Two powerful tools contributed to the creation of modern mathematics in the seventeenth century: the discovery of the general algorithms of calculus and the development and application of infinite series techniques. These two streams of discovery reinforced each other in their simultaneous development because each served to extend the range of application of the other.
The origin of the analysis and derivations of certain infinite series, notably those relating to the arctangent, sine and cosine, was not in Europe, but an area in South India that now falls within the state of Kerala. From a region covering about five hundred square kilometres north of Cochin and during the period between the 14th and 16th centuries, there emerged discoveries in infinite series that predate similar work of Gregory, Newton and Leibniz by three hundred years.
There are several questions worth exploring about the activities of this group of mathematicians/astronomers (hereafter called the Kerala School), apart from technical ones relating to the mathematical content of their work. In this paper we confine our attention to their background and the motivation underlying their interest in a particular series, the arctan series (and its special case, pi), and consider that work in a cross-cultural context by comparing it to similar work that emerged in Europe during the seventeenth century and in China during the eighteenth century.(1)
The modern derivation of the arctan power series (and the series for pi) is a simple matter using calculus. The two series may be expressed as:
(tan q)**3 (tan q)**5
q = tan q - + - ..... if |tanq| =< 1 (A)
For tan q = 1 or q = 45 degree = pi/4 radians, the above series becomes
pi/4 = 1 - 1/3 + 1/5 - ..... (B)
Series (A) was first investigated in Europe by the Scottish mathematician, James Gregory, in 1671. Two years later, the German philosopher and mathematician, Gottfried Wilhelm Leibniz, discovered Series (B) using a different method from that of Gregory. But nearly three centuries earlier, both series were known and used in astronomical work in Kerala. And in a text published in 1774 after his death, the Chinese mathematician, Ming Antu, derived nine formulae of infinite series, including 'three formulae of Master Jartoux' transmitted to China without any proofs by the French Jesuit, Pierre Jartoux, at the beginning of the eighteenth century. One of the three that Ming Antu proved, using the Chinese traditional procedures, was a variant of Series (B).
A comparison between the three discoveries of the arctan and pi power series is instructive. The discoveries occurred at a time when methods of calculus were in their infancy and when infinite series had not been fully incorporated into the main corpus of mathematical procedures. In Europe, it was the analysis of infinite series and the calculus that reinforced each other to lay the foundation of 'modern' mathematics.
A more important reason from the point of view of this paper is that, as a specific illustration of 'different ways of knowing' the same result, it is useful to examine the motivation that underpinned the apparently independent derivations of the infinite series for the arctan (and pi) in the three traditions. Before we do so, consider briefly the background of the Indian work.
There is a general belief among historians of mathematics that mathematics in India made little progress after Bhaskaracharaya in the twelfth century AD, that later scholars seemed content to 'chew the cud' as it were, writing endless commentaries on the works of those who preceded them, until the introduction of modern mathematics by the British. However, in Kerala, the period between the fourteenth and seventeenth century marked a high point in the indigenous development of mathematics and astronomy.
The story of the discovery of Kerala mathematics sheds some fascinating light on the character of the historical scholarship of the period. In 1832, Charles Whish (2) read a paper to a joint meeting of the Madras Literary Society and the Royal Asiatic Society in which he referred to five works of the period, 1450-1850: The Tantrasamgraha ('A Digest of Scientific Knowledge') of Nilakantha (1444-1545), the Yuktibhasa ['An Exposition of the Rationale'] of Jyesthadeva (fl. 1500-1610), Kriyakramakari ('Operational Techniques') of Sankara Variyar (c. 1500-1560) and Narayana (c. 1500-1575), Karanapaddati ('A Manual of Performances in the Right Sequence') of Putumana Somayajin (c. 1660-1740) and Sadratnamala ('A Garland of Bright Gems') of Sankara Varman (1800-1838).
An important feature of the last four texts is their claim to have derived their main ideas from Madhava (c. 1340-1425) and Nilakantha who are referred to as acharyas (or teachers). It is possible that Madhava wrote a comprehensive treatise on astronomy and mathematics, including sections on infinite series. And it is probably to the contents of this text that others who came after him refer to in such glowing terms. Such a work remains to be discovered.
These authors form part of a tradition of continuing scholarship in Kerala over a period four hundred years from the birth of Madhava in 1340 to the probable death of Putumana Somayajin in 1740. In the present state of knowledge of source materials it is difficult to assign many of the developments to any particular person. The results should be seen as produced by members of a school as it were, spread over several generations.(3)
As far as we know, the members of the Kerala School were predominantly Nambuthri Brahmins with a few who came from sub-castes, such as the Variyars and the Pisharotis, traditionally associated with specific duties in the temple. Within a mainly two-tier caste system, consisting of Brahmins and Nayars, the latter probably being given the status of Sudras (or the fourth in the conventional caste hierarchy), two institutions operated to strengthen and sustain the economic and social dominance of the Nambuthris to a degree not known elsewhere in India: the janmi system of land-holding and the Nambuthri control of vast tracts of land owned by temples, often situated near Brahmin villages.(4)
There were other factors that helped to strengthen the economic and social powers of the Nambuthris. The Nayars practised the marumakkattayam (matrilineal) system of descent without the formal institution of marriage. Sexual alliances between Nayar women and Nambuthiri men were permitted, indeed sometimes encouraged, with children of such unions remaining the sole responsibility of their mother's family. At the same time, the Nambuthris operated a system of patrilineal descent (makkatayam), with a form of primogeniture that allowed only the eldest son to inherit land and property and to marry Nambuthri women. The eldest son was also required to provide for the material needs of his siblings consisting of younger brothers and and umarried sisters.
Madhava and all those who knew and followed him lived and worked in large compounds called illams in villages with predominantly Nambuthri settlements. Set well away from roads to prevent contact with others, often surrounded by a high wall, each illam had its own well for water, a tank for bathing and a number of outbuildings. Many of these illams belonged to households that owned large landed properties and were very affluent. With their estates farmed by workers or tenants from lower castes and often under the management of Nayars, the Nambuthris, and particularly the younger sons, enjoyed considerable leisure and were expected to pass their time in study and ritual observances.
These illams provided a base for the education of the young in Sanskrit works, including mathematical and astronomical classics (notably from the Aryabhitiya of Aryabhata (b. AD 476) and its commentaries).(5) Not only was traditional knowledge transmitted in these illams by rote, but they also provided a centre for research and scholarship. Sometimes, the scholars wrote commentaries on the classics and in those commentaries they appended their own discoveries as additions and supplements.(6) The short distances between the illams, the role of the temple (7) and political stability combined to provide for long and stable development, usually based on generations of teacher-student relationships. A study of their interaction with certain temple personnel (especially, the ambalavasis such as Sankara Variyar and Achuta Pisharoti) may shed light both on how non-Brahmin Hindus were recruited into their circle as well the process by which a wider dissemination of the results of their work in mathematics and astronomy took place into the neigbouring areas, notably today's Tamilnadu.
The primary mathematical motivation for the Kerala work on infinite series arose from a recognition of the impossibility of arriving at an exact value for the circumference of a circle given the diameter (i.e., the incommensurability of pi). Nilakantha explained in his Aryabhatiyabhasya why only an approximate value of the circumference is obtained for a given diameter in the following words:
Why is only the approximate value (of circumference) given here? Let me explain. Because the real value cannot be obtained. If the diameter can be measured without a remainder, the circumference measured by the same unit (of measurement) will leave a remainder. Similarly, the unit which measures the circumference without a remainder will leave a remainder when used for measuring the diameter. Therefore, the two measured by the same unit will never be without a remainder. Though we try very hard, we can reduce the remainder to a small quantity but never achieve the state of 'remainderlessness'. This is the problem.
This explanation was prompted by a passage in Aryabhata's Aryabhatiya. Verse 10 of the section on Ganita reads:
Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle whose diameter is 20,000.
It was the word 'approximately' that gave food for thought. And the strategy to be followed was expressed thus in Kriyakramakari of Sankara and Narayana:
Thus even by computing the results progressively, it is impossible theoretically to come to a final value. So, one has to stop computation at that stage of accuracy that one wants and take the final result arrived at ignoring the previous results.
The approach indicated was based on trying to find the length of an arc by approximating it to a straight line. Known today as the method of direct rectification of an arc of a circle, it involves summation of very small arc segments and reducing the resulting sum to an integral. Even a cursory comparison of the modern derivation with the Kerala derivation show certain similarities, with the major difference being that algorithms of calculus are deployed in the modern derivation.(8)
The Kerala derivation uses an interesting geometric technique. The tangent is divided up into equal segments while forcing a subdivision of the arc into unequal parts. This is required since the method involves the summation of a large number of very small arc segments, traditionally achieved in European and Arab mathematics by the 'method of exhaustion.'(9) where there was a subdivision of an arc into equal parts. The adoption of this 'infinite series' technique rather than the 'method of exhaustion' for implicitly calculating pi was not through ignorance of the latter in Kerala mathematics. But, as Jysthadeva points out in the Yuktibhasa, the former avoids tedious and time-consuming root-extractions.
This may provide a valuable clue as to a foundational difference in approach between European mathematics (based on the Greek tradition) and Kerala mathematics (the heir of an earlier Indian tradition). In the Indian case, numbers were merely entities whose value depended on their efficacy for mathematical operation. In the European case (following directly from the Greeks) it was from measurability that countability and other operations stemmed.(10)
There are other aspects of the 'tool kit' used by the Kerala mathematicians that need highlighting. The derivation of the arctan series employed two results in elementary mathematics that have a long history in India: (i) the Pythagorean theorem which dates back to the earliest written evidence on Indian mathematics, the Sulbasutras (c. 800 BC)(11); and (ii) the properties of similar triangles which is little more than a geometrical version of the 'rule of three' (trairasika) of which probably the first systematic treatment is found in the Bakhshali Manuscript (variously dated from AD 200 - 700), although Verse 26 of the ganita section of Aryabhatiya is the more likely source for Kerala mathematics.
The Kerala derivation deployed an ingenious iterative re-substitution procedure to obtain the binomial expansion for the expression 1/(1 + x) and then proceeding through a number of repeated summations (varamsamkalithas) of series, arrived at what must be the most remarkable part of the derivation, an intuitive leap leading to the asymptotic formula:(12)
lim (1/n**p) * Sum r**(p-1) = 1/p
It was soon realised that the series:
pi/4 = 1 - 1/3 + 1/5 - .....
was not particularly useful for making accurate estimates of the circumference given the diameter (i.e., estimating pi ) because of the slowness of the convergence of the series. This gave impetus to developments in two directions: (a) rational approximations by applying corrections to partial sums of the series; (b) obtaining more rapidly converging series by transforming the original series. There was considerable work in both directions which are examined in detail in Yuktibhasa and Kriyakramakari. What the work exhibits is a measure of understanding of the idea of convergence, of the notion of rapidity of convergence and an awareness that convergence can be speeded up by transformations.(13)
As an illustration of the remarkable efficiency of some corrections suggested, consider the following example from Yuktibhasa. What is required is to evaluate the circumference of a circle with a diameter of 10**11. Without the correction and using Series (B) with the number of terms on the right-hand side as nineteen, the circumference is about 3.194 x 10**11. However, incorporating one of the corrections(14) gives the circumference as 3.1415926529 x 10**11; that is correct to 8 places. And this interest in increasing the accuracy of the estimate continued for a long time, so that as late as the nineteenth century, the author of Sadratnamala estimated the circumference with diameter 10**18 as: 314,159,265,358,979,324 correct to 17 places!
It is clear that in deriving the arctan series, the Kerala School showed both an awareness of the principle of integration and an intuitive perception of small quantities and operations with such quantities. However, having come so near to the formulation of the crucial concept of the 'limit' of a function, they shied away from developing the methods and algorithms of calculus, being content with a geometrical approach which the Europeans soon replaced with calculus.(15)
In Europe, the details of the circumstances and ideas leading to the discovery of the arctan series by Gregory and Leibniz are well known.(16) It was an important event because it was a precursor of calculus. In an attempt to discover an infinite series representation of any given trigonometric function and the relationship between the function and its successive derivatives, Gregory stumbled on the arctan series. He took, in terms of modern notation,
d q = d (tan q) / (1 + (tan q)**2)
and carried out term by term integration to obtain his result. Leibniz's discovery arose from his application of fresh thinking to an old problem, namely quadrature or the process of determining a square that has an area equal to the area enclosed by a circle. In applying a transformation formula (similar to the present-day rule for integration by parts) to the quadrature of the circle, he discovered the series for p. It must be pointed out, however, that the ideas of calculus such as integration by parts, change of variables and higher derivatives were not completely understood then. They were often dressed up in geometric language with, for example, Leibniz talking about 'characteristic triangles' and 'transmutation'.
The Chinese work is interesting for a different reason. Infinite series, as a mathematical object, was introduced into China divorced from its European context, i.e., calculus. The background to this introduction is interesting. The introduction of European mathematics into China began in the closing decades of the sixteenth century, when the Chinese first came in contact with the Jesuits. In their intention to spread their religion in China, the Jesuits arrived from Europe bringing with them both new technological gadgets and also scientific theories which, though not updated with more recent discoveries in Europe, proved a sufficient novelty and attraction for the educated classes. In 1601, the Italian Jesuit, Matteo Ricci (1552-1601) began his translation of the first six books of Euclid's Elements into Chinese in 1607. Later, in the last few decades of the Ming dynasty, many astronomical books were translated into Chinese. But most of the scientific books translated were pre-Newtonian publications. In early Qing dynasty, after listening to a debate between a Jesuit astronomer, Adam Schall, and a Chinese astronomer, Yang Guangxian, the Kangxi Emperor, became interested in knowing more about Western science. In answer to an invitation to send more mathematicians and astronomers, Louis XIV of France sent a group led by J. de Fontaney, 'the King's mathematician', and asked them to make astronomical observations, study the flora and fauna, and learn the technical arts of China. In 1690, the French Jesuits began teaching mathematics to the Emperor and his courtiers. Pierre Jartoux, a French Jesuit, arrived in China in 1701 and taught at the court. He introduced three results new to Chinese mathematics: the power series for sine, versed sine and for pi which was derived from arc sine function. For none of these results did Jartoux provide a proof; the calculus needed was not known in China until the middle of the nineteenth century.
Ming Antu (d. 1765) was an astronomer who had worked with the Jesuits in cartography and later reforming the astronomical system. At his death he was the director of the Imperial Board of Astronomy. In his book, Ge Yuan Mi Lu Jie Fa (Quick Methods of Trigonometry and for Determining the Precise Ratio of the Circle) contains the statement and proof of nine formulae, including the 'three formulae of Master Jartoux'. It is possible that Ming Antu was introduced to the three formulae by Jartoux himself. His proofs are based on the generalisation of a method occurring both in Chinese and European tradition: the method of the division of the circle. In China, it is found in Liu Hui's commentary on the premier text, Chiu Chang Suan Shu, from the beginning of the Christian Era. The idea of the method is to approximate the circle by inscribing polygons, the number of sides which is doubled at each step. This method was extended by using continued proportions (lu) as an algebraic language, so that it applies to the measurement of any arc.(17)
The major breakthrough in Kerala mathematics was the appearance of mathematical analysis in the form of infinite series and their finite approximations relating to circular and trigonometric functions. The primary motivation for this work was a mixture of intellectual curiosity and a requirement for greater accuracy in astronomical computations. Demonstrations of these results are not completely rigorous by today's standards, but they are nonetheless correct. And these demonstrations may well be chosen for a modern mathematics class room because the approach is more intuitive and therefore more convincing.
A historical and comparative study of infinite service provides a suitable vehicle for testing certain perceptions about different mathematical traditions. A widely accepted view among historians of mathematics is that mathematics outside the sphere of Greek influence, such as Indian or Chinese mathematical traditions, was algebraic in inclination and empirical in practice that provided a marked contrast to Greek mathematics that was geometric and anti-empirical. Again, many commonly available books on history of mathematics declare or imply that Indian mathematics, whatever are its other achievements, did not have any notion of proof. What a comparative study would indicate are the dangers of such categorisations and generalisations. And in a deeper sense it would bring home the point that between different mathematical traditions there are certain basic differences in the cognitive tructures of mathematicsdifferences in their ontological conceptions regarding the existence and nature of mathematical objects and methodological conceptions regarding the nature and ways of establishing mathematical truths.
1. This paper concentrates on the Kerala work, with only brief comparisons made with the other two traditions.
2. For further details on how Whish's excavation of Kerala mathematics was viewed in the West, see G. G. Joseph, 'Cognitive encounters in India during the age of imperialism', Race and Class, 36, 3 (1995): pp. 39-56
3. Madhava began a school that had the following direct teacher-student lineage lasting about three hundred years:
Madhava (ca. 1340-1425) =>Parameswara (c. 1380-1460) => Damodara (b.1410) => Nilakantha (1444-1543) => Chitrabhanu (1474-1550) => Narayana (c. 1525-1610) and Sankara Variyar (c. 1500-1560)
Damodara (b. 1410) => Jyesthadeva (ca. 1500-1575) => Achuta Pisharoti (ca. 1550-1621)
4. During a period of protracted war around AD 1000, the Nambuthri Brahmins became trustees of substantial properties owned by temples. The temple wealth was further augmented by transfers of land and other endowments belonging to individuals who did so as an insurance against devastation or to obtain exemption from taxation. It was under such circumstances that the Jamni system originated in Kerala. The Janmi, who was in almost all cases a Nambuthri landlord, exercised not only absolute proprietorships of land and the resulting powers to evict tenants from his land at will but even the power of life and death over them. A system of agrarian serfdom came into existence where the sale of any land meant also transfer of its tenants and workers to the new land owner.
5. The original sources of the Madhava school of mathematics and astronomy were primarily the works of Aryabhata (b. AD 476) and Bhaskara I (c. AD 600) but later these were supplemented by those of Sridhara (c. AD 900), Sripati (c. AD 1040) and most importantly of Bhaskaracharya (b. AD 1114). These works were scrutinised and some ideas contained integrated into a vigorous tradition that already existed of a South Indian astronomy. For further details, see G. G. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics, London: Penguin, 1992 (3rd Reprint, 1996).
6. There has been a long Indian tradition of providing yuktis (rationales) and upapattis (proofs) in commentaries rather than in the main texts which were often couched in cryptic verses that could be easily memorised.
7. The role of the temple as an institution creating and sustaining intellectual activities needs further emphasis. As a meeting place of those involved in the study of mathematics and astronomy, as a vehicle for receiving and disseminating scientific knowledge, as an agency for recruiting able students and practitioners outside the confines of narrow caste and regional lines, in all these cases the temple played an important part. For further discussion, see Rajan Gurukkal, The Kerala Temple and the Early Medieval Agrarian System, Sukapuram: Vallathol Vidyapeetham, 1992.
8. For such a comparison, see G. G. Joseph, 'Infinite Series across Cultures - A Study of Mathematical Motivation', School of Economic Studies Discussion Paper Series, No. 9615, May 1996.
9. The method, first suggested by Eudoxus of Cnidus (fl. 375 BC) and then extensively used by Archimedes, is based on the simple observation that if a circle is enclosed between two polygons of n sides, then, as n increases, the gap between the circumference of the circle and the perimeters of the inscribed and circumscribed polygons diminishes so that eventually the perimeters of the polygons and the circle would become identical. Or in other words, as n increases, the difference in the area between the polygons and the circle would be gradually exhausted.
10. For further discussion of this point, see G. G. Joseph, ' Different Ways of Knowing: Contrasting Styles of Argument in Indian and Greek Mathematical Traditions' in Mathematics, Education and Philosophy: An International Perspective, ed. P. Ernest, London: The Falmer Press, 1994, pp. 194-204.
11. For further details, see G. G. Joseph, 'Geometry of Vedic Altars' in NEXUS Architecture and Mathematics, ed. K. Williams, Fucecchio: Edizioni Dell'Erbi, 1996, pp. 97-114 and Joseph, 1992, ibid.,pp. 224-236
12. This asymptotic relation made its first appearance in Europe in the works of Roberval (1634) and Fermat (1636). For details of the Kerala derivation, see Joseph (1996), ibid., p. 4-7.
13. For further details, see Joseph, 1996, ibid., p. 21-26.
14. The correction used is to incorporate the following as the last term:
Fc(n) = (n2 + 1)/(4n3 + 5n) where n is the number of terms
15. It is interesting in this context to note about five hundred years after Madhava, Yesudas Ramchandra wrote a book in 1850, entitled A Treatise on the Problems of Maxima and Minima in which he claimed that he had developed a new method, consistent with the Indian tradition of mathematics, to solve all problems of maxima and minima by algebra and not calculus. This book was republished in England with the help of the British mathematician, Augustus De Morgan. For further details, see Joseph, 1995, ibid.
16. For details, see C.A H. Edwards, The Historical Development of the Calculus, New York: Springer-Verlag, 1979.
17. For details, see C. Jami, 'Western Influence and Chinese Tradition in an Eighteenth Century Chinese Mathematical Work', Historia Mathematica, 15 (1988): pp. 311-331.