Indian mathematics

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Indian mathematics emerged in the Indian subcontinent^[1] from 1200 BCE^[2] until the end of the 18th century, after which Indian mathematicians were directly part of the development of global mathematics. In the classical period of Indian mathematics (400 CE to 1600 CE), important contributions were made by scholars like Aryabhata, Brahmagupta, Mahāvīra, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji. The decimal number system in worldwide use today was first recorded in Indian mathematics.^[3] Indian mathematicians made early contributions to the study of the concept of zero as a number,^[4] negative numbers,^[5] arithmetic, and algebra.^[6] In addition, trigonometry^[7] was further advanced in India, and, in particular, the modern definitions of sine and cosine were developed there.^[8] These mathematical concepts were transmitted to the Middle East, China, and Europe^[6] 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 sutras 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 so important as the ideas involved.^[1][9] 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) and is likely from the 7th century CE,^[10][11] or may be a composite text representing three or more stages of development between the 3rd and 10th centuries CE.^[12][13][14]

A later landmark in Indian mathematics was the development of the series expansions for trigonometric functions (sine, cosine, and arc tangent) by mathematicians of the Kerala school in the 15th 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).^[15] 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.^{[16][17][18][19]}

Prehistory

Excavations at Harappa, Mohenjo-daro and other sites of the Indus Valley Civilisation have uncovered evidence of the use of "practical mathematics". The people of the Indus Valley Civilization manufactured bricks whose dimensions were in the proportion 4:2:1, considered favourable for the stability of a brick structure. They used a standardised system of weights based on the ratios: 1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200, and 500, with the unit weight equaling approximately 28 grams (and approximately equal to the English ounce or Greek uncia). They mass-produced weights in regular geometrical shapes, which included hexahedra, barrels, cones, and cylinders, thereby demonstrating knowledge of basic geometry.^[20]

The inhabitants of Indus civilisation also tried to standardise 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 or 3.4 centimetres) was divided into ten equal parts. Bricks manufactured in ancient Mohenjo-daro often had dimensions that were integral multiples of this unit of length.^[21][22]

Hollow cylindrical objects made of shell and found at Lothal (2200 BCE) and Dholavira are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.^[23]

Vedic period

Samhitas and Brahmanas

The religious texts of the Vedic Period provide evidence for the use of large numbers. By the time of the Yajurvedasaṃhitā- (1200–900 BCE), numbers as high as 10¹² were being included in the texts.^[2] For example, the mantra (sacrificial formula) at the end of the annahoma ("food-oblation rite") performed during the aśvamedha, and uttered just before-, during-, and just after sunrise, invokes powers of ten from a hundred to a trillion:^[2]

Hail to śata ("hundred," 10²), hail to sahasra ("thousand," 10³), hail to ayuta ("ten thousand," 10⁴), hail to niyuta ("hundred thousand," 10⁵), hail to prayuta ("million," 10⁶), hail to arbuda ("ten million," 10⁷), hail to nyarbuda ("hundred million," 10⁸), hail to samudra ("billion," 10⁹, literally "ocean"), hail to madhya ("ten billion," 10¹⁰, literally "middle"), hail to anta ("hundred billion," 10¹¹, lit., "end"), hail to parārdha ("one trillion," 10¹² lit., "beyond parts"), hail to the dawn (uṣas), hail to the twilight (vyuṣṭi), hail to the one which is going to rise (udeṣyat), hail to the one which is rising (udyat), hail to the one which has just risen (udita), hail to svarga (the heaven), hail to martya (the world), hail to all.^[2]

The solution to partial fraction was known to the Rigvedic People as states in the purush Sukta (RV 10.90.4):

With three-fourths Puruṣa went up: one-fourth of him again was here.

The Satapatha Brahmana (ca. 7th century BCE) contains rules for ritual geometric constructions that are similar to the Sulba Sutras.^[24]

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.^[25] Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"^[26] 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.^[26]

According to (Hayashi 2005, p. 363), 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 Babylonians."

The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) produce separately."^[27]

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.^[27]

They contain lists of Pythagorean triples,^[28] which are particular cases of Diophantine equations.^[29] They also contain statements (that with hindsight we know to be approximate) about squaring the circle and "circling the square."^[30]

Baudhayana (c. 8th century BCE) composed the Baudhayana Sulba Sutra, the best-known Sulba Sutra, which contains examples of simple Pythagorean triples, such as: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25), and (12, 35, 37),^[31] 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."^[31] 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."^[31] Baudhayana gives a formula for the square root of two:^[32]

${\displaystyle {\sqrt {2}}\approx 1+{\frac {1}{3}}+{\frac {1}{3\cdot 4}}-{\frac {1}{3\cdot 4\cdot 34}}=1.4142156\ldots }$

The formula is accurate up to five decimal places, the true value being 1.41421356...^[33] This formula is similar in structure to the formula found on a Mesopotamian tablet^[34] from the Old Babylonian period (1900–1600 BCE):^[32]

${\displaystyle {\sqrt {2}}\approx 1+{\frac {24}{60}}+{\frac {51}{60^{2}}}+{\frac {10}{60^{3}}}=1.41421297\ldots }$

which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places (after rounding).

According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written ca. 1850 BCE^[35] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,^[36] 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."^[37] 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 would not correspond directly to the overall knowledge on the topic at that time. Since, unfortunately, no other contemporaneous sources have been found it may never be possible to settle this issue satisfactorily.^[37]

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, Pāṇini (c. 520–460 BCE). His grammar includes early use of Boolean logic, of the null operator, and of context free grammars, and includes a precursor of the Backus–Naur form (used in the description programming languages).^[38][39]

Pingala (300 BCE – 200 BCE)

Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is Pingala (piṅgalá) (fl. 300–200 BCE), a music theorist who authored the Chhandas Shastra (chandaḥ-śāstra, also Chhandas Sutra chhandaḥ-sūtra), a Sanskrit treatise on prosody. There is evidence that in his work on the enumeration of syllabic combinations, Pingala stumbled upon both Pascal's triangle and binomial coefficients, although he did not have knowledge of the binomial theorem itself.^[40][41] Pingala's work also contains the basic ideas of Fibonacci numbers (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, ...^[40]

The text also indicates that Pingala was aware of the combinatorial identity:^[41]

${\displaystyle {n \choose 0}+{n \choose 1}+{n \choose 2}+\cdots +{n \choose n-1}+{n \choose n}=2^{n}}$

Kātyāyana

Kātyāyana (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.

Jain mathematics (400 BCE – 200 CE)

Although Jainism as a religion and philosophy predates its most famous exponent, the great Mahavira (6th century BCE), most Jain texts on mathematical topics were composed after the 6th century BCE. Jain 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 Jain 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 infinities led them to classify numbers into three classes: enumerable, innumerable and infinite. Not content with a simple notion of infinity, they went on to define five different types of infinity: the infinite in one direction, the infinite in two directions, the infinite in area, the infinite everywhere, and the infinite perpetually. In addition, Jain mathematicians devised notations for simple powers (and exponents) of numbers like squares and cubes, which enabled them to define simple algebraic equations (beejganita samikaran). Jain mathematicians were apparently also the first to use the word shunya (literally void in Sanskrit) to refer to zero. More than a millennium later, their appellation became the English word "zero" after a tortuous journey of translations and transliterations from India to Europe.

In addition to Surya Prajnapti, important Jain works on mathematics included the Sthananga Sutra (c. 300 BCE – 200 CE); the Anuyogadwara Sutra (c. 200 BCE – 100 CE); and the Satkhandagama (c. 2nd century CE). Important Jain 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 Jain philosophy and metaphysics, composed a mathematical work called Tattwarthadhigama-Sutra Bhashya.

Oral Tradition

Mathematicians of ancient and early medieval India were almost all Sanskrit pandits (paṇḍita "learned man"),^[42] who were trained in Sanskrit language and literature, and possessed "a common stock of knowledge in grammar (vyākaraṇa), exegesis (mīmāṃsā) and logic (nyāya)."^[42] Memorisation of "what is heard" (śruti in Sanskrit) through recitation played a major role in the transmission of sacred texts in ancient India. Memorisation 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."^[43]

Styles of memorisation

Prodigious energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity.^[44] For example, memorisation of the sacred Vedas 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 in the original order.^[45] The recitation thus proceeded as:

word1word2, word2word1, word1word2; word2word3, word3word2, word2word3; ...
In another form of recitation, dhvaja-pāṭha^[45] (literally "flag recitation") a sequence of N words were recited (and memorised) by pairing the first two and last two words and then proceeding as:

word₁word₂, word_{N − 1}word_N; word₂word₃, word_{N − 3}word_{N − 2}; ..; word_{N − 1}word_N, word₁word₂;
The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to (Filliozat 2004, p. 139), took the form:

word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4; ...
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.^[45] Similar methods were used for memorising mathematical texts, whose transmission remained exclusively oral until the end of the Vedic period (ca. 500 BCE).

The Sutra genre

Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred Vedas, which took the form of works called Vedāṇgas, or, "Ancillaries of the Veda" (7th–4th century BCE).^[46] The need to conserve the sound of sacred text by use of śikṣā (phonetics) and chhandas (metrics); 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 (astrology), gave rise to the six disciplines of the Vedāṇgas.^[46] 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.^[46]

Extreme brevity was achieved through multiple means, which included using ellipsis "beyond the tolerance of natural language,"^[46] using technical names instead of longer descriptive names, abridging lists by only mentioning the first and last entries, and using markers and variables.^[46] 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.^[47] The brevity achieved in a sūtra is demonstrated in the following example from the Baudhāyana Śulba Sūtra (700 BCE).

The design of the domestic fire altar in the Śulba Sūtra

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.^[48] 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 [cord] in three.
II.65. In another layer one places the [bricks] North-pointing.^[48]

According to (Filliozat 2004, p. 144), 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.^[48]

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.^[49]

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.^[50] However, according to (Hayashi 2003, p. 123), "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:^[50]

• Rule ('sūtra') in verse by Āryabhaṭa
• Commentary by Bhāskara I, consisting of:
  • Elucidation of rule (derivations were still rare then, but became more common later)
  • Example (uddeśaka) usually in verse.
  • Setting (nyāsa/sthāpanā) of the numerical data.
  • Working (karana) of the solution.
  • Verification (pratyayakaraṇa, literally "to make conviction") of the answer. These became rare by the 13th century, derivations or proofs being favoured by then.^[50]

Typically, for any mathematical topic, students in ancient India first memorised the sūtras, which, as explained earlier, were "deliberately inadequate"^[49] 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 characterise astronomical computations as "dust work" (Sanskrit: dhulikarman).^[51]

Numerals and the decimal number system

It is well known that the decimal place-value system in use today was first recorded in India, then transmitted to the Islamic world, and eventually to Europe.^[52] The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers.^[52] However, how, when, and where the first decimal place value system was invented is not so clear.^[53]

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 4th century BCE to the 4th century CE. Almost contemporaneously, another script, the Brāhmī script, 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.^[54]

The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE.^[55] A copper plate from Gujarat, India mentions the date 595 CE, written in a decimal place value notation, although there is some doubt as to the authenticity of the plate.^[55] Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.^[55]

There are older textual sources, although the extant manuscript copies of these texts are from much later dates.^[56] Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE.^[56] Discussing the counting pits of merchants, Vasumitra remarks, "When [the same] clay counting-piece is in the place of units, it is denoted as one, when in hundreds, one hundred."^[56] Although such references seem to imply that his readers had knowledge of a decimal place value representation, the "brevity of their allusions and the ambiguity of their dates, however, do not solidly establish the chronology of the development of this concept."^[56]

A third decimal representation was employed in a verse composition technique, later labelled Bhuta-sankhya (literally, "object numbers") used by early Sanskrit authors of technical books.^[57] Since many early technical works were composed in verse, numbers were often represented by objects in the natural or religious world that correspondence to them; this allowed a many-to-one correspondence for each number and made verse composition easier.^[57] According to Plofker 2009, the number 4, for example, could be represented by the word "Veda" (since there were four of these religious texts), the number 32 by the word "teeth" (since a full set consists of 32), and the number 1 by "moon" (since there is only one moon).^[57] So, Veda/teeth/moon would correspond to the decimal numeral 1324, as the convention for numbers was to enumerate their digits from right to left.^[57] The earliest reference employing object numbers is a ca. 269 CE Sanskrit text, Yavanajātaka (literally "Greek horoscopy") of Sphujidhvaja, a versification of an earlier (ca. 150 CE) Indian prose adaptation of a lost work of Hellenistic astrology.^[58] Such use seems to make the case that by the mid-3rd century CE, the decimal place value system was familiar, at least to readers of astronomical and astrological texts in India.^[57]

It has been hypothesized that the Indian decimal place value system was based on the symbols used on Chinese counting boards from as early as the middle of the first millennium BCE.^[59] According to Plofker 2009,

These counting boards, like the Indian counting pits, ..., had a decimal place value structure ... Indians may well have learned of these decimal place value "rod numerals" from Chinese Buddhist pilgrims or other travelers, or they may have developed the concept independently from their earlier non-place-value system; no documentary evidence survives to confirm either conclusion."^[59]

Bakhshali Manuscript

The oldest extant mathematical manuscript in India is the Bakhshali Manuscript, a birch bark manuscript written in "Buddhist hybrid Sanskrit"^[11] in the Śāradā script, which was used in the northwestern region of the Indian subcontinent between the 8th and 12th centuries CE.^[60] The manuscript was discovered in 1881 by a farmer while digging in a stone enclosure in the village of Bakhshali, near Peshawar (then in British India and now in Pakistan). Of unknown authorship and now preserved in the Bodleian Library in Oxford University, the manuscript has been variously dated—sometimes as early as the "early centuries of the Christian era."^[61] The 7th century CE is now considered a plausible date.^[62]

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.^[60] 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."^[60] Many of its problems are of a category known as 'equalisation 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.^[63]

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.^[63]

In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from 224-383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together.^[64][65][66]

Classical period (400–1600)

This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji give broader and clearer shape to many branches of mathematics. Their contributions would spread to Asia, the Middle East, and eventually to Europe. Unlike Vedic mathematics, their works included both astronomical and mathematical contributions. In fact, mathematics of that period was included in the 'astral science' (jyotiḥśāstra) and consisted of three sub-disciplines: mathematical sciences (gaṇita or tantra), horoscope astrology (horā or jātaka) and divination (saṃhitā).^[51] This tripartite division is seen in Varāhamihira's 6th century compilation—Pancasiddhantika^[67] (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.^[51]

Fifth and sixth centuries

Surya Siddhanta

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry.^{[citation needed]} Because it contains many words of foreign origin, some authors consider that it was written under the influence of Mesopotamia and Greece.^{[68][better source needed]}

This ancient text uses the following as trigonometric functions for the first time:^{[citation needed]}

• Sine (Jya).
• Cosine (Kojya).
• Inverse sine (Otkram jya).

It also contains the earliest uses of:^{[citation needed]}

• Tangent.
• Secant.

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:

• Quadratic equations
• Trigonometry
• The value of π, correct to 4 decimal places.

Aryabhata also wrote the Arya Siddhanta, which is now lost. Aryabhata's contributions include:
Trigonometry:

• Introduced the trigonometric functions.
• Defined the sine (jya) as the modern relationship between half an angle and half a chord.
• Defined the cosine (kojya).
• Defined the versine (utkrama-jya).
• Defined the inverse sine (otkram jya).
• Gave methods of calculating their approximate numerical values.
• Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
• Contains the trigonometric formula sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225)sin nx.
• Spherical trigonometry.

Arithmetic:

• Continued fractions.

Algebra:

• Solutions of simultaneous quadratic equations.
• Whole number solutions of linear equations by a method equivalent to the modern method.
• General solution of the indeterminate linear equation .

Mathematical astronomy:

• Accurate calculations for astronomical constants, such as the:
  • Solar eclipse.
  • Lunar eclipse.
  • The formula for the sum of the cubes, which was an important step in the development of integral calculus.^[69]

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:

• ${\displaystyle \sin ^{2}(x)+\cos ^{2}(x)=1}$
• ${\displaystyle \sin(x)=\cos \left({\frac {\pi }{2}}-x\right)}$
• ${\displaystyle {\frac {1-\cos(2x)}{2}}=\sin ^{2}(x)}$

Seventh and eighth centuries

Brahmagupta's theorem states that AF = FD.

In the 7th century, two separate fields, arithmetic (which included measurement) and algebra, began to emerge in Indian mathematics. The two fields would later be called pāṭī-gaṇita (literally "mathematics of algorithms") and bīja-gaṇita (lit. "mathematics of seeds," with "seeds"—like the seeds of plants—representing unknowns with the potential to generate, in this case, the solutions of equations).^[70] Brahmagupta, in his astronomical work Brāhma Sphuṭa Siddhānta (628 CE), included two chapters (12 and 18) devoted to these fields. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).^[71] In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral:^[71]

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 generalisation of Heron's formula), as well as a complete description of rational triangles (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

${\displaystyle A={\sqrt {(s-a)(s-b)(s-c)(s-d)}}\,}$

where s, the semiperimeter, given by ${\displaystyle s={\frac {a+b+c+d}{2}}.}$
Brahmagupta's Theorem on rational triangles: A triangle with rational sides ${\displaystyle a,b,c}$ and rational area is of the form:

${\displaystyle a={\frac {u^{2}}{v}}+v,\ \ b={\frac {u^{2}}{w}}+w,\ \ c={\frac {u^{2}}{v}}+{\frac {u^{2}}{w}}-(v+w)}$

for some rational numbers ${\displaystyle u,v,}$ and ${\displaystyle w}$.^[72]

Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers^[71] and is considered the first systematic treatment of the subject. The rules (which included ${\displaystyle a+0=\ a}$ and ${\displaystyle a\times 0=0}$) were all correct, with one exception: ${\displaystyle {\frac {0}{0}}=0}$.^[71] Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:

${\displaystyle \ ax^{2}+bx=c}$

To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.^[73]

This is equivalent to:

${\displaystyle x={\frac {{\sqrt {4ac+b^{2}}}-b}{2a}}}$

Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,^[74]

${\displaystyle \ x^{2}-Ny^{2}=1,}$

where ${\displaystyle N}$ is a nonsquare integer. He did this by discovering the following identity:^[74]
Brahmagupta's Identity: ${\displaystyle \ (x^{2}-Ny^{2})(x'^{2}-Ny'^{2})=(xx'+Nyy')^{2}-N(xy'+x'y)^{2}}$ which was a generalisation of an earlier identity of Diophantus:^[74] Brahmagupta used his identity to prove the following lemma:^[74]

Lemma (Brahmagupta): If ${\displaystyle x=x_{1},\ \ y=y_{1}\ \ }$ is a solution of ${\displaystyle \ \ x^{2}-Ny^{2}=k_{1},}$ and, ${\displaystyle x=x_{2},\ \ y=y_{2}\ \ }$ is a solution of ${\displaystyle \ \ x^{2}-Ny^{2}=k_{2},}$, then:

${\displaystyle x=x_{1}x_{2}+Ny_{1}y_{2},\ \ y=x_{1}y_{2}+x_{2}y_{1}\ \ }$ is a solution of ${\displaystyle \ x^{2}-Ny^{2}=k_{1}k_{2}}$

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 ${\displaystyle \ x^{2}-Ny^{2}=k}$ has an integer solution for any one of ${\displaystyle \ k=\pm 4,\pm 2,-1}$ then Pell's equation:

${\displaystyle \ x^{2}-Ny^{2}=1}$

also has an integer solution.^[75]

Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:^[74]

Example (Brahmagupta): Find integers ${\displaystyle \ x,\ y\ }$ such that:

${\displaystyle \ x^{2}-92y^{2}=1}$

In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician."^[74] The solution he provided was:

${\displaystyle \ x=1151,\ y=120}$

Bhaskara I

Bhaskara I (c. 600–680) expanded the work of Aryabhata in his books titled Mahabhaskariya, Aryabhatiya-bhashya and Laghu-bhaskariya. He produced:

• Solutions of indeterminate equations.
• A rational approximation of the sine function.
• A formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.

Ninth to twelfth centuries

Virasena

Virasena (8th century) was a Jain mathematician in the court of Rashtrakuta King Amoghavarsha of Manyakheta, Karnataka. He wrote the Dhavala, a commentary on Jain mathematics, which:

• Deals with the concept of ardhaccheda, the number of times a number could be halved, and lists various rules involving this operation. This coincides with the binary logarithm when applied to powers of two,^[76][77] but differs on other numbers, more closely resembling the 2-adic order.
• The same concept for base 3 (trakacheda) and base 4 (caturthacheda).

Virasena also gave:

• The derivation of the volume of a frustum by a sort of infinite procedure.

It is thought that much of the mathematical material in the Dhavala can attributed to previous writers, especially Kundakunda, Shamakunda, Tumbulura, Samantabhadra and Bappadeva and date who wrote between 200 and 600 CE.^[77]

Mahavira

Mahavira Acharya (c. 800–870) from Karnataka, the last of the notable Jain 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:

• Zero
• Squares
• Cubes
• square roots, cube roots, and the series extending beyond these
• Plane geometry
• Solid geometry
• Problems relating to the casting of shadows
• Formulae derived to calculate the area of an ellipse and quadrilateral inside a circle.

Mahavira also:

• Asserted that the square root of a negative number did not exist
• Gave the sum of a series whose terms are squares of an arithmetical progression, and gave empirical rules for area and perimeter of an ellipse.
• Solved cubic equations.
• Solved quartic equations.
• Solved some quintic equations and higher-order polynomials.
• Gave the general solutions of the higher order polynomial equations:
  • ${\displaystyle \ ax^{n}=q}$
  • ${\displaystyle a{\frac {x^{n}-1}{x-1}}=p}$
• Solved indeterminate quadratic equations.
• Solved indeterminate cubic equations.
• Solved indeterminate higher order equations.

Shridhara

Shridhara (c. 870–930), who lived in Bengal, wrote the books titled Nav Shatika, Tri Shatika and Pati Ganita. He gave:

• A good rule for finding the volume of a sphere.
• The formula for solving quadratic equations.

The Pati Ganita is a work on arithmetic and measurement. It deals with various operations, including:

• Elementary operations
• Extracting square and cube roots.
• Fractions.
• Eight rules given for operations involving zero.
• Methods of summation of different arithmetic and geometric series, which were to become standard references in later works.

Manjula

Aryabhata's differential equations were elaborated in the 10th century by Manjula (also Munjala), who realised that the expression^[78]

${\displaystyle \ \sin w'-\sin w}$

could be approximately expressed as

${\displaystyle \ (w'-w)\cos w}$

He understood the concept of differentiation after solving the differential equation that resulted from substituting this expression into Aryabhata's differential equation.^[78]

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:

• Numerical mathematics (Ank Ganit).
• Algebra.
• Solutions of indeterminate equations (kuttaka).

Shripati

Shripati Mishra (1019–1066) wrote the books Siddhanta Shekhara, a major work on astronomy in 19 chapters, and Ganit Tilaka, an incomplete arithmetical treatise in 125 verses based on a work by Shridhara. He worked mainly on:

• Permutations and combinations.
• General solution of the simultaneous indeterminate linear equation.

He was also the author of Dhikotidakarana, a work of twenty verses on:

• Solar eclipse.
• Lunar eclipse.

The Dhruvamanasa is a work of 105 verses on:

• Calculating planetary longitudes
• eclipses.
• planetary transits.

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:

• Interest computation
• Arithmetical and geometrical progressions
• Plane geometry
• Solid geometry
• The shadow of the gnomon
• Solutions of combinations
• Gave a proof for division by zero being infinity.

Algebra:

• The recognition of a positive number having two square roots.
• Surds.
• Operations with products of several unknowns.
• The solutions of:
  • Quadratic equations.
  • Cubic equations.
  • Quartic equations.
  • Equations with more than one unknown.
  • Quadratic equations with more than one unknown.
  • The general form of Pell's equation using the chakravala method.
  • The general indeterminate quadratic equation using the chakravala method.
  • Indeterminate cubic equations.
  • Indeterminate quartic equations.
  • Indeterminate higher-order polynomial equations.

Geometry:

• Gave a proof of the Pythagorean theorem.

Calculus:

• Conceived of differential calculus.
• Discovered the derivative.
• Discovered the differential coefficient.
• Developed differentiation.
• Stated Rolle's theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis).
• Derived the differential of the sine function.
• Computed π, correct to five decimal places.
• Calculated the length of the Earth's revolution around the Sun to 9 decimal places.

Trigonometry:

• Developments of spherical trigonometry
• The trigonometric formulas:
  • ${\displaystyle \ \sin(a+b)=\sin(a)\cos(b)+\sin(b)\cos(a)}$
  • ${\displaystyle \ \sin(a-b)=\sin(a)\cos(b)-\sin(b)\cos(a)}$

Kerala mathematics (1300–1600)

The Kerala school of astronomy and mathematics was founded by Madhava of Sangamagrama in Kerala, South India and included among its members: Parameshvara, 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 functions, 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 Yuktibhāṣā (c.1500–c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on Tantrasangraha.^[79]
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 an achievement. However, the Kerala School did not invent calculus,^[80] because, while they were able to develop Taylor series expansions for the important trigonometric functions, differentiation, term by term integration, convergence tests, iterative methods for solutions of non-linear equations, and the theory that the area under a curve is its integral, they developed neither a theory of differentiation or integration, nor the fundamental theorem of calculus.^[81] The results obtained by the Kerala school include:

• The (infinite) geometric series: ${\displaystyle {\frac {1}{1-x}}=1+x+x^{2}+x^{3}+x^{4}+\cdots {\text{ for }}|x|<1 annotation="">$^[82] This formula was already known, for example, in the work of the 10th-century Arab mathematician Alhazen (the Latinised form of the name Ibn Al-Haytham (965–1039)).^[83]
• A semi-rigorous proof (see "induction" remark below) of the result: ${\displaystyle 1^{p}+2^{p}+\cdots +n^{p}\approx {\frac {n^{p+1}}{p+1}}}$ for large n. This result was also known to Alhazen.^[79]
• Intuitive use of mathematical induction, however, the inductive hypothesis was not formulated or employed in proofs.^[79]
• Applications of ideas from (what was to become) differential and integral calculus to obtain (Taylor–Maclaurin) infinite series for sin x, cos x, and arctan x.^[80] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:^[79]

${\displaystyle r\arctan \left({\frac {y}{x}}\right)={\frac {1}{1}}\cdot {\frac {ry}{x}}-{\frac {1}{3}}\cdot {\frac {ry^{3}}{x^{3}}}+{\frac {1}{5}}\cdot {\frac {ry^{5}}{x^{5}}}-\cdots ,{\text{ where }}y/x\leq 1.}$

${\displaystyle \sin x=x-x{\frac {x^{2}}{(2^{2}+2)r^{2}}}+x{\frac {x^{2}}{(2^{2}+2)r^{2}}}\cdot {\frac {x^{2}}{(4^{2}+4)r^{2}}}-\cdots }$

${\displaystyle r-\cos x=r{\frac {x^{2}}{(2^{2}-2)r^{2}}}-r{\frac {x^{2}}{(2^{2}-2)r^{2}}}{\frac {x^{2}}{(4^{2}-4)r^{2}}}+\cdots ,}$

where, for r = 1, the series reduces to the standard power series for these trigonometric functions, for example:

${\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+{\frac {x^{5}}{5!}}-{\frac {x^{7}}{7!}}+\cdots }$

and

${\displaystyle \cos x=1-{\frac {x^{2}}{2!}}+{\frac {x^{4}}{4!}}-{\frac {x^{6}}{6!}}+\cdots }$

• Use of rectification (computation of length) of the arc of a circle to give a proof of these results. (The later method of Leibniz, using quadrature, i.e. computation of area under the arc of the circle, was not used.)^[79]
• Use of the series expansion of ${\displaystyle \arctan x}$ to obtain the Leibniz formula for π:^[79]

${\displaystyle {\frac {\pi }{4}}=1-{\frac {1}{3}}+{\frac {1}{5}}-{\frac {1}{7}}+\cdots }$

• A rational approximation of error for the finite sum of their series of interest. For example, the error, ${\displaystyle f_{i}(n+1)}$, (for n odd, and i = 1, 2, 3) for the series:

${\displaystyle {\frac {\pi }{4}}\approx 1-{\frac {1}{3}}+{\frac {1}{5}}-\cdots +(-1)^{(n-1)/2}{\frac {1}{n}}+(-1)^{(n+1)/2}f_{i}(n+1)}$

${\displaystyle {\text{where }}f_{1}(n)={\frac {1}{2n}},\ f_{2}(n)={\frac {n/2}{n^{2}+1}},\ f_{3}(n)={\frac {(n/2)^{2}+1}{(n^{2}+5)n/2}}.}$

• Manipulation of error term to derive a faster converging series for ${\displaystyle \pi }$:^[79]

${\displaystyle {\frac {\pi }{4}}={\frac {3}{4}}+{\frac {1}{3^{3}-3}}-{\frac {1}{5^{3}-5}}+{\frac {1}{7^{3}-7}}-\cdots }$

• Using the improved series to derive a rational expression,^[79] 104348/33215 for π correct up to nine decimal places, i.e. 3.141592653.
• Use of an intuitive notion of limit to compute these results.^[79]
• A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.^[81] However, they did not formulate the notion of a function, or have knowledge of the exponential or logarithmic functions.

The works of the Kerala school were first written up for the Western world by Englishman C.M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries."^[84]

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 Yuktibhāṣā given in two papers,^[85][86] a commentary on the Yuktibhāṣā's proof of the sine and cosine series^[87] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).^[88][89]

The Kerala mathematicians included Narayana Pandit^{[dubious – discuss]} (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 Sangamagrama (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.

Parameshvara (c. 1370–1460) wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His Lilavati Bhasya, a commentary on Bhaskara II's Lilavati, contains one of his important discoveries: a version of the mean value theorem. 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:

${\displaystyle {\begin{aligned}&x+y=a,\ x-y=b,\ xy=c,x^{2}+y^{2}=d,\\[8pt]&x^{2}-y^{2}=e,\ x^{3}+y^{3}=f,\ x^{3}-y^{3}=g\end{aligned}}}$

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-bhāṣā (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's take on "Ethnomathematics":

[Their work] takes on board some of the objections raised about the classical Eurocentric trajectory. The awareness [of 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 civilisations – 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"^[90]

The historian of mathematics, Florian Cajori, suggested that he and others "suspect that Diophantus got his first glimpse of algebraic knowledge from India."^[91] However, he also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".^[92]

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 and formulas for truncation error) 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.^[93] 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.^[93] 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."^[80][94]

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.^[81] However, they were not able, 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."^[81] The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own;^[81] 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."^[81] This is an active area of current research, especially in the manuscript collections of Spain and Maghreb. This research is being pursued, among other places, at the Centre National de Recherche Scientifique in Paris.^[81]

Science in the medieval Islamic world

From Wikipedia, the free encyclopedia

 

The Tusi couple, a mathematical device invented by Nasir al-Din Tusi in 1247 to model the not perfectly circular motions of the planets

Science in the medieval Islamic world was the science developed and practised during the Islamic Golden Age under the Umayyads of Córdoba, the Abbadids of Seville, the Samanids, the Ziyarids, the Buyids in Persia, the Abbasid Caliphate and beyond, spanning the period c. 800 to 1250. Islamic scientific achievements encompassed a wide range of subject areas, especially astronomy, mathematics, and medicine. Other subjects of scientific inquiry included alchemy and chemistry, botany, geography and cartography, ophthalmology, pharmacology, physics, and zoology.

Medieval Islamic science had practical purposes as well as the goal of understanding. For example, astronomy was useful for determining the Qibla, the direction in which to pray, botany had practical application in agriculture, as in the works of Ibn Bassāl and Ibn al-'Awwam, and geography enabled Abu Zayd al-Balkhi to make accurate maps. Islamic mathematicians such as al-Khwarizmi, Avicenna and Jamshid al-Kashi developed methods in algebra, geometry and trigonometry. Islamic doctors described diseases like smallpox and measles and challenged classical Greek medical theory. Al-Biruni, Avicenna and others described the preparation of hundreds of drugs made from medicinal plants and chemical compounds. Islamic physicists studied optics and mechanics (as well as astronomy) and criticised Aristotle's view of motion.

The significance of medieval Islamic science has been debated by historians. The traditionalist view holds that it lacked innovation, and was mainly important for handing on ancient knowledge to medieval Europe. The revisionist view holds that it constituted a scientific revolution. Whatever the case, science flourished across a wide area around the Mediterranean and further afield, for several centuries, in a wide range of institutions.

Context

Islamic expansion:

  under Muhammad, 622–632

  under Rashidun caliphs, 632–661

  under Umayyad caliphs, 661–750

 

The Abbasid Caliphate, 750–1261 (and later in Egypt) at its height, c. 850

The Islamic era began in 622. Islamic armies conquered Arabia, Egypt and Mesopotamia, eventually displacing the Persian and Byzantine Empires from the region. Within a century, Islam had reached the area of present-day Portugal in the west and Central Asia in the east. The Islamic Golden Age (roughly between 692 and 945) spanned the periods of the Umayyad Caliphate (661-750) and, in particular, the early phase of the succeeding Abbasid Caliphate (750-1258), with stable political structures and flourishing trade. Major religious and cultural works of the Islamic empire were translated into Arabic. Islamic culture inherited Greek, Indic, Assyrian and Persian influences. A new common civilisation formed, based on Islam. An era of high culture and innovation ensued, with rapid growth in population and cities. The Arab Agricultural Revolution in the countryside brought more crops and improved agricultural technology, especially irrigation. This supported the larger population and enabled culture to flourish.^[1][2] From the 8th century onwards, scholars such as Al-Kindi^[3] translated Indian, Assyrian, Sassanian (Persian) and Greek knowledge, including the works of Aristotle, into Arabic. These translations supported advances by scientists across the Islamic world.^[4]

Islamic science survived the initial Christian reconquest of Spain, including the fall of Seville in 1248, as work continued in the eastern centres (such as in Persia). After the completion of the Spanish reconquest in 1492, the Islamic world went into an economic and cultural decline.^[2] The Abbasid caliphate was followed by the Ottoman Empire (c. 1299-1922), centred in Turkey, and the Safavid Empire (1501-1736), centred in Persia, where work in the arts and sciences continued.^[5]

Fields of inquiry

Islamic scientific achievements encompass a wide range of subject areas, especially mathematics, astronomy, and medicine.^[4] Other subjects of scientific inquiry included physics, alchemy and chemistry, ophthalmology, and geography and cartography.^[6]

Alchemy and chemistry

Alchemy was already well established before the rise of Islam. It was based on the belief that substances were made up of the four Aristotelian elements, fire, earth, air, and water in different proportions. Alchemists supposed that gold was the noblest metal, and that other metals formed a series down to the basest, such as lead. They believed, too, that a fifth element, the elixir, could transform a base metal into gold. Jabir ibn Hayyan (8th–9th centuries) wrote on alchemy, based on his own experiments. He described laboratory techniques and experimental methods that would continue to be used when alchemy had transformed into chemistry. Ibn Hayyan identified many substances including sulphuric and nitric acids. He described processes such as sublimation, reduction and distillation. He made use of equipment such as the alembic and the retort stand.^[7][8][9]

Astronomy and cosmology

al-Biruni's explanation of the phases of the moon

Astronomy was a major discipline within Islamic science. Effort was devoted both towards understanding the nature of the cosmos and to practical purposes. One of these was determining the Qibla, the direction in which to pray. Another was astrology, predicting events affecting human life and selecting suitable times for actions such as going to war or founding a city.^[10] Al-Battani (850–922) accurately determined the length of the solar year. He contributed to the Tables of Toledo, used by astronomers to predict the movements of the sun, moon and planets across the sky. Some of his astronomic tables were later used by Copernicus.^[11]

Al-Zarqali (1028–1087) developed a more accurate astrolabe, used for centuries afterwards. He constructed a water clock in Toledo. He discovered that the Sun's apogee moves slowly relative to the fixed stars, and obtained a good estimate of its motion^[12] for its rate of change.^[13] Nasir al-Din al-Tusi (1201–1274) wrote an important revision to Ptolemy's celestial model. When he became Helagu's astrologer, he was given an observatory and gained access to Chinese techniques and observations. He developed trigonometry as a separate field, and compiled the most accurate astronomical tables available up to that time.^[14]

Quince, Cypress, and Sumac trees, in Zakariya al-Qazwini's 13th century Wonders of Creation

Botany

The study of the natural world extended to a detailed examination of plants. The work done was directly useful in the unprecedented growth of pharmacology across the Islamic world. Al-Dinawari popularised botany in the Islamic world with his six-volume Kitab al-Nabat (Book of Plants). Only volumes 3 and 5 have survived, with part of volume 6 reconstructed from quoted passages. In what survives, 637 plants are described in alphabetical order from the letters sin to ya, so the whole book must have covered several thousand kinds of plants. Al-Dinawari described the phases of plant growth and the production of flowers and fruit. Zakariya al-Qazwini's thirteenth century encyclopedia ʿAjā'ib al-makhlūqāt (The Wonders of Creation) contained, among many other topics, both realistic botany and fantastic accounts. For example, he described trees which grew birds on their twigs in place of leaves, but which could only be found in the far-distant British Isles.^[15][16][17] The use and cultivation of plants was documented in the 11th century by Muhammad bin Ibrāhīm Ibn Bassāl of Toledo in his book Dīwān al-filāha (The Court of Agriculture), and Ibn al-'Awwam al-Ishbīlī of Seville in his 12th century book Kitāb al-Filāha (Treatise on Agriculture).^[18] Ibn Bassāl had travelled widely across the Islamic world, returning with a detailed knowledge of agronomy. His practical and systematic book describes over 180 plants and how to propagate and care for them. It covered leaf and root vegetables, herbs, spices and trees.^[19] Abū l-Khayr described in minute detail how olive trees should be grown, grafted, treated for disease, and harvested. He gave similar detail for crops such as cotton.^[18]

Geography and cartography

Surviving fragment of the first World Map of Piri Reis (1513)

The swift spread of Islam across Western Asia and North Africa encouraged an unprecedented growth in trade and travel by land and sea as far away as Southeast Asia, China, much of Africa, Scandinavia and even Iceland. Geographers worked to create increasingly accurate maps of the known world, starting from many existing but fragmentary sources.^[20] Abu Zayd al-Balkhi (850–934), founder of the Balkhī school of cartography in Baghdad, wrote an atlas called Figures of the Regions (Suwar al-aqalim).^[21] Al-Biruni (973–1048) measured the radius of the earth using a new method. It involved observing the height of a mountain at Nandana (now in Pakistan).^[22] Al-Idrisi (1100–1166) created a map of the world for Roger, the Norman King of Sicily. He also wrote the Book of Roger, a geographic study of the peoples, climates, resources and industries of the whole of the world known at that time.^[23] The Ottoman admiral Piri Reis (c. 1470–1553) made a map of the New World and West Africa in 1513. He made use of maps from Greece, Portugal, Muslim sources, and perhaps one made by Christopher Columbus. He was part of a major tradition of
Ottoman cartography.^[24]

 
Modern copy of al-Idrisi's 1154 Tabula
Rogeriana, upside-down, North at top

Mathematics

A page from al-Khwarizmi's Algebra

Islamic mathematicians gathered, organised and clarified the mathematics they inherited from ancient Egypt, Greece, India, Mesopotamia and Persia, and went on to make innovations of their own. Islamic mathematics can be divided into algebra, geometry and arithmetic. Algebra was mainly used for recreation: it had few practical applications at that time. Geometry was studied at different levels. Some texts contain practical geometrical rules for surveying and for measuring figures. Theoretical geometry was a necessary prerequisite for understanding astronomy and optics, and it required years of concentrated work. Early in the Abbasid caliphate, soon after Baghdad was founded in the mid-eighth century, some mathematical knowledge was assimilated from the pre-Islamic Persian tradition in astronomy. Astronomers from India were invited to the court of the caliph in the late eighth century; they explained the rudimentary trigonometrical techniques used in Indian astronomy. Ancient Greek works such as Ptolemy's Almagest and Euclid's Elements were translated into Arabic. By the second half of the ninth century, Islamic mathematicians were already making contributions to the most sophisticated parts of Greek geometry. Islamic mathematics reached its apogee in the Eastern part of the Islamic world between the tenth and twelfth centuries. Most mathematical works were written in Arabic, others in Persian.^[25][26][27]

Omar Khayyam's "Cubic equation and intersection of conic sections"

al-Khwarizmi (8th–9th centuries), considered the greatest mathematician of Islamic civilization, was instrumental in the adoption of the Indian numbering system. He developed algebra, which also had Indian antecedents, introduced methods of simplifying equations, and used Euclidean geometry in his proofs.^[28][29] Ibn Ishaq al-Kindi (801–873) worked on cryptography for the caliphate.^[30] Avicenna (ca. 980–1037) contributed to mathematical techniques such as casting out nines.^[31] Thabit ibn Qurra (835–901) calculated the solution to a chessboard problem involving an exponential series.^[32] al-Farabi (ca. 870–950) attempted to describe, geometrically, the repeating patterns popular in Islamic decorative motifs in his book Spiritual Crafts and Natural Secrets in the Details of Geometrical Figures.^[33] Omar Khayyam (1048–1131), known in the West as a poet, calculated the length of the year to within 5 decimal places. He found geometric solutions to all 13 forms of cubic equations. He developed some quadratic equations still in use.^[34] Jamshid al-Kashi (ca. 1380–1429) is credited with several theorems of trigonometry including the law of cosines, also known as Al-Kashi's Theorem. He is often credited with the invention of decimal fractions, and a method like Horner's to calculate roots. He calculated π correct to 17 significant figures.^[35]

Medicine

A coloured illustration from Mansur's Anatomy, c. 1450

Islamic society paid careful attention to medicine, following a hadith enjoining the preservation of good health. Its physicians inherited knowledge and traditional medical beliefs from the civilisations of classical Greece, Rome, Syria, Persia and India. These included the writings of Hippocrates such as the theory of the four humours, and the theories of Galen.^[36] al-Razi (ca. 854–925/935) identified smallpox and measles, and recognized that fever was a part of the body's defenses. He wrote a 23-volume compendium of Chinese, Indian, Persian, Syriac and Greek medicine. al-Razi questioned the classical Greek medical theory of how the four humours regulate life processes. He challenged Galen's work on several fronts, including the treatment of bloodletting, arguing that it was effective.^[37] al-Zahrawi (936–1013) was a surgeon whose most important surviving work is referred to as al-Tasrif (Medical Knowledge). It is a 30 volume set mainly discussing medical symptoms, treatments, and pharmacology. The last volume, on surgery, describes surgical instruments, supplies, and pioneering procedures.^[38] Avicenna (ca. 980–1037) wrote the major medical textbook, The Canon of Medicine.^[31] ibn al-Nafis (1213–1288) wrote an influential book on medicine; it is believed to have replaced Avicenna's Canon in the Islamic world. He wrote commentaries on Galen and Avicenna's works. One of these commentaries, discovered in 1924, described the circulation of blood through the lungs.^[39][40]

Optics and ophthalmology

The eye according to Hunayn ibn Ishaq, c. 1200

Optics developed rapidly in this period. By the ninth century, there were works on physiological, geometrical and physical optics. Topics covered included mirror reflection. Hunayn ibn Ishaq (809–873) wrote the book Ten Treatises on the Eye; this was influential in the West until the 17th century.^[41] Abbas ibn Firnas (810–887) developed lenses for magnification and the improvement of vision.^[42] Ibn Sahl (ca. 940–1000) discovered the law of refraction known as Snell's law. He used the law to produce the first Aspheric lenses that focused light without geometric aberrations.^[43][44]

In the eleventh century, Ibn al-Haytham (Alhazen, 965–1040) rejected the Greek ideas about vision, whether the Aristotelian tradition that held that the form of the perceived object entered the eye (but not its matter), or that of Euclid and Ptolemy that held that the eye emitted a ray. Al-Haytham proposed in his Book of Optics that vision occurs by way of light rays forming a cone with its vertex at the center of the eye. He suggested that light was reflected from different surfaces in different directions, thus causing objects to look different.^{[45][46][47][48]} He argued further that the mathematics of reflection and refraction needed to be consistent with the anatomy of the eye.^[49]

Pharmacology

Ibn Sina teaching the use of drugs. 15th century Great Canon of Avicenna

Advances in botany and chemistry in the Islamic world encouraged developments in pharmacology. Muhammad ibn Zakarīya Rāzi (Rhazes) (865–915) promoted the medical uses of chemical compounds. Abu al-Qasim al-Zahrawi (Abulcasis) (936–1013) pioneered the preparation of medicines by sublimation and distillation. His Liber servitoris provides instructions for preparing "simples" from which were compounded the complex drugs then used. Sabur Ibn Sahl (d 869), was the first physician to describe a large variety of drugs and remedies for ailments. Al-Biruni (973–1050) wrote the Kitab al-Saydalah (The Book of Drugs), describing in detail the properties of drugs, the role of pharmacy and the duties of the pharmacist. Ibn Sina (Avicenna) described 700 preparations, their properties, mode of action and their indications. He devoted a whole volume to simples in The Canon of Medicine. Works by Masawaih al-Mardini (c. 925–1015) and Ibn al-Wafid (1008–1074) were printed in Latin more than fifty times, appearing as De Medicinis universalibus et particularibus by Mesue the younger, and the Medicamentis simplicibus by Abenguefit respectively. Peter of Abano (1250–1316) translated and added a supplement to the work of al-Mardini under the title De Veneris. Al-Muwaffaq, in the 10th century, wrote The foundations of the true properties of Remedies, describing chemicals such as arsenious oxide and silicic acid. He distinguished between sodium carbonate and potassium carbonate, and drew attention to the poisonous nature of copper compounds, especially copper vitriol, and also lead compounds.^[50][16]

Physics

Self trimming lamp in Ahmad ibn Mūsā ibn Shākir's treatise on mechanical devices, c. 850

The fields of physics studied in this period, apart from optics and astronomy which are described separately, are aspects of mechanics: statics, dynamics, kinematics and motion. In the sixth century John Philoponus rejected the Aristotelian view of motion. He argued instead that an object acquires an inclination to move when it has a motive power impressed on it. In the eleventh century, Ibn Sina adopted roughly the same idea, namely that a moving object has force which is dissipated by external agents like air resistance.^[51] Ibn Sina distinguished between 'force' and 'inclination' (mayl); he claimed that an object gained mayl when the object is in opposition to its natural motion. He concluded that continuation of motion depends on the inclination that is transferred to the object, and that the object remains in motion until the mayl is spent. He also claimed that a projectile in a vacuum would not stop unless it is acted upon. That view is consistent with Newton's first law of motion, on inertia.^[52] As a non-Aristotelian suggestion, it was essentially abandoned until it was described as "impetus" by Jean Buridan (c. 1295–1363), who was influenced by Ibn Sina's Book of Healing.^[51]

In Abū Rayḥān al-Bīrūnī's (973–1048) Shadows, non-uniform motion is described as the result of acceleration.^[53] Ibn-Sina's theory of mayl tried to relate the velocity and weight of a moving object, a precursor of the concept of momentum.^[54] Aristotle's theory of motion stated that a constant force produces a uniform motion; Abu'l-Barakāt al-Baghdādī (c. 1080 – 1164/5) disagreed, arguing that velocity and acceleration are two different things, and that force is proportional to acceleration, not to velocity.^[55]

Ibn Bajjah (Avempace, c. 1085–1138) proposed that for every force there is a reaction force. While he did not specify that these forces be equal, it was still an early version of Newton's third law of motion.^[56]

The Banu Musa brothers, Jafar-Muhammad, Ahmad and al-Hasan (ca. early 9th century) created automated devices described in their Book of Ingenious Devices.^[57][58][59]

Zoology

Page from the Kitāb al-Hayawān by Al-Jahiz

Many classical works including those of Aristotle were transmitted from Greek to Syriac, then to Arabic, then to Latin in the Middle Ages. Aristotle's zoology remained dominant in its field for the next two thousand years.^[60] The Kitāb al-Hayawān (كتاب الحيوان, English: Book of Animals) is a 9th-century Arabic translation of History of Animals: 1–10, On the Parts of Animals: 11–14,^[61] and Generation of Animals: 15–19.^[62][63]

The book was mentioned by Al-Kindī (d. 850), and commented on by Avicenna (Ibn Sīnā) in his The Book of Healing. Avempace (Ibn Bājja) and Averroes (Ibn Rushd) commented on and criticised On the Parts of Animals and Generation of Animals.^[64]

Significance

Historians of science differ in their views of the significance of the scientific accomplishments in the medieval Islamic world. The traditionalist view, exemplified by Bertrand Russell,^[65] holds that Islamic science, while admirable in many technical ways, lacked the intellectual energy required for innovation and was chiefly important for preserving ancient knowledge, and handing it on to medieval Europe. The revisionist view, exemplified by Abdus Salam,^[66] George Saliba^[67] and John M. Hobson^[68] holds that a Muslim scientific revolution occurred during the Middle Ages.^[69] Scholars such as Donald Routledge Hill and Ahmad Y Hassan argue that Islam was the driving force behind these scientific achievements.^[70]

According to Ahmed Dallal, science in medieval Islam was "practiced on a scale unprecedented in earlier human history or even contemporary human history".^[71] Toby E. Huff^[72][73] takes the view that, although science in the Islamic world did produce innovations, it did not lead to a scientific revolution, which in his view required an ethos that existed in Europe in the twelfth and thirteenth centuries, but not elsewhere in the world.^[74] Will Durant, Fielding H. Garrison, Hossein Nasr and Bernard Lewis held that Muslim scientists helped in laying the foundations for an experimental science with their contributions to the scientific method and their empirical, experimental and quantitative approach to scientific inquiry.^{[75][76][77][78]}

James E. McClellan III and Harold Dorn, reviewing the place of Islamic science in world history, comment that the positive achievement of Islamic science was simply to flourish, for centuries, in a wide range of institutions from observatories to libraries, madrasas to hospitals and courts, both at the height of the Islamic golden age and for some centuries afterwards. It plainly did not lead to a scientific revolution like that in Early Modern Europe, but in their view, any such external comparison is just an attempt to impose "chronologically and culturally alien standards" on a successful medieval culture.^[2]