Mathematics in India

Mathematics in India: A Survey* As early Indian astronomers tried to quantify the paths of the sun, the moon, the planets and the stars on the celestial sphere with ever more accuracy, or to predict the occurrence of eclipses, they were naturally led to develop mathematical tools. Astronomy and mathematics were thus initially regarded as inseparable, the latter being the maid-servant of the former. Indeed, about 1400 BCE, the Vedāṅga Jyotiṣa, the first extant Indian text of astronomy, states in two different versions: Like the crest on the head of a peacock, like the gem on the hood of a cobra, jyotiṣa (astronomy) / gaṇita (mathematics) is the crown of the Vedāṅga śāstras [texts on various branches of knowledge]. In fact, jyotiṣa initially referred to astronomy and mathematics combined; only later did it come to mean astronomy alone (and much later did it include astrology). First Steps India’s first urban development, the Indus or Harappan civilization (2600-1900 BCE), involved a high degree of town planning. A mere glance at the plan of Mohenjo-daro’s acropolis (or upper city), Dholavira (in the Rann of Kachchh) or Kalibangan (Rajasthan), reveals fortifications and streets generally aligned to the cardinal directions and exhibiting right How much knowledge of geometry would you need to plan a city? angles. Specific proportions in the dimensions of major structures have also been pointed out. All this implies a sound knowledge of basic * The author gratefully acknowledges valuable suggestions for improvement received from Dr. M.D. Srinivas. 247 geometric principles and an ability to measure angles, which the discovery of a few cylindrical compasses made of shell, with slits cut every 45°, has confirmed. Besides, for trading purposes the Harappans developed a standardized system of weights in which, initially, each weight was double the preceding one, then, 10, 100 or 1,000 times the value of a smaller weight. This shows that the Harappans could not only multiply a quantity by such factors, but also had an inclination for a decimal system of multiples. However, there is no agreement among scholars regarding the numeral system used by Harappans. A few Harappan weights made of chert, from Dholavira, Gujarat (Courtesy: ASI) There is no scholarly consensus on the dates of the four Vedas, India’s most ancient texts, except that they are over 3,000 years old at the very least. We find in them frequent mentions of numbers by name, in particular multiples of tens, hundreds and thousands, all the way to a million millions in the Yajur Veda — a number called parārdha. (By comparison, much later, the Greeks named numbers only up to 10,000, which was a ‘myriad’; and only in the 13th century CE would the concept of a ‘million’ be adopted in 248 Europe.) The Brāhmanas, commentaries on the Vedas, knew the four arithmetical operations as well as basic fractions. Early Historical Period The first Indian texts dealing explicitly with mathematics are the Śulbasūtras, dated between the 8th and 6th centuries BCE. They were written in Sanskrit in the highly concise sūtra style and were, in effect, manuals for the construction of fire altars (called citis or vedis) intended for specific rituals and made of bricks. The altars often had five layers of 200 bricks each, the lowest layer symbolizing the earth, and the highest, heaven; they were thus symbolic representations of the universe. The first layer of one kind of śyenaciti or falcon altar described in the Śulbasūtras, made of 200 bricks of six shapes or sizes, all of them adding up to a specified total area. Because their total area needed to be carefully defined and constructed from bricks of specified shapes and size, complex geometrical calculations followed. The Śulbasūtras, for instance, are the earliest texts of geometry offering a general statement, in geometric 249 form, of the so-called Pythagoras theorem (which was in fact formulated by Euclid around 300 BCE). What is meant by ‘transcendental’ and why should this nature of π preclude exact geometrical solutions to the squaring of a circle? The geometrical expression of the Pythagoras theorem found in the Śulbasūtras. They spelt out elaborate geometric methods to construct a square resulting from the addition or subtraction of two other squares, or having the same area as a given circle, and vice-versa — the classic problems of the squaring of a circle or the circling of a square (which, because of π’s transcendental nature, cannot have exact geometrical solutions, only approximate ones). All these procedures were purely geometrical, but led to interesting corollaries; for instance, √2 was given a rational approximation which is correct to the fifth decimal! 250 The Śulbasūtras also introduced a system of linear units, most of them based on dimensions of the human body; they were later slightly modified and became the traditional units used across India. The chief units were: ¾ 14 aṇus (grain of common millet) = 1 aṅgula (a digit) ¾ 12 aṅgulas = 1 prādeśa (the span of a hand, later vitasti) ¾ 15 aṅgulas = 1 pada (or big foot) ¾ 24 aṅgulas = 1 aratni (or cubit, later also hasta) ¾ 30 aṅgulas = 1 prakrama (or step) ¾ 120 aṅgulas = 1 puruṣa (or the height of a man with his arm extended over his head) A few centuries later, Piṅgala’s Chandasūtras, a text on Sanskrit prosody, made use of a binary system to classify the metres of Vedic hymns, whose syllables may be either light (laghu) or heavy (guru); rules of calculation were worked out to relate all possible combinations of light and heavy syllables, expressed in binary notation, to numbers in one-to-one relationships, which of course worked both ways. In the course of those calculations, Piṅgala referred to the symbol for śūnya or zero. About the same time, Jaina texts indulged in cosmological speculations involving colossal numbers, and dealt with geometry, combinations and permutations, fractions, square and cube powers; they were the first in India to come up with the notion of an unknown (yāvat-tāvat), and introduced a value of π equal to √10, which remained popular in India for quite a few centuries. 251 Notice how, in columns 2 to 4, multiples of hundreds are represented through a single sign. What does this imply? Numerals as they appeared in early inscriptions, from the 3rd century BCE to the 1st century CE. Note that they do not yet follow a decimal positional system; for instance, in the first column, 40 is written as ‘20, 20’, 60 as ‘20, 20, 20’. (Adapted from INSA) With the appearance of the Brāhmī script a few centuries BCE, we come across India’s first numerals, on Ashoka’s edicts in particular, but as yet without any decimal positional value. These numerals will evolve in shape; eventually borrowed by Arabs scholars, they will be transmitted, with further alterations, to Europe and become our modern ‘Arabic’ numerals. 252 Evolution of Indian numerals, as evidenced by inscriptions. The first script, Brāhmī, was used by Aśoka in his Edicts; the last is an antecedent of the Devanagari script. (Adapted from J.J. O’Connor & E.F. Robertson) The Classical Period Together with astronomy, Indian mathematics saw its golden age during India’s classical period, beginning more or less with the Gupta age, i.e. from about 400 CE. (See module Astronomy in India for a map of Indian astronomers and mathematicians.) Shortly before that period, the full-fledged place-value system of numeral notation — our ‘modern’ way of noting numbers, unlike non-positional systems such as those depicted above or Roman numbers — had been worked out, integrating zero with the nine numerals. It is a pity that we shall never know who conceived of it. Amongst the earliest known references to it is a first-century CE work by the Buddhist philosopher Vasumitra, and it is worked out more explicitly in the Jain cosmological work Lokavibhāga, written in 458 CE. Soon it was adopted across India, and later taken to 253 Europe by the Arabs. This was a major landmark in the world history of science, since it permitted rapid developments in mathematics. One of the first attested inscriptions (from Sankheda, Gujarat) recording a date written with the place-value system of numeral notation. The date (highlighted) reads 346 of a local era, which corresponds to 594 CE. (Adapted from Georges Ifrah) About 499 CE, living near what is today Patna, Āryabhaṭa I (born 476 CE) authored the Āryabhaṭīya, the first extant siddhānta (or treatise) attempting a systematic review of the knowledge of mathematics and astronomy prevailing in his days. The text is so concise (just 121 verses) as to be often obscure, but between the 6th and the 16th century, no fewer than twelve major commentaries were authored to explicate and build upon its contents. It was eventually translated into Arabic about 800 CE (under the title Zīj alĀrjabhar), which in turn led to a Latin translation in the 13th century (in which Āryabhaṭa was called ‘Ardubarius’). 254 Why should the study of the half­chord of an arc of circle be an advance over that of the full chord? Āryabhaṭa introduced the notion of a half-chord, a substantial advance over Greek trigonometry, which considered the full chord of an arc of circle. The mathematical content of Āryabhaṭīya ranges from a very precise table of sines and an equally precise value for π (3.1416, stated to be ‘approximate’) to the area of a triangle, the sums of finite arithmetic progressions, algorithms for the extraction of square and cube roots, and an elaborate algorithm called kuṭṭaka (‘pulverizing’) to solve indeterminate equations of the first degree with two unknowns: ax + c = by. By ‘indeterminate’ is meant that solutions should be integers alone, which rules out direct algebraic methods; such equations came up in astronomical problems, for example to calculate a whole number of revolutions of a planet in a given number of years. It is worth mentioning that despite its great contributions, the Āryabhaṭīya is not free of errors: its formulas for the volumes of a pyramid and a sphere were erroneous, and would be later corrected by Brahmagupta and Bhāskarācārya respectively. 255 The Classical Period, post-Āryabhaṭa Born in 598 CE, considerable Brahmagupta was an imposing figure, with achievements in mathematics. In his Brahmasphuta Siddhānta, he studied cyclic quadrilaterals (i.e., inscribed in a circle) and supplied the formula for their area (a formula rediscovered in 17th-century Europe): if ABCD has sides of lengths a, b, c, and d, and the semiperimeter is s = (a + b +c + d)/2, then the area is given by: Area ABCD = √[(s – a) (s –b) (s – c) (s – a)] Brahmagupta boldly introduced the notion of negative numbers and ventured to Would Brahmagupta’s definition of the mathematical infinite be acceptable to modern mathematics? define the mathematical infinite as khacheda or ‘that which is divided by kha’, kha being one of the many names for zero. He discovered the bhāvanā algorithm for integral solutions to second-order indeterminate equations (called varga prakriti) of the type Nx2 + 1 = y2. He was in many ways one of the founders of modern algebra, and his works were translated into Persian and later Latin. Dated around the 7th century, the Bakhshali manuscript, named after the village (now in northern Pakistan) where it was found in 1881 in the form of 70 leaves of birch bark, gives us a rare insight into extensive mathematical calculation techniques of the times, involving in particular fractions, progressions, measures of time, weight and money. A few leaves from the Bakhshali manuscript (Courtesy: Wikipedia) 256 Other brilliant mathematicians of the siddhāntic era included Bhāskara I, a contemporary of Brahmagupta, who did pioneering work in trigonometry (proposing a remarkably accurate rational approximation for the sine function), Śrīdhara and Mahāvīra. The last, a Jain scholar who lived in the 9th century in the court of a Rashtrakuta king (in today’s Karnataka), authored the first work of mathematics that was not as part of a text on astronomy. In it, Mahāvīra dealt with finite series, expansions of fractions, permutations and combinations (working out, for the first time, some of the standard formulas in the field), linear equations with two unknowns, quadratic equations, and a remarkably close approximation for the circumference of an ellipse, among other important results. Graph showing the high accuracy of Bhāskara I’s rational approximation for the sine function from 0° to 180° (in blue). The sine function (in read) had to be shifted upward by 0.05 to make the two curves distinguishable. (Courtesy: IFIH) Bhāskara II, often known as Bhāskarācārya, lived in the 12th century. His Siddhāntaśiromani (literally, the ‘crest jewel of the siddhāntas’) broke new ground as regards cubic and biquadratic equations. He built upon Brahmagupta’s work on indeterminate equations to produce a still more effective algorithm, the chakravāla (or 257 ‘cyclic method’); with it he showed, for instance, that the smallest integral solutions to 61x2 + 1 = y2 are x = 226153980, y = 1766319049 (interestingly, five centuries later, the French mathematician Fermat offered the same equation as a challenge to some of his contemporaries). Bhāskarācārya also grasped the notion of integration as a limit of finite sums: by slicing a sphere into ever smaller rings, for instance, he was able to calculate its area and volume. He came close to the modern notion of derivative by discussing the notion of instant speed (tātkālika gati) and understood that the derivative of the sine function is proportional to the cosine. The first part of Bhāskarācārya’s Siddhāntaśiromani is a collection of mathematical problems called Līlāvatī, named after an unknown lady to whom Bhāskara puts problems in an often poetical language. Līlāvatī became so popular with students of mathematics across India that four centuries later, Akbar had it translated into Persian by a court poet. The Kerala School of Mathematics Along with astronomy, mathematics underwent a revival in the Kerala School, which flourished there from the 14th to the 17th century. Its pioneer, Mādhava (c. 1340–1425), laid some of the foundations of calculus by working out power series expansions for the sine and cosine functions (the so-called Newton series), and by spelling out this fundamental expansion of π: This is known as the Gregory–Leibniz series, but ought one day to be named after Mādhava. He went on to propose a more rapidly convergent series for π: 258 which enabled him to calculate π to 11 correct decimals. Nīlakaṇṭha Somayāji (c. 1444–1545) and Jyeṣṭhadeva (c. 1500–1600) built on such results and considerably enriched what might be called the Indian foundations of calculus. The latter, for instance, worked out the binomial expansion: Features of Indian mathematics As elsewhere, mathematics in India arose from practical needs: constructing fire altars according to precise specifications, tracking the motion of planets, predicting eclipses, etc. But India’s approach remained essentially pragmatic: rather than developing an axiomatic method such as that of the Greek (famously introduced by Euclid for geometry), it focused on obtaining formulas and algorithms that yielded precise and reliable results. Nevertheless, Indian mathematicians did often provide logically rigorous justifications for their results, especially in the longer texts. Indeed, Bhāskarācārya states that presenting proofs (upapattis) is part of the teaching tradition, and Jyeṣṭhadeva devotes considerable space to them in his Yukti Bhāṣā. The shorter texts, on the other hand, often dispensed with the development of proofs. In the same spirit, the celebrated S. Ramanujan produced many important theorems but did not take time to supply proofs for them, leaving this for others to do! Whether those specificities limited the further growth of Indian mathematics is open to debate. Other factors have been discussed by historians of science, such as historical disruptions of centres and networks of learning (especially in north India), limited royal patronage, or the absence of a conquering impulse (which, in Europe, did 259 fuel the growth of science and technology). Be that as it may, India’s contribution in the field was enormous by any standard. Through the Arabs, many Indian inputs, from the decimal place-value system of numeral notation to some of the foundations of algebra and analysis, travelled on to Europe and provided crucial ingredients to the development of modern mathematics. *** Match the following Śulbasūtras kuṭṭaka Āryabhaṭa expansions of trigonometric functions Bhāskara I Chakravāla Brahmagupta Pythagoras theorem Bhāskara II negative numbers Mādhava rational approximation for the sine Comprehension questions 1. Write a few sentences on the inception of mathematics in India. 2. How would a rational approximation for the sine function be useful, when tables of sine were already available? 3. The Jain mathematicians used √10 for the value of the ratio of a circle’s circumference to its diameter (π). Āryabhaṭa offered a value (62832/2000) which, he said, was ‘approximate’. Bhāskarācārya proposed 22/7 for a ‘rough approximation’, and 3927/1250 for a ‘good approximation’. And Mādhava’s work on π is summarized above. What conclusions can you draw from these various results? 260 4. Consider the following statement by the French mathematician Pierre Simon de Laplace in 1814: ‘It is to India that we owe the ingenious method of expressing every possible number using a set of ten symbols, each symbol having a positional as well as an absolute value. A profound and important idea, it now appears to us so simple that we fail to appreciate its true merit. But its real simplicity and the way it has facilitated all calculations has placed our arithmetic foremost among useful inventions. We will appreciate the greatness of this invention all the more if we remember that it eluded the genius of the two greatest men of Antiquity, Archimedes and Apollonius.’ Discuss this statement and its implications. Why does Laplace find the Indian positional system of numeral notation ‘simple’? Project ideas ¾ Prepare a PowerPoint presentation on some of the important contributions of the ‘siddhāntic’ period of mathematics, i.e. from Āryabhaṭa to Bhāskarācārya. ¾ Prepare a PowerPoint presentation on some of the important contributions of the Kerala School of mathematics. ¾ Using Internet resources such as the website of University of St. Andrews, Scotland (http://www-history.mcs.st-andrews.ac.uk/history/Indexes/Indians.html http://www-history.mcs.st-andrews.ac.uk/history/Indexes/HistoryTopics.html), and draw a timeline for Indian as well as Babylonian, Greek, Arabic and Chinese mathematics. ¾ Consider the following four basic operations: 227 + 109; 128 – 77; 56 x 83; 45 ÷ 12. Work them out in full, but with those numbers expressed exclusively with Roman numerals: CCXXVII + CIX, etc. Spell out the rules involved clearly and follow them consistently. State your conclusions. 261 Exercises* ¾ Work out the value of √2 given in the Śulbasūtras, compare to the true value and calculate the margin of error. ¾ Āryabhaṭa I expressed the formula for the volume of a sphere thus: ‘Half the circumference multiplied by half the diameter is the area of a circle. That area multiplied by its own square root is the exact volume of a sphere’ (Āryabhaṭīya, 2.7). Show that this works out to π3/2 r3. Mahāvīra (c. 850 volume of a sphere is 9/2 r3. Śrīdhara (c. 900 CE) CE) proposed that the and Āryabhaṭa II (c. 950 CE) both proposed 38/9 r3. Tabulate these three formulas and calculate their margins of error with respect to the correct formula. Add a column for Bhāskarācārya’s formula: ‘[The sphere’s surface] multiplied by its diameter and divided by 6’ (Līlāvatī 109); work it out and conclude. ¾ Work out how many terms beyond 1 are required in Mādhava’s ‘rapidly convergent series’ given above to reach 11 correct decimals for π. Use a calculator, but provide an estimate of the time it would have taken you to do the calculations by hand — which is what Mādhava’s did! General bibliography 1. A.K. Bag, Mathematics in Ancient and Medieval India, Chaukhambha Orientala, Delhi, 1979 2. S. Balachandra Rao, Indian Mathematics and Astronomy: Some Landmarks, Jnana Deep Publications, Bangalore, 3rd edn 2004 3. D.M. Bose, S.N. Sen & B.V. Subbarayappa, eds, A Concise History of Science in India, Universities Press, Hyderabad, 2nd edn, 2009 * Twenty more exercises are found in the selections from Primary Texts further below. 262 4. Bibhutibhushan Datta, Ancient Hindu Geometry: The Science of the Śulba, 1932, repr. Cosmo Publications, New Delhi, 1993 5. Bibhutibhushan Datta & Avadhesh Narayan Singh, History of Hindu Mathematics, 1935, repr. Bharatiya Kala Prakashan, Delhi, 2004 6. G.G. Emch, R. Sridharan, M.D. Srinivas, eds, Contributions to the History of Indian Mathematics, Hindustan Book Agency, Gurgaon, 2005 7. George Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer, Penguin Books, New Delhi, 2005, 3 vols 8. George Gheverghese Joseph, The Crest of the Peacock, Penguin Books, London & New Delhi, 2000 9. George Gheverghese Joseph, A Passage to Infinity: Medieval Indian Mathematics from Kerala and its Impact, Sage, New Delhi, 2009 10. S. Parameswaran, The Golden Age of Indian Mathematics, Swadeshi Science Movement – Kerala, Kochi, 1998 11. Kim Plofker, Mathematics in India, Princeton University Press, Princeton, 2009 12. T.R.N. Rao & Subhash Kak, eds, Computing Science in Ancient India, Center for Advanced Computer Studies, Louisiana, 1998, and Munshiram Manoharlal, New Delhi, 2000 13. T.A. Sarasvati Amma, Geometry in Ancient and Medieval India, Motilal Banarsidass, New Delhi, 1999 14. Helaine Selin, & Roddam Narasimha, eds, Encyclopaedia of Classical Indian Sciences, Universities Press, Hyderabad, 2007 15. C.S. Seshadri, ed., Studies in the History of Mathematics, Hindustan Book Agency, New Delhi, 2010 16. B.S. Yadav & Man Mohan, eds, Ancient Indian Leaps into Mathematics, Birkhäuser, Boston, 2011 Internet resources (all URLs accessed in June 2012) ¾ Indian Journal of History of Science published by INSA: www.insa.nic.in/INSAuth/OurPublications.aspx ¾ A major resource on Indian mathematicians by J.J. O’Connor & E.F. Robertson: 263 http://www-history.mcs.st-andrews.ac.uk/history/Indexes/Indians.html ¾ An overview of Indian mathematics by J.J. O’Connor & E.F. Robertson: http://www-history.mcs.st-and.ac.uk/HistTopics/Indian_mathematics.html ¾ Indian Mathematics: Redressing the balance, by Ian G. Pearce: http://www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/index.html ¾ To download an earlier translation of Āryabhaṭa’s Āryabhaṭīya: http://archive.org/download/The_Aryabhatiya_of_Aryabhata_Clark_1930/The_Aryabhatiy a_of_Aryabhata_Clark_1930.pdf ¾ A huge resource of primary texts on the history of science, including many Indian texts (with or without translation): www.wilbourhall.org/ ¾ A fascinating film by the BBC, ‘The Story of Maths’: www.youtube.com/watch?v=i9x4cTJ18is&feature=related ¾ A resource on the history of mathematics in ancient cultures: http://www-history.mcs.st-andrews.ac.uk/history/Indexes/HistoryTopics.html ¾ Papers on the history of mathematics: www.journals.elsevier.com/historia-mathematica/open-archive/ ¾ A resource on the history of mathematics, but with very little on India: www.math.tamu.edu/~dallen/masters/hist_frame.htm ¾ A history of mathematics (limited content, but interesting as it is prepared by students): http://library.thinkquest.org/22584/ ¾ A resource on the history of mathematics, with many links: http://archives.math.utk.edu/topics/history.html ¾ Useful references on the history of mathematics: http://aleph0.clarku.edu/~djoyce/mathhist/ ¾ More references: http://www.dcs.warwick.ac.uk/bshm/resources.html YZ 264 Primary Texts on Mathematics in India: A Selection (All figures courtesy IFIH) White Yajur-Veda (tr. adapted from R.T.H. Griffith) O Agni, may these [sacrificial] bricks be my own milch cows: one, and ten, and ten tens, a hundred, and ten hundreds, a thousand, and ten thousand, and a hundred thousand, and a million, and a hundred millions, and a thousand millions, and a hundred thousand millions, and a million millions. May these bricks be my own milch cows in the world beyond and in this world. (Yajurveda Vājasaneyisaṃhitā, 17.2) Note: The Yajur-Veda is one of the four Vedas and exists in two versions (the White and the Black); it is dedicated to the conduct of sacrifices. Here, the priest constructs an altar for a fire sacrifice (Agni is the fire-god) and prays for each brick to become the equivalent of a milk-giving cow, a symbol of wealth (whether material or spiritual). *** Rāmāyaṇa (tr. Gita Press) The wise speak of a hundred thousand multiplied by hundred as a crore, while a lakh [100,000] of crores is called a śaṅku. A lakh of śaṅkus is known as a mahāśaṅku. A lakh of mahāśaṅkus is spoken of as a vṛnda in this context. A lakh of vṛndas is known as a mahāvṛnda. A lakh of mahāvṛndas is spoken of in this context as a padma. A lakh of padmas is known as a mahāpadma. A lakh of mahāpadmas is spoken of in this context as a kharva. A lakh of kharva is known as a mahākharva. A lakh of mahākharvas is called a samudra. A lakh of samudras 265 is called an ogha. A lakh of oghas is popularly known as a mahaugha. Surrounded according to this computation by a thousand crore and one hundred śaṅkus and a thousand mahāśaṅkus and likewise by a hundred vṛndas, even so by a thousand mahāvṛndas and a hundred padmas, in the same manner by a thousand mahāpadmas and a hundred kharvas, nay, by a hundred samudras and similarly by a hundred mahaughas and by a hundred crore mahaughas [of monkey warriors] as well as by the gallant Vibhīṣaṇa and his own ministers, Sugrīva, the ruler of monkeys, is following you for waging war — Sugrīva, who is [thus] surrounded by a huge army and ever endowed with extraordinary might and prowess. Carefully observing, O great king [Rāvaṇa], this army ranged like a blazing planet, a supreme effort may now be put forth so that your victory may be ensured and no discomfiture may follow at the hands of the enemies. (Yuddhakāṇda, 28.33–42) Note: The Rāmāyaṇa is a famous Indian epic which narrates the abduction of Sītā by the demon-king Rāvaṇa and the resulting war waged against him by Rāma, assisted by Sugrīva, the king of monkeys. In this passage, Rāvaṇa is given a description of Sugrīva’s army and its immense numbers of warriors as it prepares to attack Lanka. It is noteworthy that all numbers given are multiples of ten. The total number of warriors adds up to an astronomical number close to 1071, which is clearly intended to be taken metaphorically. Exercise ¾ In the above extract, write out the value of each named number. What is the value of the highest named number? Can you justify our statement that ‘The total number of warriors adds up to an astronomical number close to 1071’? *** 266 Baudhāyana’s Śulbasūtras (tr. S.N. Sen & A.K. Bag) Having desired [to construct] a square, one is to take a cord of length equal to the [side of the] given square, make ties at both ends and mark it at its middle. The [east-west] line [equal to the cord] is drawn and a pole is fixed at its middle. The two ties [of the cord] are fixed in it [pole] and a circle is drawn with the mark [in the middle of the cord]. Two poles are fixed at both ends of the diameter [east-west line]. With one tie fastened to the eastern [pole], a circle is drawn with the other. A similar [circle] about the western [pole]. The second diameter is obtained from the points of intersection of these two [circles]; two poles are fixed at two ends of the diameter [thus obtained]. With two ties fastened to the eastern [pole] a circle is drawn with the mark. The same [is to be done] with respect to the southern, the western and the northern [poles]. The end points of intersection of these [four circles] produce the [required] square. (1.4) 267 Note: This extract spells out a simple method to draw a square on the ground with nothing more than a rope and two poles or pegs. Given the importance of the square as a basic shape of fire altar, such a construction was fundamental (and a few more methods are proposed by other Śulbasūtras authors). Baudhāyana’s method can be summarized through the following diagram: axis EW is a given, while axis NS is obtained from the intersections of two larger circles drawn from points E and W; four more circles can now be drawn from the resulting intersections of the two axis with a circle drawn from point O. The intersections of those four circles form the desired square ABCD. Exercise ¾ With a one-metre-long rope and two pegs, replicate Baudhāyana’s method. Can you propose any other purely method to construct a square with the same apparatus? Baudhāyana’s Śulbasūtras (tr. S.N. Sen & A.K. Bag) The areas [of the squares] produced separately by the length and the breadth of a rectangle together equal the area [of the square] produced by the diagonal. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36. (1.12–13) Note: The first part of this extract is a geometrical statement of the so-called Pythagoras theorem (see a figure in the survey text). The second part lists a few of the so-called ‘Pythagoras triplets’, i.e. triplets of integers satisfying the Pythagoras theorem, for instance 32 + 42 = 52, 122 + 52 = 132. (Here, of course, Baudhāyana takes the result for granted and omits the third term.) 268 Baudhāyana’s Śulbasūtras (tr. S.N. Sen & A.K. Bag) If it is desired to combine two squares of different measures, a [rectangular] part is cut off from the larger [square] with the side of the smaller; the diagonal of the cut-off [rectangular] part is the side of the combined square. If it is desired to remove a square from another, a [rectangular] part is cut off from the larger [square] with the side of the smaller one to be removed; the [longer] side of the cut-off [rectangular] part is placed across so as to touch the opposite side; by this contact [the side] is cut off. With the cutoff [part] the difference [of the two squares] is obtained. (2.1–2) Note: The Śulbasūtras deal with transformations of one geometrical figure into another with no change in the figure’s area: for instance, a square into a rectangle and vice-versa, of a rectangle into an isosceles trapezium, or a square into a circle and vice-versa (see next extract). Here, Baudhāyana gives a method to geometrically construct a square having an area the sum or difference of the areas of two given squares. 269 The first case is illustrated in the above figure, in which an application of the Pythagoras theorem to triangle AEF shows that the area of the red square is equal to the sum of the area of square ABCD and that of the blue square. Exercise ¾ Work out the geometrical method conveyed in the second extract above (2.2), which deals with subtracting the areas of two squares. Draw a figure explaining the process. Baudhāyana’s Śulbasūtras (tr. S.N. Sen & A.K. Bag) If it is desired to transform a square into a circle [having the same area], [a cord of length] half the diagonal [of the square] is stretched from the centre to the east [a part of it lying outside the eastern side of the square]; with onethird [of the part lying outside] added to the remainder [of the half diagonal], the [required] circle is drawn. (2.9) 270 Note: The circling of a square, i.e. producing a circle having the same area as a given square, as well as the reverse problem (see next extract), exercised the best mathematical minds the world over from antiquity to medieval times. Baudhāyana’s above method is simple and is summarized through the above figure, in which point F is chosen such that FG = 1/3 GE. The resulting circle (in blue) has nearly the same area as the square ABCD. But how nearly? Calculations show that the resulting circle is too large by about 1.7%. Exercise ¾ Justify our statement that the circle resulting from Baudhāyana’s method to circle a square is too large by about 1.7%. Can you propose a more accurate geometrical method to construct such a circle? Baudhāyana’s Śulbasūtras (tr. S.N. Sen & A.K. Bag) To transform a circle into a square, the diameter is divided into eight parts; one [such] part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth [of the part left] less the eighth [of the sixth part]. Alternatively, divide [the diameter] into fifteen parts and reduce it by two of them; this gives the approximate side of the square [desired]. (2.10–11) Note: Baudhāyana offers two different methods for the reverse problem, i.e. the ‘quadrature (or squaring) of a circle’. If the circle’s radius is r, its diameter d and the desired square’s side a, the first method can be expressed as: 7 8 8 28 8 29 8 29 271 6 8 29 6 8 and the second as: 2 15 26 15 Calculations show that the resulting square is too small by about 1.7% and 4.4% respectively. Exercise ¾ Work out the calculations involved in the above two methods and justify our statement that the resulting square is too small by about 1.7% and 4.4% respectively. Can you propose an approximation of the type , where integers m and n are both less than 25, with a within 0.2% of the ideal value? *** 272 Āryabhaṭa I, Āryabhaṭīya (tr. K.S. Shukla) 225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 143, 131, 119, 106, 93, 79, 65, 51, 37, 22 and 7 — these are the Rsine-differences [at intervals of 225 minutes of arc] in terms of minutes of arc. (1.12) Note: This is a table of sine values from 3.75° to 90°, for every 3.75°. But this table is unusual in several respects for a 21st-century maths student: 5. Āryabhaṭa does not consider our modern sine but, in accordance to the usage of his times, the sine multiplied by the radius R of a circle where the angle is considered (see diagram in our survey text). This is generally referred to as Rsine. R is given an arbitrary value, which differs from one author to another; Āryabhaṭa adopted 3438’ (taking 360° or 21600’ as the circle’s circumference, and dividing it by 2π to get 3438’). Unlike the non-dimensional sine, Rsine is a linear dimension. One advantage of the Rsine is that it can have high values even with small angles. 6. The values given are incremental, that is, each value has to be added to all preceding ones in order to get the absolute value; for instance, the Rsine value given for 7° is 225’ + 224’ = 449’; for 90°, it will be the sum of all the values, i.e. 3438’ (which is the value of R, since sin 90° = 1). 7. Āryabhaṭa’s table was actually not expressed with numerals as above, but in a coded language of his own, in which each letter corresponds to a number or to a power of ten. The first few sine values read makhi, bhakhi, phakhi, dhakhi, etc. As calculations show, Āryabhaṭa’s values are highly accurate — to within 0.02%. 273 Exercise ¾ Work out in full Āryabhaṭa’s table of sines from 3.75° to 90°, comparing with the modern values for every angle, and the error involved. Justify our statement that it never exceeds 0.02%. Āryabhaṭa I, Āryabhaṭīya (tr. K.S. Shukla) Having subtracted the greatest possible cube root from the last cube place and then having written down the cube root of the number subtracted in the line of the cube root], divide the second non-cube place [standing on the right of the last cube place] by thrice the square of the cube root [already obtained]; [then] subtract from the first non-cube place [standing on the right of the second non-cube place] the square of the quotient multiplied by thrice the previous [cube root]; and [then subtract] the cube [of quotient] from the cube place [standing on the right of the first non-cube place] [and write down the quotient on the right of the previous cube root in the line of the cube root, and treat this as the new cube root. Repeat the process if there are still digits on the right]. [2.5] Note: Āryabhaṭa gives here an algorithm for the extraction of a cube root. We reproduce K.S. Shukla’s explanation (reformulated by M.S. Sriram): 274 We consider the cube root of 17,71,561 as an example. Beginning from the units place, the notational places are called cube place (c), first non-cube place (n), second noncube place(n’), cube place (c), first non-cube place (n), second non-cube place (n’), and so on. The process ends and the cube root is 121. The algorithm is obviously based on the algebraic identity: (a + b)3 = a3 + 3a2b + 3ab2 + b3. Exercise ¾ Following this algorithm, extract the cube root of 970,299. Āryabhaṭa I, Āryabhaṭīya (tr. Kim Plofker) Divide the distance between [the centres of] the earth and the sun, multiplied by [the diameter of] the earth, by the difference between [the diameters of] the sun and the earth. The quotient is the length of the earth’s shadow [measured] from the [perpendicular] diameter of the earth. (4.39) 275 Note: Āryabhaṭa gives a simple formula to calculate the length of the shadow cast by the earth; such a calculation is essential to the prediction of a lunar eclipse (when the moon passes in the cone of the earth’s shadow), its totality, duration, etc. Let d be the distance between the centres of the sun and the earth, s the distance from the centre of the earth to the tip of its shadow, DS and DE the diameters of the sun and the earth, as shown in the above figure. Āryabhaṭa’s formula can be expressed as: Exercise ¾ Prove the above formula. *** 276 Bhāskara I, Mahābhāskarīyam (tr. K.S. Shukla) I briefly state the rule [for finding Rsine values] without making use of the Rsine differences 225 etc. [as given by Āryabhaṭa, see above]. Subtract the degrees of the bhuja or koṭi [lateral and vertical sides of a right-angled triangle, i.e. cosine or sine] from the degrees of half a circle [i.e. from 0° to 180°]. Then multiply the remainder by the degrees of the bhuja and put down the result at two places. At one place subtract the result from 40,500. By onefourth of the remainder [thus obtained] divide the result at the other place as multiplied by the antyaphala [i.e. the epicyclic radius]. Thus is obtained the entire bāhuphala (or koṭiphala) for the Sun, Moon or the star-planets. So also are obtained the direct and inverse Rsines. (7.17–19) Note: Leaving aside the astronomical terms, we can see that Bhāskara proposes the following rational approximation for the Rsine: sin 4 180° 40500 180° where Ɵ is in degrees. The high accuracy of the formula is illustrated by the curve found in our survey text. The maximum absolute error in this range is a tiny 0.0016. The scholar K.S. Shukla provided a geometric ‘rationale’ for the above approximation. *** Severus Sebokht, Syria, 662 CE I will omit all discussion of the science of the Indians, a people not the same as the Syrians; of their subtle discoveries in astronomy, discoveries that are more ingenious than those of the Greeks and the Babylonians; and of their 277 valuable methods of calculation which surpass description. I wish only to say that this computation is done by means of nine signs. If those who believe, because they speak Greek, that they have arrived at the limits of science, [would read the earlier texts], they would perhaps be convinced, even if a little late in the day, that there are others also who know something of value. (From George Gheverghese Joseph, The Crest of the Peacock) Note: Severus Sebokht, a Nestorian bishop from Syria, wrote on geography and astronomy. Piqued by the arrogance of Greek scholars who thought their science was superior to that of other cultures, he authored this well-known praise of the Indian place-value system of numeral notation. It is also a useful chronological marker, since it shows that this system had reached the Mediterranean world by the 7th century CE. *** Brahmagupta, Brāhmasphuṭasiddhānta (tr. Kim Plofker) [The sum] of two positives is positive, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. [If] a smaller [positive] is to be subtracted from a larger positive, [the result] is positive; [if] a smaller negative from a larger negative, [the result] is negative; [if] a larger [negative or positive is to be subtracted] from a smaller [positive or negative, the algebraic sign of] their difference is reversed — negative [becomes] positive and positive negative. 278 A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root. The sum [of two quantities] increased or diminished by [their] difference [and] divided by two is [their] mixture. The difference of [two] squares [of the quantities] divided by the difference [of the quantities themselves] is increased and diminished by the difference [and] divided by two; [this] is the operation of unlikes. (18.30–36) Note: Brahmagupta, who has sometimes been called the ‘father of Indian algebra’, lays down here rules for operations with negative numbers and with zero. Exercises ¾ Express all the above rules in algebraic notation. Can you spot any rule that would not be accepted by modern mathematics? 279 ¾ Remove all the supplementary phrases added by the translator within square brackets; will you be able to make out all meanings? This will give you a feel of the concise style generally adopted by scientific authors of those times (the same applies to Āryabhaṭa’s method for cube root extraction above). Brahmagupta, Brāhmasphuṭasiddhānta (tr. Kim Plofker) The nature of squares: [Put down] twice the square-root of a given square multiplied by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. The sum of the thunderbolt-products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rūpas [known quantity or constant]. (18.64– 65) Note: The so-called ‘square-nature’ methods are ways of solving second-degree indeterminate equations. Here, Brahmagupta explains how to find a solution for what is now commonly known as ‘Pell’s Equation’ [Nx2 + 1 = y2]; we will illustrate the procedure using one of his examples below, namely 83x2 + 1 = y2. The key is to find, for the given ‘multiplier’ N, a solution (a, b) to an auxiliary equation Na2 ± k = b2 where k ≠ 1, and then manipulate a and b to provide a solution to the original equation. If we take our ‘given square’ to be 1 and multiply it by the ‘multiplier’ 83, we want to increase or diminish the result by some quantity to give a perfect square. E.g., 83 x 12 – 2 = 92. After we ‘put down twice’ the chosen roots, 1 1 9 9 , we take the ‘sum of the thunderbolt-products’ (apparently a technical term for cross-multiplication): 1 x 9 + 9 x 1 = 18. That is the ‘first’ quantity, 280 and the ‘last’ is 83 x 12 + 92 = 164. The new ‘additive’ is the square of the previous one: 2 x 2 = 4, giving 83 x 182 + 4 = 1642. Then the desired x and y are found by dividing the ‘first’ and ‘last’ by the previous ‘subtractive’: x = 18/2 = 9, y = 164/2 = 82. The same technique can also be used to form a new solution from two distinct previous solutions, instead of from one solution ‘put down twice.’ [This note is borrowed from the translator, Kim Plofker. By ‘indeterminate equation’ is meant an equation for which integral solutions alone are desired. This method of Brahmagupta is called the bhāvanā.] Exercises ¾ Solve the above second-degree indeterminate equation again, carefully following the step-by-step instructions and comparing them with Brahmagupta’s brief explanation. ¾ By applying the above bhāvanā, show that a set of integral solutions to 18x2 + 1 = y2 is (4, 17). ¾ One of Brahmagupta’s corollaries to his bhāvanā is that if (a, b) is a solution to Nx2 + 1 = y2, then (2ab, b2 + Na2) is also a solution. Give a proof for this corollary, and use it to produce two more sets of solutions to the above equation. *** Bhāskarācārya, Bījagaṇita (tr. adapted from S.K. Abhyankar) Multiply both sides [of an equation] by a known quantity equal to four times the coefficient of the square of the unknown; add to both sides a known quantity equal to the square of the [original] coefficient of the unknown: then [extract] the root. (116) 281 Note: Bhāskarācārya here spells out a method to solve the equation ax2 + bx = c, where it is tacitly assumed that the coefficients a, b and c are positive; in fact, he attributes this solution to Śrīdhara, an earlier mathematician whose work on algebra is lost. We first multiply the above equation by 4a and then add b2 to both the sides: 4a2x2 + 4abx + b2 = 4ac + b2, which becomes: (2ax + b)2 = 4ac + b2. Taking roots, we obtain: 2 4 , from which we get one of the two standard solutions (the other deriving from the negative root of the first member of the above equation). This strategy is extended to methods for solving the cubic and the biquadratic (or fourth-degree) equations. Bhāskarācārya, Bījagaṇita (tr. adapted from S.K. Abhyankar) One man says to the other, ‘If you will give me 100 rupees, I shall be twice as rich as you are.’ The other man says, ‘If you give me 10 rupees, I shall be six times as rich as you are.’ Tell me the wealth of each of them. (93) A merchant started with a sum. Entering a city, he paid Rs. 10 as duty. After trading his amount became double. From that he spent Rs. 10 on dinner and left the city after paying Rs. 10 as duty. He went to two other cities; the same was the case in both of them. After coming back his amount had trebled. What was the [initial] sum? (101) 282 Note: Bhāskarācārya enjoyed mathematical brainteasers, as both his Bījagaṇita and his Līlāvatī testify (following the example of earlier authors, such as Mahāvīra). They usually involve systems of linear equations, occasionally quadratic ones. Exercise ¾ Solve the above two brainteasers. Bhāskarācārya, Līlāvatī (tr. KS Patwardhan et al.) Arjuna became furious in the war and, in order to kill Karṇa, picked up some arrows. With half of the arrows, he destroyed all of Karṇa’s arrows. He killed all of Karṇa’s horses with four times the square root of the arrows. He destroyed the spear with six arrows. He used one arrow each to destroy the top of the chariot, the flag, and the bow of Karṇa. Finally he cut off Karṇa’s head with another arrow. How many arrows did Arjuna discharge? (76) Note: This brainteaser from Līlāvatī involves a simple quadratic equation (which assumes that Arjuna discharged all the arrows he had picked up). The equation is deftly woven into a famous episode of the Mahābhārata war. A king had a beautiful palace with eight doors. Skilled engineers had constructed four open squares which were highly polished and huge. In order to get fresh air, 1 door, 2 doors, 3 doors, ... are opened. How many different types of breeze arrangements are possible? How many kinds of relishes can be made by using 1, 2, 3, 4, 5 or 6 types from sweet, bitter, astringent, sour, salty and hot substances? (122) Note: These two examples involve standard combinatorics. 283 Exercise ¾ Solve the above three brainteasers. *** Jyeṣṭhadeva, Gaṇita-Yukti-Bhāṣā (tr. K.V. Sarma) Now, the method to ascertain two numbers if any two of the following five, viz., the sum, difference, product, sum of squares, and difference of squares of the two numbers, are known. Qn. 1. Here, if the difference of two numbers is added to their sum, the result obtained will be twice the bigger number. Then, if the difference is subtracted from the sum, the result obtained will be twice the smaller number. Then, when the two results, as obtained above, are halved, the two numbers, respectively, will result. Qn. 2. Now, to ascertain the numbers when their sum and product are known: Here, in accordance with the rationale explained earlier, if four times the product is subtracted from the square of the sum, and the root of the result found, it will be the difference between the numbers. Using this [and the sum of the numbers], the two numbers can be got as explained above. Qn. 3. Now, [given] the sum and the sum of the squares [of the numbers]: There, when the square of the sum is subtracted from twice the sum of the squares and the root of the result found, it will be the difference between the numbers. 284 Qn. 4. Then, when the difference between the squares is divided by the sum [of the numbers], the result will be the difference between the numbers, as per the rationale explained earlier. Qn. 5. Then, [given] the difference and the product of the numbers: There, if the product is multiplied by four and the square of the difference added and the root of the result found, it will be the sum of the numbers. Qn. 6. Then, given the difference and the sum of squares: When the square of the difference is subtracted from double the sum of the squares, and the root of the result found, it will be the sum of the numbers. Qn. 7. Then, when the difference of the squares is divided by the difference [of the numbers], the result will be the sum of the numbers. Qn. 8. Then, [given] the product and the sum of the squares [of the numbers]: Here, subtract twice the product from the sum of the squares, and find the root of the result. This will be the difference [between the numbers]. When the product is multiplied by 4 and the square of the difference added, the root of the result is the sum [of the numbers]. Qn. 9. Then, [given] the product and the difference of the squares [of the numbers]: Now, we obtain the squares of the two numbers. Here, the calculations done using the numbers can be done using the squares of the numbers. The distinction here would be that the results will also be in terms of squares. There, when the product is squared, it will be the product of the squares, [since] there is no difference in the result of multiplication when the sequence [of the steps] is altered. Hence, taking that the product and the difference of the squares are known, the sum of the squares can be derived by the same method used for calculating the sum [of two numbers] given their 285 product and difference. Here, when the square of the product is multiplied by four and added to the square of the difference in the squares, the root of the result will be the sum of the squares. Then placing this sum of the squares in two places, add to one the difference of the squares and subtract it from the other. Then divide both by 2. The results will be the squares of the two numbers. Qn. 10. Then, the tenth [question] is when the sum of the squares and the difference of the squares are known. This too has been answered above. These are the ten questions. These have been stated here since they are made use of in several places. Cube roots have no use in planetary computation. Hence they are not stated here. Thus [have been explained] a way of computation. Note: This is chapter 2 of Jyeṣṭhadeva’s Gaṇita-Yukti-Bhāṣā, a Malayalam work of the 16th century, divided in two major parts, one on mathematics and the other on astronomy. Here Jyeṣṭhadeva lays down ten commonly used rules of algebra. Exercise ¾ How many of these ‘ten questions’ can you express in algebraic form and prove? YZ 286