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MATHEMATICAL HERITAGE OF INDIA: SOME REMEDIAL ISSUES:
MATHEMATICS HISTORY IN TEACHING
Vinod Mishra
Sant Longowal Institute of Engineering & Technology, India
[email protected]
Abstract. Indian mathematics has its glorious root in the civilization. This article explores the
various causes of lacking of its expansion during ancient and medieval periods. Its pedagogical
advantages in teaching are well known and, therefore, necessary steps for upliftment of history of
Indian mathematics have been suggested. At the end, utility of history of mathematics in
mathematical teaching has been experimented and a course on history of Indian mathematics has
been proposed.
1. Introduction
Mathematics has the largest history of more than 5000 years developed in isolation in the
beginning of different cultures-later exchanged the ideas somehow till the emergence of modern
era-and what we observe today is the synthesis of fruitful concepts of ancient, medieval and, of
course, later modern periods in its major share.
Post independent India has paid special attention towards the better understanding of
history of Indian science by initiation of Government of India by constituting a National
Commission (1969) functional under the Indian National Science Academy (INSA, New Delhi)
and establishing National Institute of Science, Technology and Development Studies (New
Delhi). In this context it is worthy to mention that the first major activity in the form of a
symposium was held at University of Delhi in 1950. Since then a number of theme oriented
programs (lectures/seminars/symposia/workshops) on history of mathematics in global
perspective have been organized. Doubtless to say that the first International Congress on
History of Science (July 8-14, 1929), Mexico City, under the auspices of International
Commission on History of Science proved to be a pace-setter for the whole world.
INSA has brought record publications of books and project based monographs on history of
mathematics. It also brings a journal Indian Journal on History of Science. Yet another
organization Indian Society for History of Mathematics constituted in 1978, successfully brings
out journal Ga𝑛𝑖̇ta Bh𝑎̅rat𝑖̅.
2. Mathematics in India-Some Remarks
In Indian subcontinent, mathematics flourished under the banner of Hindu astronomy in
the early centuries and after the emergence of Jaina philosophy in about 300 B.C., Indian
mathematics grew interacting each other. Interfusion of Indian mathematics with other cultural
mathematics after the siddh𝑎̅ntic age cannot be ruled out, in fact on small scale. The concept of
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permutation and combination, infinity among others are of course some of the notable
contributions of the Jaina sects.
The ancient civilization of India differs from those of Egypt, Mesopotamia and Greece, in
that of its traditions has been preserved without a break down to the present day. Until the advent
of the archaeologist, the peasant of Egypt or Iraq had no knowledge of the culture of his
forefathers, and it is doubtful whether his Greek counterpart had any but the vagvest ideas about
the glory of Peri-clean Athens. In each case there had been an almost complete break with the
past. On the other hand, the earliest Europeans to visit which indeed exaggerated that antiquity,
and claimed not to have fundamentally changed for many thousands of years. To this day
legends known to the humblest India recall the names of shadowy chieftains who lived nearly a
thousand years before Christ, and the orthodox Brahman in his daily worship repeats hymns
composed earlier. India and China have, in fact, the oldest continuous cultural traditions in the
world [2, p. 4].
Regarding transmission of Buddhist culture and Indian scientific culture outside India,
Gupta's view is worth noticeable (cf. Ga𝑛𝑖̇ta Bh𝑎̅rat𝑖̅ 11 (1989), 38-39):
The rock edicts of king Aś oka (3rd century B.C.) show that he had already paved the way for the
expansion of Buddhism outside India. Subsequently, Buddhist missionaries took Buddhism to
Central Asia, China, Korea, Japan and Tibet in the North, and to Burma, Ceylon, Thailand,
Cambodia and other countries of the South. This helped in spreading Indian culture to these
countries. It is well said that “Buddhism was, in fact, a spring wind flowing from one end of the
garden of Asia to the other and causing to bloom not only the lotus of India, but the rose of
Persia, the temple flower of Ceylon, the zebina of Tibet, the chrysanthemum of China and the
cherry of Japan. It is also said that Asian culture is, as a whole, Buddhist culture”. Moreover,
some of these countries received with Buddhism not only their region but practically the whole
of their civilization and culture.
The tradition of Buddhist education system gave birth to large-scale monastic
universities. Some of these famous universities were Nalanda, Valabhi, Vikramsila, Jagaddala
and Odantapuri. They attracted students and scholars from all parts of Asia. Of these, the
Nalanda University was most famous with about 10000 students and 1500 teachers. The range of
studies covered both sacred and secular subjects of Buddhist as well as Br𝑎̅hminical learning.
The monks eagerly studied, besides Buddhist works (including Abhidharmako𝑠́ a), the Vedas,
medicine, arithmetic, occult sciences and other popular subjects. There was a special provision
for the study of astronomy and an astronomical observatory is said to be a part thereof [op. cit,
p. 39].
Today we know more about western contribution than Indian. What are the reasons
behind? Why did they impose their superiority? and put stamps even on the problems / ideas
which were well-known and well-defined in India hundreds year ago. A few instances are:
1. The theorem “the diagonal rope of an oblong produces both (areas) which its side and
length produce separately” (cf. [7]) was the contribution of Baudha̅yana of India who
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lived in about 800 B.C. It was dedicated to Pythagoras (ca. 540 A.D.) of Greek which had
been little known to him.
2. Needham’s observation that the “Rule of Three though generally attributed to India is
found in the Han Chiu Chang earlier than any Sanskrit texts” ([11, p. 146], cf. [10, p.1]).
Very recently, after a lengthy discussion, Maiti on the basis of T.S. Kuppana Shastri’s
work tried to trace that the antiquity of the Rule of Three’ (trair𝑎̅𝑠́ ika) dates back to
Veda𝑛̇ gajyoti𝑠̇ a (ca. 600 B.C.).
3. The equivalent of the so-called Leibniz’s series (Leibniz, 1646-1716 A.D.)

1 1
1
 1        (1) n 1

4
3 5
2n  1
had been obtained by M𝑎̅dhava (fl. 1360-1425 A.D.).
4. The result
r
1 (ab  cd )( ac  bd )( ad  bc)
4 ( s  a)( s  b)( s  c)( s  d )
for a cyclic quadrilateral, wherein r = circum-radius, a, b, c, d, the lengths of sides and s
= (a+b+c+d)/2, has been given by Parame𝑠́ vara (fl. 1360-1455) more than two centuries
earlier than what is stated to have been rediscovered in Europe by S.A.J. L’Huilier (1782
A.D.).
Others record correctly. A few quotes are:
1. “I shall not now speak of the knowledge of the Hindus,------ of their subtle discoveries in the
science of astronomy –discoveries even more ingenious than those of the Greaks and
Babylonians – of their method of calculation which no words can praise strongly enough –I
mean the system using nine symbols. If these things were known by the people who think that
they alone have mastered the sciences because they speak Greek they would perhaps, be
convinced, though little late in the day, that other folk, not only Greeks, but men of a different
tongue, know something as well as they”.
The Syrian astronomer–monk Severus Sebokht (writing A.D.662) (cf. [2, p. VI ])
2. “Most of the popular history of mathematics texts gives the impression that contemporary
mathematical though stems directly from resurgence of western creativity after the dark age.
However, mathematical ideas continued to develop in India, the Middle East and Orient from
their ancient beginnings. There is evidence that some of modern mathematics travelled from
these sources.
Examples of Hindu mathematics appear in extant manuscripts from 8th and 9th centuries.
Many of the permutation and combination formulas attributed to Cardano, Tartaglia and Pascal
were known to the Hindus-----. It is clear that the algebra of combinations in the 12th century and
perhaps earlier was considerably more advanced in Hindu mathematics than in Western Circles”.
Sharon Kunoff (1990/91)
3. “The development of our system of notation for integers was one of the two most influential
contributions of India to the history of mathematics. The other was the introduction of an
equivalent of the sine function in trigonometry to replace the Greek tables of chords.”
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C.B. Boyer [p. 241]
4. “The Rule of Three which originated among the Hindus is a device used by oriental merchants
to secure results to certain numerical problems.”
Lam Lay Yong (p.329)(cf. Ga𝑛𝑖̇ta Bh𝑎̅rat𝑖̅ 18(1996), p.7)
Other few examples are the (so-called) Pascal’s triangle, the notion of differential
calculus, the basic trigonometry, Pell’s equation etc. Behind these discoveries they sometimes
claim that Indians took inspiration from a common source, though it might not be.
3. Root Cause of Lacking of Indian Mathematics Expansion
Let us see the root cause of lacking of Indian mathematics expansion during ancient and
medieval periods.
1) The proofs were not explicitly mentioned through they were understood well. In
Hayashi’s view, “Neither the 𝐴̅ryabhat𝑖̅ya nor the Brahmasphutasiddh𝑎̅nta contains proof of
their mathematical rules, but this does not necessarily mean that their authors did not prove them.
It was probably a matter of the style of exposition. In fact, later prose commentaries contain a
number of demonstrations or derivations, together with underlying principles ------. The
recognition of the importance of proofs dates back at least to the time of Bh𝑎̅skara I (around
A.D. 600), who, in this commentary on the 𝐴̅ryabhat𝑖̅ya, rejected the Jain value of  , 10 ,
saying that it was only a tradition and there was no derivation of it. He also emphasized the
importance of verifications of solutions to mathematical problems. ------ verifications are still
found in Siṁhatilakasu̅ri’s commentary (in the thirteenth century) on Sri̅pati's Ga𝑛̇ itatilaka, but
occur very rarely thereafter, in contradiction to the growing popularity of derivations”
2) Chronological order had not been strictly adhered to. “Chronology is the back bone of
history and its knowledge is essential for a historian dealing with any period, culture-area or
subject. There cannot be a coherent history without a chronological order. Proper historical
writing is not possible unless there is a sound chronology”, says Gupta (1989). For the case of
India he further emphasis that “the problem of chronology continues to be very serious especially
with regard to the prehistoric and ancient periods. The dates of most of the important events and
literacy sources are full of serious controversies and divergent opinions. What to say about the
absolute chronology, even a relative chronology is not free from challenges.” Chronology
problem has arisen because whatever Indian writers wrote they dedicated to God and did not
want credit by mentioning authorship, date etc. and more so in some cases they generally
attributed as if the things (matters) were enunciated in antiquity.
3) There was no tradition of educating enmasses through proper writings. While writing
proto- historic period of India, Balshaw (p. 30) argues that “among the many people who
entered India in the 2nd millennium B.C.# was a group of related tribes whose priests has
perfected a very advanced poetic technique, which they used for the composition of hymns to be
sung in praise of their gods at sacrifices. These tribes, chief of which was that of the Bh𝑎̅ratas,
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settled mainly in East Punjab and in the region between the Satlaj and the Jamna which later
became known as Brahmavarta. The hymns composed by their priests in their new home were
carefully handed down by word of mouth, and early in the first millennium B.C. were collected
and arranged. They were still not committed to writing, but by now they were looked on as so
sacred that even minor alternations in their text were not permitted, and the priestly schools
which preserved then devise the most remarkable and effective system of checks and counter
checks to ensure their purity. Even when the art of writing was widely known in India the hymns
were rarely written, but, thanks to the brilliant feats of memory of many generations of
Brahmans and the extreme sanctity which the hymns were thought to possess, they have survived
to the present day in a form which, from internal evidence, appears not to have been seriously
tampered with for nearly three thousand years. This great collection of hymns is Rg Veda, still in
theory the most sacred of the numerous sacred texts of the Hindus.”
In connection with “s𝑢̅tra period, an age of specialization”, Bag [p. 4] writes “------- that
the study of mathematics started with the s𝑢̅tra period. At first, the study was strictly
surbservient to the primary needs and education meant only the transmission of traditions from
the teacher to the pupil and the committing to memory the sacred texts. In course of time,
however, the contents of this education began to widen out and each one of the several angas
(limbs) of the Veda began to develop. It is in this connection with the construction of sacrificial
altars of proper size and shape that the problems of geometry and perhaps also of arithmetic and
algebra were evolved. The study of astronomy arose out of the necessity for fixing the proper
time for sacrifices.” It will be worth noticing that the 𝐴̅ryabhat𝑖̅ya is the oldest Indian work on
astronomy with a chapter on mathematics having mentioned the date of compilation 499 A.D.
4) Till the middle of eighteenth century A.D. Europeans were unaware of Indian culture
and its contributions, though they knew it little from various sources. The main reason was
the lack of knowledge of Sanskrit, the principal Indian language, in which most of the oriental
books had been written. “Until the last half of the 18th century Europeans made no real attempt to
study India’s ancient past, and her early history was known only from brief passages in the
works of Greek and Latin authors. A few devoted missionaries in the Pennisula gained deep
understanding of contemporary Indian life, and a brilliant mastery of the vernaculars, but they
made no real attempt to understand the historical background of the culture of the people among
whom they worked. They accepted that culture at its face value, as very ancient and unchanging,
and their only studies of Indian’s past were in the nature of speculations linking the Indians with
the descendants of Noah and the vanished empires of the Bible” [2, p. 4]. Meanwhile Father
Hanxleden was one of a few Jesuits who succeeded in mastering Sanskrit in the year from 1699
to 1732 and compiled the first ever Sanskrit grammar in European language. Since then more
scholars including Father Coeurdoux (1767) (a Jesuit) and Sir William Jones (1783) (an
Englishman) took keen interest and recognized the kinship of Sanskrit. [2, pp. 4-5].
“Of the little band of Englishmen who administered Bengal for the Honourable East India
Company only one, Charles Wilkins (1749-1836), had managed to learn Sanskrit. With the aid of
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Wilkins and friendly Bengali pandits Jones began to learn the language. On the first day of 1784
the Asiatic Society of Bengal was founded on Jones’ initiative and with himself as president. In
the Journal of this society, Asiatic Researches, the first real steps in revealing India’s past were
taken. In November 1784 the first direct translation of a Sanskrit work into English, Wilkins’
Several important translations of works appeared in successive issues of Asiatic
Researches and elsewhere. Jones and Wilkins were truly the fathers of Indology. They were
followed in Calcutta by Henry Colebrooke (1765-1837) and Horace Hayman Wilson (17891860). To the works of these pioneers must be added that of the Frenchman Anquetil-Duperron,
a Persian scholar who, in 1786, published a translation of four Upani𝑠̇ ads from a 17th –century
Persian version- the translation of the whole manuscript, containing 50 Upani𝑠̇ ads, appearing in
1801.
Interest in Sanskrit began to grow in Europe as a result of those translations. In 1795, the
government of the French Republic founded the Ecole des Languages Orientales Vivantes, and
there Alexander Hamilton (1762-1824), one of the founding members of the Asiatic Society of
Bengal, held prisoner on parole in France at the end of the Peace of Amiens in 1803, became the
first person to teach Sanskrit in Europe. It was from Hamilton that Friedrich Schlegel, the first
German Sanskritist, learnt the languages. The first university chair of Sanskrit was founded at the
College de France in 1814 and held by Leonard de Chezy, while from 1818 onwards the larger
German universities set up professorships. Sanskrit was first taught in England in 1805 at the
training college of the East India Company at Hertfort. The earliest English chair was the Boden
Professorship at Oxford, first filled in 1832, when it was conferred upon H.H. Wilson, who had
been an important member of the Asiatic Society of Bengal. Chairs were afterwards founded at
London, Cambridge and Edinburgh, and at several universities of Europe and America” [2, pp.
5-6].
Afterwards the establishment of French Societe Asiatique (1821), Paris and Royal Asiatic
Society (1823), London beginnings the work of editing and study of ancient Indian literature
went on space throughout the 19th century [2, p. 6].
4. Why do Research on Indian Mathematics?
Is it obligatory for an Indian to do research on Indian mathematics if one makes a mind of
research in history of mathematics and its pedagogical advantages in teaching/education? The
reason is obvious because:
a) It is the mirror of our cultural heritage and, therefore, it becomes a duty of every nation to
correctly understand its heritage, to preserve it and to interpret it for outside world.
b) Indians are fortunate to have Sanskrit, Persian and Arabic scholars who made valuable
contributions by editing/translating a number of manuscripts and placed before them for
mathematicians who are not expert in these languages. However, some of them may not
have that level of mathematical knowledge. To rectify the errors in editing/translating the
manuscripts and to find their inter-relationships and comparative evolution, the study
becomes even more important.
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c) It is often pinpointed that some of the westerners have not properly interpreted our
manuscripts, may be due to difficulty of language, and therefore it becomes important
that we should provide the correct interpretations. While doing so we should rather avoid
introducing our own biased statements any way.
5. The Necessary Steps for Upliftment of History of Indian Mathematics
For the benefit of mathematical community the following steps may be initiated/taken care of:
a) A bulk of Indian manuscripts which are either untouched or unearthed. A proper survey
of such works in India and abroad is imminent task.
b) Archival records in history of science/mathematics need to be maintained. Recently a
central unit for cataloguing science archives has been constituted at Lucknow [Ga𝑛𝑖̇ta
Bh𝑎̅rat𝑖̅].
c) Exhibitions on paintings and sculptures certainly give many impressions regarding lifestyle of mathematicians and reflects ethnological culture of that time.
d) Documentaries/films on history of mathematics highlighting the contributions of eminent
mathematicians and their lives should be produced. As the first step, film on Ramanujan
has been produced.
e) The showing of mathematical dramas creates excitement in the subject-study. Ms P.
Sharda (TIFR, Mumbai) and associates have taken such a lead in showing the drama of
Indian mathematics at various placed of India (including that in International Conference
on History of Mathematical Sciences (INSA, 2001).
f) A course on history of mathematics should be part of every university mathematics
degree and selected areas of history of mathematics should form a suitable component of
‘wider-studies’ courses and that it should be an essential component in the training of
teachers of mathematics.
g) Suitable teaching materials continue to be developed and every mathematics text book
should highlight the contributions of both ancient and modern mathematicians and their
work, related to the subject.
h) There are advantages in presenting ancient problems in a modern context. In turn, this
makes the subject more understandable. Modern generalizations should be avoided.
i) Two aspects which stimulate students’ interest are (i) emphasis on ‘story-line’, and (ii)
some introduction of top personalities of individual mathematicians including their
historical anecdotes.
j) Every mathematical concept has its origin in some physical phenomenon/problem,
though today it may be many generations of abstractions away from its sources.
Knowledge of this connection will not only strengthen the understanding of the concepts
but also enables the concept to find application in future.
k) Stress should be given to translations, editions and interpretations of available
manuscripts by experts. INSA is doing a right job in this pretext. While translating the
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verses / proses from ancient texts / manuscripts few words may be added in brackets at
appropriate places so as to make the matter tangible.
l) Analytical database of information sources must be created. There is immense need of
documentation for highlighting the history of mathematics activities of the other
countries, institutions and societies.
6. History of Mathematics as a Course of Study
Hardly a University in India has adopted a course of study in History of Mathematics at
undergraduate and postgraduate level. Might be due to shortage of expertise, due to lack of
interest or non-availability of resource materials. The best course could be (1) History of Ancient
and Medieval Mathematics, (2) History of Modern Mathematics or (3) Combination of (1) &
(2).
Below is a proposed outline syllabus of ‘History of Indian Mathematics’ course from the
exhaustive content. Its parts could be suitably adopted after minor changes as per need at B.Sc.
and M.Sc. programmes with mathematics as a major subject. Each chapter has to contain origin
and development of the concepts, their necessasity, proofs /rationale, parallelism with other
culture areas and relation to modern mathematics. Sufficient problems / examples should be
included.
Course Outline
Progressive series
Arithmetic progression. Geometrical and symbological interpretation. Complex series.
Geometric progression and their geometrical interpretation.
Rule of Compound Proportions
Rule of three terms and their inversion. Rules of seven and nine.
Series for Pie
Values of Pie. Various types of Pie series and their convergence.
Indeterminate Equations
Indeterminate equations of first order of the type 𝑎𝑥 − 𝑏𝑦 = ±𝑐, 𝑎𝑥 + 𝑏𝑦 = 𝑐. Kuttaka method.
Linear simultaneous indeterminate equations.
Second degree indeterminate equations: 𝑁𝑥 2 + 𝑘 = 𝑦 2 . Cakraval method. Evaluation of √𝑁 .
Triangles and Quadrilaterals
Theorem of square on the diagonal. Geometrical proof. Some geometrical constructions.
Application of right triangle in Mah𝑎̅vedi structure.
Convex quadrilaterals.
Incircumscribing triangles and Cyclic quadrilaterals
Incircumscribing triangle. Area. Circum-radius.
Cyclic quadrilateral. Area. Third diagonal concept.
Similarity of Plane Figures
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Theorem of square on half chord. Examples: Problems of broken bamboo, Lotus, Hawk and rat,
Crane and fish, Two bamboos. Brahmagupta’s theorem. Shadow problems.
Combinatorics and Binomial Theorem
Combinatorics and Meru representation.
Permutation of unlike and some as like things. Sum of permutations.
The so called Binomial theorem. Meru way of presentation.
Plane Trigonometry and Calculus
𝑅 sin 𝜃, 𝑅 cos 𝜃. Sines and cosines of multiple and submultiple arcs. 𝑠𝑖𝑛18° .
Notion of differential and integral calculus.
The other topics which could be included are: quadratic equations, frustum like solids, circles
and spheres, spherical trigonometry and many more.
7. History of Mathematics in Teaching-Some Issues
The recent years have witnessed a growing attraction towards the role of history of mathematics
not only in improving the teaching and learning of mathematics in its many facets at all levels
but also in training of teachers and educational research. How to utilize history of mathematics in
teaching? obviously becomes an important component of research in mathematical education,
and forces us to evolve a method or a program that develops “a deeper understanding of the
factors involved in the relations between history and pedagogy of mathematics, in different areas
of mathematics, and with pupils at different stages with different environments and
backgrounds”.
8. Experimentation in Teaching
I would like to share teaching experience using history of mathematics in a Semester Course
“Numerical Methods” for Bachelor of Engineering class at my parent institution. Students were
to be taught the concept of numerical solutions of algebraic and transcendental equations
particularly using Newton-Raphson (N-R) method for finding the nth root of N, 𝑁 > 0, a
positive non-square integer. Instead of providing the students only the modern theory and
applications of N-R method, I thought to equip them right from the effort of computing √𝑁 in
the Indian as well as other culture areas and their necessasity through algebraic and geometrical
means. And ultimately relating the evaluation of √𝑁 to that of N-R method as a particular case
of it. Even they were taught some extended modern generalization of √𝑁 . The class started with
the basic introduction to algebraic and transcendental equations followed by computational
efforts for evaluating √2 and√3 in various cultural areas of India, Babylonia and Greek. Then
followed the Lemma as [see Vinod Mishra, A Study in Vedic Geometry and Its Relevance to
Science and Technology, Ph.D. Thesis, GKU Hardwar, 1998, Chapter II]:
Lemma 1: Let 𝑁 > 0 be a non square integer such that 𝑁 = 𝐴12 + 𝑟, where 𝐴1 is the largest
positive integer and |𝑟| the smallest integer. Then
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𝐴𝑖 =
2
2
𝑁𝑑𝑖−1
+𝑛𝑖−1
2𝑑𝑖−1 𝑛𝑖−1
, 𝑖 = 2, 3, ….
are the successive convergents to the series
√𝑁 = 𝐴1 + 𝑚𝑟 + ∑∞
𝑖=3
2
2
𝑁𝑑𝑖−1
−𝑛𝑖−1
2𝑑𝑖−1 𝑛𝑖−1
.
(1)
Where 𝑑𝑖−1and 𝑛𝑖−1 stand for the sum of denominator and numerator up to (𝑖 − 1) terms
respectively and m for a positive integer.
𝑛
The above Lemma can be easily extended to √𝑁.
Yet another concept from Nara̅yana’s method was also explored.
The following Lemma due to Brahmagupta is less known:
Lemma2: Let N be a positive integer. If two integral solutions of the equation
𝑁𝑥 2 + 1 = 𝑦 2
(2)
Are known then any number of other solutions can be found.
For example if (𝑝, 𝑞) and (𝑝́ , 𝑞́ ), then (𝑝𝑞́ + 𝑝́ 𝑞 , 𝑞𝑞́ + 𝑁𝑝𝑝́ ) is also a solution of (2). If we take
(𝑝, 𝑞)=(𝑝́ , 𝑞́ ) = (𝑎, 𝑏) in this Lemma, then (2𝑎𝑏, 𝑏 2 + 𝑁𝑎2 ) is also a solution. For finding an
approximate value of square root, Nar𝑎̅yana in his Bijaga𝑛̇ ita, gives the rule which is put as:
𝑏
If (𝑎, 𝑏) is a solution of (2), then √𝑁 ≅ 𝑎 (𝑎 ≠ 0), the first approximation. From (2), √𝑁 =
𝑦
𝑥
1
√1 − 𝑦 2. Nara̅yana anticipates that if y (so also x) is large, 𝑦𝑥 is a close approximation of √𝑁.
2
2
+𝑁𝑎
The second approximation would be 𝑏 2𝑎𝑏
.
The above two results are exactly derived from Newton-Raphson iterative method. Let
f ( x)  N  x 2 .
Then
approximations of
f ( A1 )  N  A1 and
2
f ( A1 )  2 A1 .
N are respectively A1 and A2  A1 
So
f ( A1 )
f  ( A1 )
the
 A1 
first
N  A12
2 A1

and
N  A12
2 A1
second

Nd i2  ni2
2 d i ni
.
Continuing of above technique yield third and subsequent approximations.
At the end they were explored some examples in old as well as modern areas of concern, not
much beyond the scope of syllabus. With the little effort, the outcomes were:
1) Students were surprised to know the methods which were hardly known to them through
modern available resources.
2) They have generated a sense of interest in subject and shown eagerness to know more of
concept beginning from origin to varied applications.
3) Since the subject was not purely history of computing in isolation but related to modern
one, only glimpse of it was explored to the students due to tight semester course and time
constraint. This created interest among the students to the extent that in the other
lecturers of the same subject their sincerity in the class and eagerness to know the
concept in depth from all points of view of understanding increased, even their
attendance in class were better. They anticipated that in the lecturers to come they would
be exposed with such surprising extra notes. And in their point of view I as a teacher was
more recognized among the class students.
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9. Suggestions
Every teaching has different and unique strategy. The origin of concepts, their necessasity and
development in various cultures, various proof of a particular aspect, their applications in varied
areas and efforts made by various mathematicians are the major components to be included. The
effort of introducing some glimpse of old theory and applications, wherever possible, must
continue in the class in order to maintain their interest towards mathematics course. This in turn
will strengthen their base.
Acknowledgement
Author feels gratefulness to Dr. S.L. Singh, retired Professor of Gurukula Kangri University
(Hardwar) for some fruitful suggestions.
REFERENCES
1. Bag, A.K., Mathematics in Ancient and Medieval India. Chaukhamba Orientalia, Varanasi,
1979.
2. Basham, A.L., The Wonder That Was India, Sendwick and Jackson Limited, London, 1985.
3. Boyer, C.B., A History of Mathematics (revised by Uta. C. Merzbach), John Wiley & Sons,
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4. Gupta, R.C., Reports on History of Mathematics in Mathematical Teaching, Ga𝑛𝑖̇ta
Bh𝑎̅rat𝑖̅
1(1979), 36-38.
5. Gupta, R.C., The Chronic Problem of Ancient Indian Chronology, Ga𝑛𝑖̇ta Bh𝑎̅rat𝑖̅12(1990),
17-26.
6. Gupta, R.C., Sino-Indian Interaction and the Great Chinese Buddhist AstronomerMathematician I-Hsing, Ga𝑛𝑖̇ta Bh𝑎̅rat𝑖̅ 11 (1989), 38-49.
7. Hayashi, Takao, Indian Mathematics, in: Companion Encyclopaedia of the History and
Philosophy of Mathematical Sciences (Ed. I. Grattan-Guinness), Routledge, London, 1984.
8. Kapur, J.N., Fascinating World of Mathematical Sciences, Volume VII: Biography and
History of Mathematics, Mathematical Sciences Trust Society, New Delhi, 1994.
9. Kapur, J.N., Some Random Thoughts on Mathematics Education and Research, University
News, Feb. 24, 1977, 15-18.
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10. Maiti, N.L., Antiquity of Trairaś ika in India, Ga𝑛𝑖̇ta Bh𝑎̅rat𝑖̅ 18 (1996), 1-8.
11.Needham, Joseph, Science and Civilization in China, Volume III, University Press,
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Additional Resources
Mishra, Vinod and S.L. Singh Theorem of Square on the Diagonal and its Application, Indian
Journal of History of Science 31(1996) 157-166.
Mishra, Vinod and S.L. Singh First Degree Indeterminate Analysis in Ancient India and its
Application by Virasena, Indian Journal of History of Science 32(1997), 127-133.
Mishra, Vinod and S.L. Singh Values of  from antiquity to Ramanujan, Sugakushi Kenkyu, No.
157, 1998, 12-25.
Mishra, Vinod and S.L. Singh, Incircumscribing Triangles & Cyclic Quadrilaterals in Ancient &
Medieval Indian Geometry, Sugakushi Kenkyu, No. 161, 1999,1-11
Mishra, Vinod, Geometry Relating to Circles
Medicine and Science 18 (2002), 47-106, No. 1.
and
Spheres, Studies
in
History of
Mishra, Vinod, Combinatorics and the (So-called) Binomial Theorem, Studies
History of Medicine and Science 18 (2002), 59-121, No.2.
in
Mishra, Vinod, Similarity of Plane Figures: A Reflection in Indian Geometry, Studies
History of Medicine and Science 19 (2003), 95-114, No. 1-2.
in
# This argument is rejected by many Indian and foreign scholars through archaeological findings
and other ways, and concludes that Aryans were actually Indians. The report recently appears in
the National News Paper The Hindustan Times, New Delhi Edition, Sept.2, 1998, front cover,
under the title: ‘Aryans Were Not Invaders from Central Asia: Archaeologists Debunk Earlier
Theories’.
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Vinod Mishra
Department of Mathematics
Sant Longowal Institute of Engineering & Technology
Longowal 148106 Punjab. INDIA
[email protected]
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