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ऊँ सहनाववतू । सह नौभभुनक्तु ।सहवीर्यँ करवावहै ।
तेजस्वि नावधीतमस्तु । मा विद्विषावहै ।।
ऊँ शान्तिः शान्तिः शान्तिः ।।


References on Mathematical Sciences

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

http://www.infinityfoundation.com/sourcebook.htm
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.
1328.
4. A. Seidenberg, 1978. The origin of Mathematics. Archive
for History of Exact Sciences 18.4, pp. 30142.
5. Frits Staal, 1965. Euclid and Panini. Philosophy East and
West 15.2, pp. 99116.
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. 199204.
8. C.O. Selenius, 1975. Rationale of the chakravala process
of Jayadeva and Bhaskara II. Historia Mathematica, 2, pp.
167184.
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. 73127.
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. 4766.
12. Saroja Bhate and Subhash Kak, 1993. Panini's grammar and
computer science. ABORI, 72, pp. 7994.
Section 3: Astronomy
13. Subhash Kak, 1992. The astronomy of the Vedic altars and
the Rgveda. Mankind Quarterly, 33, pp. 4355.
14. Subhash Kak, 1995. The astronomy of the age of geometric
altars. Quarterly Journal of the Royal Astronomical Society,
36, pp. 385395.
15. Subhash Kak, 1996. Knowledge of planets in the third
millennium BC. Quarterly Journal of the Royal Astronomical
Society, 37, pp. 709715.
16. Subhash Kak, 1998. Early theories on the distance to the
sun. Indian Journal of History of Science, 33, pp. 93100.
17. B.N. Narahari Achar, 1998. Enigma of the fiveyear yuga
of Vedanga Jyotisa, Indian Journal of History of Science,
33, pp. 101109.
18. B.N. Narahari Achar, 2000. On the astronomical basis of
the date of Satapatha Brahmana, Indian Journal of History of
Science, 35, pp. 119.
19. B.L. van der Waerden, 1980. Two treatises on Indian
astronomy, Journal for History of Astronomy 11, pp.5058.
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.
784790.

What EleventhCentury 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.

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

Gupta, R.C. (1980),
"Indian Mathematics and Astronomy in Eleventh Century
Spain," gaNitabhAratI 2:537
, (1982), "Indian Mathematics Abroad upto the Tenth
Century AD," gaNitabhAratI 4:106
, (1989), "SinoIndian Interaction and the Great Chinese
Buddhist AstronomerMathematician IHsing (AD 683727),"
gaNitabhAratI 11:3949

Hayashi, Takao (1994),
"Indian Mathematics," in: GrattanGuinness: 11830
, (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).
 Pingree, David. 1970. Census
of the Exact Sciences in Sanskrit. Philadelphia:
American Philosophical Society. Four volumes.
 _____________. 1981.
JyotiAA.stra : Astral and Mathematical Literature .
Wiesbaden: Harrassowitz. History of Indian Literature vol. 6
fasc. 4.

Joseph, George Ghevarughese, "Foundations of
Eurocentrism in Mathematics," Race and Class, XXVII,
3(1987), p.1328.

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
Books:
K Shankar Shukla, Bhaskara I, Bhaskara I and his works II.
MahaBhaskariya (Sanskrit) (Lucknow, 1960).
K Shankar Shukla, Bhaskara I, Bhaskara I and his works III.
LaghuBhaskariya (Sanskrit) (Lucknow, 1963).
Articles:
R C Gupta, Bhaskara I's approximation to sine, Indian J. History
Sci. 2 (1967), 121136.
R C Gupta, On derivation of Bhaskara I's formula for the sine,
Ganita Bharati 8 (14) (1986), 3941.
T Hayashi, A note on Bhaskara I's rational approximation to
sine, Historia Sci. No. 42 (1991), 4548.
P K Majumdar, A rationale of Bhaskara I's method for solving ax
± c = by, Indian J. Hist. Sci. 13 (1) (1978), 1117.
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), 200205.
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), 119129.
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), 5368.
K S Shukla, Hindu mathematics in the seventh century as found in
Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1)
(1971), 115130.
K S Shukla, Hindu mathematics in the seventh century as found in
Bhaskara I's commentary on the Aryabhatiya II, Ganita 22 (2)
(1971), 6178.
K S Shukla, Hindu mathematics in the seventh century as found in
Bhaskara I's commentary on the Aryabhatiya III, Ganita 23 (1)
(1972), 5779
K S Shukla, Hindu mathematics in the seventh century as found in
Bhaskara I's commentary on the Aryabhatiya IV, Ganita 23 (2)
(1972), 4150.
I I Zaidullina, Bhaskara I and his work (Russian), Istor.
Metodol. Estestv. Nauk No. 36 (1989), 4549.
References for The Indian
Sulvasutras
http://wwwgap.dcs.stand.ac.uk/~history/HistTopics/References/Indian_sulbasutras.html
Books
B Datta, The science of the Sulba
(Calcutta, 1932).
G G Joseph, The crest of the peacock
(London, 1991).
Articles
R C Gupta, New Indian values of from the
Manava sulba sutra, Centaurus 31 (2)
(1988), 114125.
R C Gupta, Baudhayana's value of square
root of 2,
Math. Education 6 (1972), B77B79.
S C Kak, Three old Indian values of ,
Indian J. Hist. Sci. 32 (4) (1997),
307314.
R P Kulkarni, The value of known to
Sulbasutrakaras, Indian J. Hist. Sci. 13
(1) (1978), 3241.
G Kumari, Some significant results of
algebra of preAryabhata era, Math. Ed.
(Siwan) 14 (1) (1980), B5B13.
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),
5368.
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), 220222; 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
yearsOriginal
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),
p.8.1
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 (18841960)
Book
ref: ISBN 0 8426 0967 9
Published by Motilal Banarasidas
From the Introduction by Smti Manjula Trivedi
16031965.
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
Extracts:
We may however, at this point draw the earnest attention
of every one concerned to the following salient items
thereof:

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.

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’!

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.

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.).

On
seeing this kind of work actually being performed by
the little children, the doctors, professors and
other ‘bigguns’ of mathematics are wonderstruck
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!

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.

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 ultraeasy 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’.

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 afterdeath 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.

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, 22031965
An extract:
Vedic Mathematics by the late Shankaracharya (Bharati
Krsna Tirtha) of Govardhan Pitha is a monumental work.
In his deeplayer 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 ‘navelgazers’ or ‘nosetip
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 cooperate 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
http://tinyurl.com/n2f55
Books:
Boyer, C. B. (1968). A History of Mathematics. USA: John
Wiley and Sons, INC.
AlDaffa, 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,
nonEuropean roots of Mathematics. Princeton and Oxford:
Princeton University Press. [Denoted by GJ]
Katz, V. J. (1998). A History of Mathematics (an
introduction). USA: AddisonWesley.
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, 77104. [All references
denoted by DA/JJ/AZ1]
Channabasappa, M. N. (1984). Mathematical Terminology
Peculiar to the Bakhshali Manuscript. GanitaBharati
(Bulletin of the Indian Society of the History of
Mathematics) 6, 1318.
Dwary, N. N. (1991). Mathematics in Ancient and Medieval
India. The Mathematics Education (Historical) 8, 3941.
[Denoted by ND1]
Gupta, R. C. (1990). The Chronic Problem of Ancient
Indian Chronology. GanitaBharati 12, 1726. [Denoted by
RG1]
Gupta, R. C. (1986). Highlights of Mathematical
Developments in India. The Mathematics Education 20,
131138. [Denoted by RG2]
Gupta, R. C. (1976). Aryabhata, Ancient India's Greatest
Astronomer and Mathematician. The Mathematics Education
10, B69B73. [Denoted by RG3]
Gupta, R. C. (1982). Indian Mathematics Abroad upto the
tenth Century A.D. GanitaBharati 4, 1016. [Denoted by
RG4]
Gupta, R. C. (1974). An Indian Approximation of Third
Order Taylor Series Approximation of the Sine. Historica
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Gupta, R. C. (1983). Spread and Triumph of Indian
Numerals. Indian Journal of History of Science 18, 2338
Gupta, R. C. (1987). South Indian Achievements in
Medieval Mathematics. GanitaBharati 9, 1540.
Gupta, R. C. (1976). Development of Algebra. The
Mathematics Education 10, B53B61.
Gupta, R. C. (1980). Indian Mathematics and Astronomy in
the Eleventh Century Spain. GanitaBharati 2, 5357.
Gupta, R. C. (1977). On Some Mathematical Rules from the
Aryabhatiya. Indian Journal of History of Science 12,
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Kak, S. C. (1993). Astronomy of the Vedic Altars. Vistas
in Astronomy 36, 117140.
Kak, S. C. (1988). A Frequency Analysis of the Indus
Script. Cryptologia 12, 129143.
Kak, S. C. (1999). The Solar Numbers In Angkor Wat.
Indian Journal of history of Science 34, 117126.
Kak, S. C. (1987). The Paninian Approach to Natural
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Kak, S. C. (1997). Three Old Indian Values of . Indian
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in Ancient India. GanitaBharati 4, 1725. [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 SarasvatiSindhu Civilization (c.
3000 B.C.). Article retrieved from Indology ListServe.

(aryabhattiya)

