu Civilization (c. 3000 B.C.). Article retrieved from Indology List-Serve.






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