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Indian mathematics—which here is the mathematics that emerged in , and Algebra . In addition, Trigonometry , having evolved in the Hellenistic World and having been introduced into Ancient India through the translation of Greek works, Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. In fact, the Indian astronomers in the third or fourth century, using a pre-Ptolemaic Greek table of chords, produced tables of sines and versines, from which it was trivial to derive cosines. This new system of trigonometry, produced in India, was transmitted to the Arabs in the late eighth century and by them, in an expanded form, to the Latin West and the Byzantine East in the twelfth century." was further advanced in India, and, in particular, the modern definitions of , Hipparchus , Ptolemy ) who establish the fundamental relations between the sides and angles of a right angled triangle (plane or spherical) and draw up the first tables (they consist of tables giving the ''chord'' of the arc cut out by an angle on a circle of radius ''r'', in other words the number ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)." These mathematical concepts were transmitted to the Middle East , China , and Europe "algebra" 2007. ''Britannica Concise Encyclopedia'' . Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra." and led to further developments that now form the foundations of many areas of mathematics. Ancient and medieval Indian mathematical works, all composed in Sanskrit , usually consisted of a section of '' Sutra s'' in which a set of rules or problems were stated with great economy in verse in order to aid memorization by a student. This was followed by a second section consisting of a prose commentary (sometimes multiple commentaries by different scholars) that explained the problem in more detail and provided justification for the solution. In the prose section, the form (and therefore its memorization) was not considered as important as the ideas involved. All mathematical works were orally transmitted until approximately 500 BCE; thereafter, they were transmitted both orally and in manuscript form. The oldest extant mathematical ''document'' produced on the Indian Subcontinent is the birch bark Bakhshali Manuscript , discovered in 1881 in the village of Bakhshali, near Peshawar (modern day Pakistan ); the manuscript is likely from the seventh century CE. A later landmark in Indian mathematics was the development of the Series expansions for Trigonometric Function s (sine, cosine, and Arc Tangent ) by mathematicians of the Kerala School in the fifteenth century CE. Their remarkable work, completed two centuries before the invention of Calculus in Europe, provided what is now considered the first example of a Power Series (apart from geometric series). However, they did not formulate a systematic theory of Differentiation and Integration , nor is there any direct evidence of their results being transmitted outside Kerala . (However, see Charges Of Eurocentrism below for recent research in this area.) FIELDS OF INDIAN MATHEMATICS Some of the areas of mathematics studied in ancient and medieval India include the following:
HARAPPAN MATHEMATICS (2600 BCE - 1700 BCE) See Also: Indus Valley Civilization The earliest evidence of the use of mathematics in shapes, which included Hexahedra , Barrel s, Cone s, and Cylinder s, thereby demonstrating knowledge of basic Geometry . The inhabitants of Indus civilization also tried to standardize measurement of length to a high degree of accuracy. They designed a ruler—the ''Mohenjo-daro ruler''—whose unit of length (approximately 1.32 inches) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length. THE ORAL MATHEMATICAL TRADITION Mathematicians of ancient and early medieval India were almost all Sanskrit Pandit s (''paṇḍita'' "learned man"), who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar ( ''vyākaraṇa'' ), Exegesis ( ''mīmāṃsā'' ) and logic ( ''nyāya'' )." Memorization of "what is heard" ('' śruti '' in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorization and recitation was also used to transmit philosophical and literary works, as well as treatises on ritual and grammar. Modern scholars of ancient India have noted the "truly remarkable achievements of the Indian pandits who have preserved enormously bulky texts orally for millennia." ;Styles of Memorization Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity. For example, memorization of the sacred '' Veda s'' included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the ''jaṭā-pāṭha'' (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order. The recitation thus proceeded as: In another form of recitation, ''dvaja-pāṭha'' (literally "flag recitation") a sequence of ''N'' words were recited (and memorized) by pairing the first two and last two words and then proceeding as: The most complex form of recitation, ''ghana-pāṭha'' (literally "dense recitation"), according to , took the form: That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the '' Ṛgveda '' ( Ca. 1500 BCE), as a single text, without any variant readings. Similar methods were used for memorizing mathematical texts, whose transmission remained exclusively oral until the end of the Vedic Period (ca. 500 BCE). ;The ''Sūtra'' Genre Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Veda s, which took the form of works called ''Vedāṇgas'' , or, "Ancillaries of the Veda" (7th-4th century BCE). The need to conserve the sound of sacred text by use of ''śikṣā'' ( Phonetics ) and '' Chandas '' ( Metric s); to conserve its meaning by use of ''vyākaraṇa'' ( Grammar ) and '' Nirukta '' ( Etymology ); and to correctly perform the rites at the correct time by the use of '' Kalpa '' ( Ritual ) and ''jyotiṣa'' ( Astronomy ), gave rise to the six disciplines of the ''Vedāṇgas''. Mathematics arose as a part of the last two disciplines, ritual and astronomy (which also included astrology). Since the ''Vedāṇgas'' immediately preceded the use of writing in ancient India, they formed the last of the exclusively oral literature. They were expressed in a highly compressed mnemonic form, the ''sūtra'' (literally, "thread"): The knowers of the ''sūtra'' know it as having few phonemes, being devoid of ambiguity, containing the essence, facing everything, being without pause and unobjectionable. Extreme brevity was achieved through multiple means, which included using Ellipsis "beyond the tolerance of natural language," using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables. The ''sūtras'' create the impression that communication through the text was "only a part of the whole instruction. The rest of the instruction must have been transmitted by the so-called ''Guru-shishya Parampara'' , 'uninterrupted succession from teacher (''guru'') to the student (''śisya''),' and it was not open to the general public" and perhaps even kept secret. The brevity achieved in a ''sūtra'' is demonstrated in the following example from the Baudhāyana ''Śulba Sūtra'' (700 BCE). The domestic fire-altar in the Vedic Period was required by ritual to have a square base and be constituted of five layers of bricks with 21 bricks in each layer. One method of constructing the altar was to divide one side of the square into three equal parts using a cord or rope, to next divide the transverse (or perpendicular) side into seven equal parts, and thereby sub-divide the square into 21 congruent rectangles. The bricks were then designed to be of the shape of the constituent rectangle and the layer was created. To form the next layer, the same formula was used, but the bricks were arranged transversely. The process was then repeated three more times (with alternating directions) in order to complete the construction. In the Baudhāyana ''Śulba Sūtra'', this procedure is described in the following words: "II.64. After dividing the quadri-lateral in seven, one divides the transverse in three.II.65. In another layer one places the [bricks North-pointing." According to , the officiant constructing the altar has only a few tools and materials at his disposal: a cord (Sanskrit, ''rajju'', f.), two pegs (Sanskrit, ''śanku'', m.), and clay to make the bricks (Sanskrit, ''iṣṭakā'', f.). Concision is achieved in the ''sūtra'', by not explicitly mentioning what the adjective "transverse" qualifies; however, from the feminine form of the (Sanskrit) adjective used, it is easily inferred to qualify "cord." Similarly, in the second stanza, "bricks" are not explicitly mentioned, but inferred again by the feminine plural form of "North-pointing." Finally, the first stanza, never explicitly says that the first layer of bricks are oriented in the East-West direction, but that too is implied by the explicit mention of "North-pointing" in the ''second'' stanza; for, if the orientation was meant to be the same in the two layers, it would either not be mentioned at all or be only mentioned in the first stanza. All these inferences are made by the officiant as he recalls the formula from his memory. VEDIC PERIOD (1500 BCE - 400 BCE) See Also: Vedanga Vedas The religious texts of the (9th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.A. Seidenberg, 1978. The origin of mathematics. Archive for the history of Exact Sciences, vol 18. ;Śulba Sūtras The '' Śulba Sūtras '' (literally, "Aphorisms of the Chords" in Vedic Sanskrit ) (c. 700-400 BCE) list rules for the construction of sacrificial fire altars. Most mathematical problems considered in the ''Śulba Sūtras'' spring from "a single theological requirement," that of constructing fire altars which have different shapes but occupy the same area. The altars were required to be constructed of five layers of burnt brick, with the further condition that each layer consist of 200 bricks and that no two adjacent layers have congruent arrangements of bricks. According to , the ''Śulba Sūtras'' contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonian s." The diagonal rope (''akṣṇayā-rajju'') of an oblong (rectangle) produces both which the flank (''pārśvamāni'') and the horizontal (''tiryaṇmānī'')Since the statement is a ''sūtra'', it is necessarily compressed and what the ropes ''produce'' is not elaborated on, but the context clearly implies the square areas constructed on their lengths, and would have been explained so by the teacher to the student. They contain lists of .: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others." They also contain statements (that with hindsight we know to be approximate) about Squaring The Circle and "circling the square.": "The requirement of three altars of equal areas but different shapes would explain the interest in transformation of areas. Among other transformation of area problems the Hindus considered in particular the problem of squaring the circle. The ''Bodhayana Sutra'' states the converse problem of constructing a circle equal to a given square. The following approximate construction is given as the solution.... this result is only approximate. The authors, however, made no distinction between the two results. In terms that we can appreciate, this construction gives a value for π of 18 (3 − 2√2), which is about 3.088." Baudhayana (c. 8th century BCE) composed the ''Baudhayana Sulba Sutra'', the best-known ''Sulba Sutra'', which contains examples of simple Pythagorean triples, such as: , , , , and as well as a statement of the Pythagorean theorem for the sides of a square: "The rope which is stretched across the diagonal of a square produces an area double the size of the original square." It also contains the general statement of the Pythagorean theorem (for the sides of a rectangle): "The rope stretched along the length of the diagonal of a rectangle makes an area which the vertical and horizontal sides make together." Baudhayana gives a formula for the Square Root Of Two , :: The formula is accurate up to five decimal places, the true value being The value of this approximation, 577/408, is the seventh in a sequence of increasingly accurate approximations 3/2, 7/5, 17/12, ... to √2, the numerators and denominators of which were known as "side and diameter numbers" to the ancient Greeks, and in modern mathematics are called the Pell Numbers . If ''x''/''y'' is one term in this sequence of approximations, the next is (''x''+2''y'')/(''x''+''y''). These approximations may also be derived by truncating the Continued Fraction representation of √2. This formula is similar in structure to the formula found on a Mesopotamian tablet Neugebauer, O. and A. Sachs. 1945. ''Mathematical Cuneiform Texts'', New Haven, CT, Yale University Press. p. 45. from the Old Babylonian period (1900-1600 BCE ): ::: which expresses in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding). According to mathematician S. G. Dani, the Babylonian cuneiform tablet for details. indicating, in particular, that there was sophisticated understanding on the topic" in Mesopotamia in 1850 BCE . "Since these tablets predate the Sulbasutras period by several centuries, taking into account the contextual appearance of some of the triples, it is reasonable to expect that similar understanding would have been there in India." Dani goes on to say: "As the main objective of the ''Sulvasutras'' was to describe the constructions of altars and the geometric principles involved in them, the subject of Pythagorean triples, even if it had been well understood may still not have featured in the ''Sulvasutras''. The occurrence of the triples in the ''Sulvasutras'' is comparable to mathematics that one may encounter in an introductory book on architecture or another similar applied area, and In all three ''Sulba Sutras'' were composed. The remaining two, the ''Manava Sulba Sutra'' composed by Manava (fl. 750-650 BCE) and the ''Apastamba Sulba Sutra'', composed by Apastamba (c. 600 BCE), contained results similar to the ''Baudhayana Sulba Sutra''. ;Vyakarana An important landmark of the Vedic period was the work of Sanskrit Grammarian , (c. 520-460 BCE). His grammar includes early use of Boolean Logic , of the Null operator, and of Context Free Grammar s, and includes a precursor of the Backus–Naur Form (used in the description Programming Languages ). JAINA MATHEMATICS (400 BCE - 200 CE) Although Jainism as a religion and philosophy predates its most famous exponent, Mahavira ( 6th Century BC ), who was a contemporary of Gautama Buddha , most Jaina texts on mathematical topcs were composed after the 6th century BCE. Jain a mathematicians are important historically as crucial links between the mathematics of the Vedic period and that of the "Classical period." A significant historical contribution of Jaina mathematicians lay in their freeing Indian mathematics from its religious and ritualistic constraints. In particular, their fascination with the enumeration of very large numbers and .) In addition to ''Surya Prajnapti'', important Jaina works on mathematics included the '' Vaishali Ganit'' (c. 3rd century BCE); the ''Sthananga Sutra'' (fl. 300 BCE - 200 CE); the ''Anoyogdwar Sutra'' (fl. 200 BCE - 100 CE); and the ''Satkhandagama'' (c. 2nd century CE). Important Jaina mathematicians included Bhadrabahu (d. 298 BCE), the author of two astronomical works, the ''Bhadrabahavi-Samhita'' and a commentary on the ''Surya Prajinapti''; Yativrisham Acharya (c. 176 BCE), who authored a mathematical text called ''Tiloyapannati''; and Umasvati (c. 150 BCE), who, although better known for his influential writings on Jaina philosophy and Metaphysics , composed a mathematical work called ''Tattwarthadhigama-Sutra Bhashya''. ;Pingala Among other scholars of this period who contributed to mathematics, the most notable is Pingala (') ( Fl. 300-200 BCE), a Musical Theorist who authored the '' Chandas Shastra ''''' ('''', also '''''Chandas Sutra''''' ''''), a Sanskrit treatise on Prosody . There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both the Pascal Triangle and Binomial Coefficients , although he did not have knowledge of the Binomial Theorem itself. Pingala's work also contains the basic ideas of Fibonacci Number s (called ''maatraameru''). Although the ''Chandah sutra'' hasn't survived in its entirety, a 10th century commentary on it by Halāyudha has. Halāyudha, who refers to the Pascal triangle as '' Meru -prastāra'' (literally "the staircase to Mount Meru "), has this to say: "Draw a square. Beginning at half the square, draw two other similar squares below it; below these two, three other squares, and so on. The marking should be started by putting '''1''' in the first square. Put '''1''' in each of the two squares of the second line. In the third line put '''1''' in the two squares at the ends and, in the middle square, the sum of the digits in the two squares lying above it. In the fourth line put '''1''' in the two squares at the ends. In the middle ones put the sum of the digits in the two squares above each. Proceed in this way. Of these lines, the second gives the combinations with one syllable, the third the combinations with two syllables, ..." The text also indicates that Pingala was aware of the Combinatorial identity: ;Katyayana Though not a Jaina mathematician, Katyayana (c. 3rd century BCE) is notable for being the last of the Vedic mathematicians. He wrote the ''Katyayana Sulba Sutra'', which presented much Geometry , including the general Pythagorean Theorem and a computation of the square root of 2 correct to five decimal places. THE WRITTEN TRADITION: PROSE COMMENTARY With the increasing complexity of mathematics and other exact sciences, both writing and computation were required. Consequently, many mathematical works began to be written down in manuscripts that were then copied and re-copied from generation to generation. "India today is estimated to have about thirty million manuscripts, the largest body of handwritten reading material anywhere in the world. The literate culture of Indian science goes back to at least the fifth century B.C. ... as is shown by the elements of Mesopotamian omen literature and astronomy that entered India at that time and (were) definitely not ... preserved orally." The earliest mathematical prose commentary was that on the work, '' Āryabhaṭīya '' (written 499 CE), a work on astronomy and mathematics. The mathematical portion of the ''Āryabhaṭīya'' was composed of 33 ''sūtras'' (in verse form) consisting of mathematical statements or rules, but without any proofs. However, according to , "this does not necessarily mean that their authors did not prove them. It was probably a matter of style of exposition." From the time of Bhaskara I (600 CE onwards), prose commentaries increasingly began to include some derivations (''upapatti''). Bhaskara I's commentary on the ''Āryabhaṭīya'', had the following structure:
Typically, for any mathematical topic, students in ancient India first memorized the ''sūtras'', which, as explained earlier, were "deliberately inadequate" in explanatory details (in order to pithily convey the bare-bone mathematical rules). The students then worked through the topics of the prose commentary by writing (and drawing diagrams) on chalk- and dust-boards (''i.e.'' boards covered with dust). The latter activity, a staple of mathematical work, was to later prompt mathematician-astronomer, Brahmagupta ( Fl. 7th century CE), to characterize astronomical computations as "dust work" (Sanskrit: ''dhulikarman''). NUMERALS AND THE DECIMAL NUMBER SYSTEM The earliest extant Script used in India was the Kharoṣṭhī script used in the Gandhara culture of the north-west. It is thought to be of Aramaic origin and it was in use from the fourth century BCE to the fourth century CE. Almost contemporaneously, another script, the Brahmi , appeared on much of the sub-continent, and would later become the foundation of many scripts of South Asia and South-east Asia. Both scripts had numeral symbols and numeral systems, which were initially ''not'' based on a place-value system. The first datable evidence of the use of the decimal place-value system in India is found in the ''Yavanajātaka'' ( Ca. 270 CE) of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology. BAKHSHALI MANUSCRIPT The oldest extant mathematical manuscript in South Asia is the '''' writing had been developed two or three centuries before the ''Gupta'' writing itself appeared. ''Gupta'' only began to evolve into ''Shāradā'' style around the ninth century CE. In other words, the Bak(h)shali manuscript cannot have been written earlier than the ninth century CE. However, in the light of certain characteristic indications, it could not have been written any later than the twelfth century CE." The 7th century CE is now considered a plausible date, Quote:"The dates so far proposed for the Bakhshali work vary from the third to the twelfth centuries AD, but a recently made comparative study has shown many similarities, particularly in the style of exposition and terminology, between Bakhshalī work and Bhāskara I's commentary on the ''Āryabhatīya''. This seems to indicate that both works belong to nearly the same period, although this does not deny the possibility that some of the rules and examples in the Bakhshālī work date from anterior periods." albeit with the likelihood that the "manuscript in its present-day form constitutes a commentary or a copy of an anterior mathematical work." The surviving manuscript has seventy leaves, some of which are in fragments. Its mathematical content consists of rules and examples, written in verse, together with prose commentaries, which include solutions to the examples. The topics treated include arithmetic (fractions, square roots, profit and loss, simple interest, the Rule Of Three , and '' Regula Falsi '') and algebra (simultaneous linear equations and Quadratic Equations ), and arithmetic progressions. In addition, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." Many of its problems are the so-called equalization problems that lead to systems of linear equations. One example from Fragment III-5-3v is the following: "One merchant has seven ''asava'' horses, a second has nine ''haya'' horses, and a third has ten camels. They are equally well off in the value of their animals if each gives two animals, one to each of the others. Find the price of each animal and the total value for the animals possessed by each merchant."Anton, Howard and Chris Rorres. 2005. ''Elementary Linear Algebra with Applications.'' 9th edition. New York: John Wiley and Sons. 864 pages. ISBN 0471669598. The prose commentary accompanying the example solves the problem by converting it to three (under-determined) equations in four unknowns and assuming that the prices are all integers. CLASSICAL PERIOD (400 - 1200) This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as sixth century compilation—''Pancasiddhantika'' (literally ''panca'', "five," ''siddhānta'', "conclusion of deliberation", dated 575 CE )—of five earlier works, Surya Siddhanta , Romaka Siddhanta , Paulisa Siddhanta , Vasishtha Siddhanta and Paitamaha Siddhanta , which were adaptations of still earlier works of Mesopotamian, Greek, Egyptian, Roman and Indian astronomy. As explained earlier, the main texts were composed in Sanskrit verse, and were followed by prose commentaries. Fifth and Sixth Centuries ;Surya Siddhanta Though its authorship is unknown, the '' Surya Siddhanta '' (c. 400) contains the roots of modern Trigonometry . Due to the large number of foreign words in the documents, Historians have concluded that its roots are in Mesopotamia and Greece.1 It uses the following as trigonometric functions for the first time:
It also contains the earliest uses of:
Later Indian mathematicians such as Aryabhata made references to this text, while later Arabic and Latin translations were very influential in Europe and the Middle East. ;Chhedi calendar This Chhedi calendar (594) contains an early use of the modern Place-value Hindu-Arabic Numeral System now used universally (see also Hindu-Arabic Numerals ). ;Aryabhata I Aryabhata (476-550) wrote the ''Aryabhatiya.'' He described the important fundamental principles of mathematics in 332 Shlokas . The treatise contained:
Aryabhata also wrote the ''Arya Siddhanta'', which is now lost. Aryabhata's contributions include: Trigonometry:
Arithmetic: Algebra:
Mathematical astronomy:
Calculus:
;Varahamihira Varahamihira (505-587) produced the ''Pancha Siddhanta'' (''The Five Astronomical Canons''). He made important contributions to Trigonometry , including sine and cosine tables to 4 decimal places of accuracy and the following formulas relating Sine and Cosine functions: Seventh and Eighth Centuries In the seventh century, two separate fields, : Brahmagupta's theorem: If a Cyclic Quadrilateral has diagonals that are Perpendicular to each other, then the perpendicular line drawn from the point of intersection of the diagonals to any side of the quadrilateral always bisects the opposite side. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's Formula ), as well as a complete description of Rational Triangle s (''i.e.'' triangles with rational sides and rational areas). Brahmagupta's formula: The area, ''A'', of a Cyclic Quadrilateral with sides of lengths ''a'', ''b'', ''c'', ''d'', respectively, is given by : where ''s'', the Semiperimeter , given by: Brahmagupta's Theorem on Rational Triangle s: A triangle with rational sides and rational area is of the form: : for some rational numbers and . Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers and is considered the first systematic treatment of the subject. The rules (which included and ) were all correct, with one exception: . Later in the chapter, he gave the first explicit (although still not completely general) solution of the Quadratic Equation : : }} This is equivalent to: : Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's Equation , : where is a nonsquare integer. He did this by discovering the following identity: Brahmagupta's Identity: which was a generalization of an earlier identity of Diophantus : Brahmagupta used his identity to prove the following lemma: Lemma (Brahmagupta): If is a solution of and, is a solution of , then: : is a solution of He then used this lemma to both generate infinitely many (integral) solutions of Pell's equation, given one solution, and state the following theorem: Theorem (Brahmagupta): If the equation has an integer solution for any one of then Pell's Equation : : also has an integer solution. Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was: Example (Brahmagupta): Find integers such that: : In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician." The solution he provided was: : ;Bhaskara I Bhaskara I (c. 600-680) expanded the work of Aryabhata in his books titled ''Mahabhaskariya'', ''Aryabhattiya Bhashya'' and ''Laghu Bhaskariya''. He produced:
Ninth to Twelfth Centuries ;Virasena Virasena (9th century) was a Jaina mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta , Karnataka. He wrote the ''Dhavala'', a commentary on Jaina mathematics, which:
Virasena also gave: ;Mahavira Mahavira Acharya (c. 800-870) from Karnataka , the last of the notable Jaina mathematicians, lived in the 9th Century and was patronised by the Rashtrakuta king Amoghavarsha . He wrote a book titled ''Ganit Saar Sangraha'' on numerical mathematics, and also wrote treatises about a wide range of mathematical topics. These include the mathematics of:
Mahavira also:
;Shridhara Shridhara (c. 870-930), who lived in Bengal , wrote the books titled ''Nav Shatika'', ''Tri Shatika'' and ''Pati Ganita''. He gave:
The ''Pati Ganita'' is a work on arithmetic and Mensuration . It deals with various operations, including:
;Manjula Aryabhata's differential equations were elaborated in the 10th century by Manjula (also ''Munjala''), who realised that the expression could be approximately expressed as He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation. ;Aryabhata II Aryabhata II (c. 920-1000) wrote a commentary on Shridhara, and an astronomical treatise '' Maha-Siddhanta ''. The Maha-Siddhanta has 18 chapters, and discusses:
;Shripati Shripati Mishra (1019-1066) wrote the books ''Siddhanta Shekhara'', a major work on astronomy in 19 chapters, and ''Ganit Tilaka'', an incomplete Arithmetic al treatise in 125 verses based on a work by Shridhara . He worked mainly on:
He was also the author of ''Dhikotidakarana'', a work of twenty verses on: The ''Dhruvamanasa'' is a work of 105 verses on: ;Nemichandra Siddhanta Chakravati Nemichandra Siddhanta Chakravati (c. 1100) authored a mathematical treatise titled ''Gome-mat Saar''. ;Bhaskara II Bhāskara II (1114-1185) was a mathematician-astronomer who wrote a number of important treatises, namely the ''Siddhanta Shiromani'', '' Lilavati '', ''Bijaganita'', ''Gola Addhaya'', ''Griha Ganitam'' and ''Karan Kautoohal''. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include: Arithmetic:
Algebra:
Geometry:
Calculus:
Trigonometry:
KERALA MATHEMATICS (1300 - 1600) See Also: Kerala school of astronomy and mathematics The , Neelakanta Somayaji , Jyeshtadeva , Achyuta Pisharati , Melpathur Narayana Bhattathiri and Achyuta Panikkar. It flourished between the 14th and 16th Centuries and the original discoveries of the school seems to have ended with Narayana Bhattathiri ( 1559 - 1632 ). In attempting to solve astronomical problems, the Kerala school astronomers ''independently'' created a number of important mathematics concepts. The most important results, series expansion for Trigonometric Function s, were given in Sanskrit verse in a book by Neelakanta called ''Tantrasangraha'' and a commentary on this work called ''Tantrasangraha-vakhya'' of unknown authorship. The theorems were stated without proof, but proofs for the series for ''sine'', ''cosine'', and inverse ''tangent'' were provided a century later in the work ''Yuktibhasa'' (c.1500-c.1610), written in Malayalam , by Jyesthadeva, and also in a commentary on ''Tantrasangraha''. Their discovery of these three important series expansions of Calculus —several centuries before calculus was developed in Europe by Isaac Newton and Gottfried Leibniz —was a landmark achievement in mathematics. However, the Kerala School cannot be said to have invented ''calculus'', because, while they were able to develop Taylor series expansions for the important trigonometric functions, they developed neither a comprehensive theory of Differentiation or Integration , nor the Fundamental Theorem Of Calculus . The results obtained by the Kerala school include:
: where : : where, for , the series reduce to the standard power series for these trigonometric functions, for example:
:
: ::where
:
However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in ''Yuktibhasa'' given in two papers,Rajagopal, C. and M. S. Rangachari. 1949. "A Neglected Chapter of Hindu Mathematics." ''Scripta Mathematica''. 15:201-209.Rajagopal, C. and M. S. Rangachari. 1951. "On the Hindu proof of Gregory's series." ''Ibid.'' 17:65-74. a commentary on the ''Yuktibhasa'''s proof of the sine and cosine seriesRajagopal, C. and A. Venkataraman. 1949. "The sine and cosine power series in Hindu mathematics." ''Journal of the Royal Asiatic Society of Bengal (Science)''. 15:1-13. and two papers that provide the Sanskrit verses of the ''Tantrasangrahavakhya'' for the series for arctan, sin, and cosine (with English translation and commentary).Rajagopal, C. and M. S. Rangachari. 1977. "On an untapped source of medieval Keralese mathematics." ''Archive for the History of Exact Sciences''. 18:89-102.Rajagopal, C. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics." ''Archive for the History of Exact Sciences''. 35:91-99. The Kerala mathematicians included Narayana Pandit (c. 1340-1400), who composed two works, an arithmetical treatise, ''Ganita Kaumudi'', and an algebraic treatise, ''Bijganita Vatamsa''. Narayana is also thought to be the author of an elaborate commentary of Bhaskara II 's Lilavati , titled ''Karmapradipika'' (or ''Karma-Paddhati''). Madhava Of Sangamagramma (c. 1340-1425) was the founder of the Kerala School . Although it is possible that he wrote ''Karana Paddhati'' a work written sometime between 1375 and 1475, all we really know of his work comes from works of later scholars. . Nilakantha Somayaji (1444-1544) composed the ''Tantra Samgraha'' (which 'spawned' a later anonymous commentary ''Tantrasangraha-vyakhya'' and a further commentary by the name ''Yuktidipaika'', written in 1501 ). He elaborated and extended the contributions of Madhava. Citrabhanu (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two Simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms: For each case, Citrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric. Jyesthadeva (c. 1500-1575) was another member of the Kerala School. His key work was the ''Yukti-bhasa'' (written in Malayalam , a regional language of Kerala ). Jyesthadeva presented proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other Kerala School mathematicians. CHARGES OF EUROCENTRISM It has been suggested that Indian contributions to mathematics have not been given due acknowledgement in modern history and that many discoveries and inventions by Indian Mathematicians are presently culturally attributed to their Western counterparts, as a result of Eurocentrism . According to G. G. Joseph: work takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness Indian and Arabic mathematics is all too likely to be tempered with dismissive rejections of their importance compared to Greek mathematics. The contributions from other civilizations - most notably China and India, are perceived either as borrowers from Greek sources or having made only minor contributions to mainstream mathematical development. An openness to more recent research findings, especially in the case of Indian and Chinese mathematics, is sadly missing" Joseph, G. G. 1997. "Foundations of Eurocentrism in Mathematics." In ''Ethnomathematics: Challenging Eurocentrism in Mathematics Education'' (Eds. Powell, A. B. et al.). SUNY Press. ISBN 0791433528. p.67-68. More recently, as discussed in the above section, the infinite series of Calculus for trigonometric functions (rediscovered by Gregory, Taylor, and Maclaurin in the late 17th century) were described (with proofs) in India, by mathematicians of the Kerala School , remarkably some two centuries earlier. Some scholars have recently suggested that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China and Arabia , and, from around 1500 , with Europe. The existence of communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission actually took place.Almeida, D. F., J. K. John, and A. Zadorozhnyy. 2001. "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications." ''Journal of Natural Geometry'', 20:77-104. Indeed, according to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century." Gold, D. and D. Pingree. 1991. "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine." ''Historia Scientiarum''. 42:49-65. Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. However, they were not able to, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the Derivative and the Integral , show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate ''predecessors'' of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources we are not now aware." This is an active area of current research, especially in the manuscripts collections of Spain and Maghreb , research that is now being pursued, among other places, at the Centre National de Recherche Scientifique in Paris . SEE ALSO
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