MATHEMATICAL HERITAGE OF INDIA: SOME REMEDIAL ISSUES: MATHEMATICS HISTORY IN TEACHING Vinod Mishra
Sant Longowal Institute of Engineering & Technology, India
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.
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 Gata Bhrat.
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 siddhntic age cannot be ruled out, in fact on small scale. The concept of 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. Gata Bhrat11 (1989), 38-39):
The rock edicts of king Aoka (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 Brhminical learning. The monks eagerly studied, besides Buddhist works (including Abhidharmakoa), 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:
The theorem “the diagonal rope of an oblong produces both (areas) which its side and length produce separately” (cf. ) was the contribution of Baudhyana of India who lived in about 800 B.C. It was dedicated to Pythagoras (ca. 540 A.D.) of Greek which had been little known to him.
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’ (trairika) dates back to Vedagajyotia (ca. 600 B.C.).
The equivalent of the so-called Leibniz’s series (Leibniz, 1646-1716 A.D.)
had been obtained by Mdhava (fl. 1360-1425 A.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 Paramevara (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.”
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. Gata Bhrat18(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.
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 ryabhatya nor the Brahmasphutasiddhnta 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 Bhskara I (around A.D. 600), who, in this commentary on the ryabhatya, rejected the Jain value of ,, 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 Sihatilakasri’s commentary (in the thirteenth century) on Srpati's Gaitatilaka, 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 Bhratas, 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 “stra period, an age of specialization”, Bag [p. 4] writes “------- that the study of mathematics started with the stra 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 ryabhatya 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 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 (1789-1860). 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 Upaniads from a 17th –century Persian version- the translation of the whole manuscript, containing 50 Upaniads, 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].
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:
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.
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.
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.
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 bulk of Indian manuscripts which are either untouched or unearthed. A proper survey of such works in India and abroad is imminent task.
Archival records in history of science/mathematics need to be maintained. Recently a central unit for cataloguing science archives has been constituted at Lucknow [Gata Bhrat].
Exhibitions on paintings and sculptures certainly give many impressions regarding life-style of mathematicians and reflects ethnological culture of that time.
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.
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).
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.
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.
There are advantages in presenting ancient problems in a modern context. In turn, this makes the subject more understandable. Modern generalizations should be avoided.
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.
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.
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 verses / proses from ancient texts / manuscripts few words may be added in brackets at appropriate places so as to make the matter tangible.
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.
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.
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 of first order of the type Kuttaka method. Linear simultaneous indeterminate equations.
Second degree indeterminate equations: Cakraval method. Evaluation of .
Triangles and Quadrilaterals
Theorem of square on the diagonal. Geometrical proof. Some geometrical constructions. Application of right triangle in Mahvedi structure.
Incircumscribing triangles and Cyclic quadrilaterals
Incircumscribing triangle. Area. Circum-radius.
Cyclic quadrilateral. Area. Third diagonal concept.
Similarity of Plane Figures
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
, . Sines and cosines of multiple and submultiple arcs.
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.
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”.
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, , 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 in the Indian as well as other culture areas and their necessasity through algebraic and geometrical means. And ultimately relating the evaluation of to that of N-R method as a particular case of it. Even they were taught some extended modern generalization of . The class started with the basic introduction to algebraic and transcendental equations followed by computational efforts for evaluating and 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 be a non square integer such that where is the largest positive integer and the smallest integer. Then
Yet another concept from Naryana’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
Are known then any number of other solutions can be found.
For example if and then is also a solution of (2). If we take = in this Lemma, then is also a solution. For finding an approximate value of square root, Naryana in his Bijagaita, gives the rule which is put as:
If is a solution of (2), then (), the first approximation. From (2), . Naryana anticipates that if y (so also x) is large, is a close approximation of
The second approximation would be .
The above two results are exactly derived from Newton-Raphson iterative method. Let Then and So the first and second approximations of are respectively and 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:
Students were surprised to know the methods which were hardly known to them through modern available resources.
They have generated a sense of interest in subject and shown eagerness to know more of concept beginning from origin to varied applications.
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.
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.
Author feels gratefulness to Dr. S.L. Singh, retired Professor of Gurukula Kangri University (Hardwar) for some fruitful suggestions.
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11.Needham, Joseph, Science and Civilization in China, Volume III, University Press, Cambridge, 1959.
12. Rao, S.B., Indian Mathematics and Astronomy (Some Landmarks), Jnana Deep Publications Bangalore, 1994.
13. Seal, B.N., The Positive Sciences of Ancient Hindus, Motilal Banarasidass, Delhi, 1991.
14. Young, Lam Lay, A Critical Study of the Yang Hui Swan Fa, Singapore University Press, 1977.
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 p 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 and Spheres, Studies in History of Medicine and Science 18 (2002), 47-106, No. 1.
Mishra, Vinod, Combinatorics and the (So-called) Binomial Theorem, Studies in History of Medicine and Science 18 (2002), 59-121, No.2.
Mishra, Vinod, Similarity of Plane Figures: A Reflection in Indian Geometry, Studies inHistory ofMedicine and Science 19 (2003), 95-114, No. 1-2.
# 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’.
Department of Mathematics
Sant Longowal Institute of Engineering & Technology