Indian mathematics
Indian mathematics emerged in the
was further advanced in India, and, in particular, the modern definitions ofAncient and medieval Indian mathematical works, all composed in
A later landmark in Indian mathematics was the development of the
Prehistory
Excavations at
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.[19][20]
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.[21]
Vedic period
History of science and technology in the Indian subcontinent |
By subject |
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Samhitas and Brahmanas
The religious texts of the
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 uṣas (dawn) , hail to the vyuṣṭi (twilight), hail to udeṣyat (the one which is going to rise), hail to udyat (the one which is rising), hail udita (to the one which has just risen), 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
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.[23] Most mathematical problems considered in the Śulba Sūtras spring from "a single theological requirement,"[24] 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.[24]
According to Hayashi, the Śulba Sūtras contain "the earliest extant verbal expression of the
The diagonal rope (akṣṇayā-rajju) of an oblong (rectangle) produces both which the flank (pārśvamāni) and the horizontal (tiryaṇmānī) <ropes> produce separately."[25]
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.[25]
They contain lists of
The expression is accurate up to five decimal places, the true value being 1.41421356...
which expresses √2 in the sexagesimal system, and which is also accurate up to 5 decimal places.
According to mathematician S. G. Dani, the Babylonian cuneiform tablet Plimpton 322 written c. 1850 BCE[34] "contains fifteen Pythagorean triples with quite large entries, including (13500, 12709, 18541) which is a primitive triple,[35] 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."[36] 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.[36]
In all, three Sulba Sutras were composed. The remaining two, the Manava Sulba Sutra composed by
- Vyakarana
An important landmark of the Vedic period was the work of
Pingala (300 BCE – 200 BCE)
Among the scholars of the post-Vedic period who contributed to mathematics, the most notable is
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, ...[39]
The text also indicates that Pingala was aware of the combinatorial identity:[40]
- 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
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
In addition to Surya Prajnapti, important Jain works on mathematics included the
Oral tradition
Mathematicians of ancient and early medieval India were almost all
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.
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:
The most complex form of recitation, ghana-pāṭha (literally "dense recitation"), according to Filliozat,[45] took the form:
That these methods have been effective is testified to by the preservation of the most ancient Indian religious text, the Ṛgveda (c. 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 (c. 500 BCE).
The Sutra genre
Mathematical activity in ancient India began as a part of a "methodological reflexion" on the sacred
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
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,[49] 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.[50]
The earliest mathematical prose commentary was that on the work,
- 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.[51]
Typically, for any mathematical topic, students in ancient India first memorised the sūtras, which, as explained earlier, were "deliberately inadequate"[50] 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).[53]
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.[54] The Syrian bishop Severus Sebokht wrote in the mid-7th century CE about the "nine signs" of the Indians for expressing numbers.[54] However, how, when, and where the first decimal place value system was invented is not so clear.[55]
The earliest extant
The earliest surviving evidence of decimal place value numerals in India and southeast Asia is from the middle of the first millennium CE.[57] 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.[57] Decimal numerals recording the years 683 CE have also been found in stone inscriptions in Indonesia and Cambodia, where Indian cultural influence was substantial.[57]
There are older textual sources, although the extant manuscript copies of these texts are from much later dates.[58] Probably the earliest such source is the work of the Buddhist philosopher Vasumitra dated likely to the 1st century CE.[58] 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."[58] 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."[58]
A third decimal representation was employed in a verse composition technique, later labelled
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.[62] According to Plofker,[60]
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."[62]
Bakhshali Manuscript
The oldest extant mathematical manuscript in India is the
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.
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.[65]
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.[65]
In 2017, three samples from the manuscript were shown by radiocarbon dating to come from three different centuries: from 224 to 383 AD, 680-779 AD, and 885-993 AD. It is not known how fragments from different centuries came to be packaged together.[66][67][68]
Classical period (400–1600)
This period is often known as the golden age of Indian Mathematics. This period saw mathematicians such as
Fourth to sixth centuries
- Surya Siddhanta
Though its authorship is unknown, the
This ancient text uses the following as trigonometric functions for the first time:[citation needed]
- Sine (Jya).
- Cosine (Kojya).
- Inverse sine(Otkram jya).
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
- Aryabhata I
- 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:
(See also :
- 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 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:
Algebra:
- Solutions of simultaneous quadratic equations.
- Whole number solutions of linear equationsby 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.[71]
- Varahamihira
Seventh and eighth centuries
In the 7th 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 generalisation of
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 triangles: A triangle with rational sides and rational area is of the form:
for some rational numbers and .[74]
Chapter 18 contained 103 Sanskrit verses which began with rules for arithmetical operations involving zero and negative numbers[73] and is considered the first systematic treatment of the subject. The rules (which included and ) were all correct, with one exception: .[73] Later in the chapter, he gave the first explicit (although still not completely general) solution of the quadratic equation:
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.[75]
This is equivalent to:
Also in chapter 18, Brahmagupta was able to make progress in finding (integral) solutions of Pell's equation,[76]
where is a nonsquare integer. He did this by discovering the following identity:[76]
Brahmagupta's Identity: which was a generalisation of an earlier identity of Diophantus:[76] Brahmagupta used his identity to prove the following lemma:[76]
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.[77]
Brahmagupta did not actually prove the theorem, but rather worked out examples using his method. The first example he presented was:[76]
Example (Brahmagupta): Find integers such that:
In his commentary, Brahmagupta added, "a person solving this problem within a year is a mathematician."[76] The solution he provided was:
- Bhaskara I
- 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
- 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 2-adic order.
Virasena also gave:
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.[79]
- Mahavira
- 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 arithmetical progression, and gave empirical rules for area and perimeterof 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:
- Solved indeterminate quadratic equations.
- Solved indeterminate cubic equations.
- Solved indeterminate higher order equations.
- Shridhara
- 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 equations were elaborated in the 10th century by Manjula (also Munjala), who realised that the expression[80]
could be approximately expressed as
This was elaborated by his later predecessor Bhaskara ii thereby finding the derivative of sine.[80]
- Aryabhata II
- Numerical mathematics (Ank Ganit).
- Algebra.
- Solutions of indeterminate equations (kuttaka).
- Shripati
- Permutations and combinations.
- General solution of the simultaneous indeterminate linear equation.
He was also the author of Dhikotidakarana, a work of twenty verses on:
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
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:
- Preliminary concept of differentiation
- Discovered the differential coefficient.
- Stated early form of 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 although didn't deceive the notion of derivative.
- Computed π, correct to five decimal places.
- Calculated the length of the Earth's revolution around the Sun to 9 decimal places.[81]
Trigonometry:
- Developments of spherical trigonometry
- The trigonometric formulas:
Kerala mathematics (1300–1600)
The
Their discovery of these three important series expansions of
- The (infinite) geometric series: [84]
- A semi-rigorous proof (see "induction" remark below) of the result: for large n.[82]
- Intuitive use of mathematical induction, however, the inductive hypothesis was not formulated or employed in proofs.[82]
- 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.[83] The Tantrasangraha-vakhya gives the series in verse, which when translated to mathematical notation, can be written as:[82]
- where, for r = 1, the series reduces to the standard power series for these trigonometric functions, for example:
- and
- 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.)[82]
- Use of the series expansion of to obtain the Leibniz formula for π:[82]
- A rational approximation of error for the finite sum of their series of interest. For example, the error, , (for n odd, and i = 1, 2, 3) for the series:
- Manipulation of error term to derive a faster converging series for :[82]
- Using the improved series to derive a rational expression,[82] 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.[82]
- A semi-rigorous (see remark on limits above) method of differentiation of some trigonometric functions.[71] 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
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,[86][87] a commentary on the Yuktibhāṣā's proof of the sine and cosine series[88] and two papers that provide the Sanskrit verses of the Tantrasangrahavakhya for the series for arctan, sin, and cosine (with English translation and commentary).[89][90]
Parameshvara (c. 1370–1460) wrote commentaries on the works of
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.
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
[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"[91]
The historian of mathematics, Florian Cajori, suggested that he and others "suspect that Diophantus got his first glimpse of algebraic knowledge from India."[92] However, he also wrote that "it is certain that portions of Hindu mathematics are of Greek origin".[93]
More recently, as discussed in the above section, the infinite series of
Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus.
See also
Notes
- ^ a b (Kim Plofker 2007, p. 1)
- ^ a b c d (Hayashi 2005, pp. 360–361)
- ^ (Ifrah 2000, p. 346): "The measure of the genius of Indian civilisation, to which we owe our modern (number) system, is all the greater in that it was the only one in all history to have achieved this triumph. Some cultures succeeded, earlier than the Indian, in discovering one or at best two of the characteristics of this intellectual feat. But none of them managed to bring together into a complete and coherent system the necessary and sufficient conditions for a number-system with the same potential as our own."
- ^ (Plofker 2009, pp. 44–47)
- ^ (Bourbaki 1998, p. 46): "...our decimal system, which (by the agency of the Arabs) is derived from Hindu mathematics, where its use is attested already from the first centuries of our era. It must be noted moreover that the conception of zero as a number and not as a simple symbol of separation) and its introduction into calculations, also count amongst the original contribution of the Hindus."
- Leonardo of Pisa wrote that compared to method of the Indians all other methods is a mistake. This method of the Indians is none other than our very simple arithmetic of addition, subtraction, multiplication and division. Rules for these four simple procedures was first written down by Brahmaguptaduring 7th century AD. "On this point, the Hindus are already conscious of the interpretation that negative numbers must have in certain cases (a debt in a commercial problem, for instance). In the following centuries, as there is a diffusion into the West (by intermediary of the Arabs) of the methods and results of Greek and Hindu mathematics, one becomes more used to the handling of these numbers, and one begins to have other "representation" for them which are geometric or dynamic."
- ^ a b "algebra" 2007. Britannica Concise Encyclopedia Archived 29 September 2007 at the Wayback Machine. 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."
- ^ (Pingree 2003, p. 45) Quote: "Geometry, and its branch trigonometry, was the mathematics Indian astronomers used most frequently. Greek mathematicians used the full chord and never imagined the half chord that we use today. Half chord was first used by Aryabhata which made trigonometry much more simple. 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."
- ^ (Bourbaki 1998, p. 126): "As for trigonometry, it is disdained by geometers and abandoned to surveyors and astronomers; it is these latter (Aristarchus, 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)."
- ^ (Filliozat 2004, pp. 140–143)
- ^ (Hayashi 1995)
- ^ a b (Kim Plofker 2007, p. 6)
- ^ (Stillwell 2004, p. 173)
- ^ (Bressoud 2002, p. 12) Quote: "There is no evidence that the Indian work on series was known beyond India, or even outside Kerala, until the nineteenth century. Gold and Pingree assert [4] that by the time these series were rediscovered in Europe, they had, for all practical purposes, been lost to India. The expansions of the sine, cosine, and arc tangent had been passed down through several generations of disciples, but they remained sterile observations for which no one could find much use."
- ^ (Plofker 2001, p. 293) Quote: "It is not unusual to encounter in discussions of Indian mathematics such assertions as that "the concept of differentiation was understood [in India] from the time of Manjula (... in the 10th century)" [Joseph 1991, 300], or that "we may consider Madhava to have been the founder of mathematical analysis" (Joseph 1991, 293), or that Bhaskara II may claim to be "the precursor of Newton and Leibniz in the discovery of the principle of the differential calculus" (Bag 1979, 294). ... The points of resemblance, particularly between early European calculus and the Keralese work on power series, have even inspired suggestions of a possible transmission of mathematical ideas from the Malabar coast in or after the 15th century to the Latin scholarly world (e.g., in (Bag 1979, 285)). ... It should be borne in mind, however, that such an emphasis on the similarity of Sanskrit (or Malayalam) and Latin mathematics risks diminishing our ability fully to see and comprehend the former. To speak of the Indian "discovery of the principle of the differential calculus" somewhat obscures the fact that Indian techniques for expressing changes in the Sine by means of the Cosine or vice versa, as in the examples we have seen, remained within that specific trigonometric context. The differential "principle" was not generalised to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of its derivative or an algorithm for taking the derivative, is irrelevant here"
- Charles Matthew Whish, in the 1830s, it was heralded as the Indians' discovery of the calculus. This claim and Mādhava's achievements were ignored by Western historians, presumably at first because they could not admit that an Indian discovered the calculus, but later because no one read anymore the Transactions of the Royal Asiatic Society, in which Whish's article was published. The matter resurfaced in the 1950s, and now we have the Sanskrit texts properly edited, and we understand the clever way that Mādhava derived the series without the calculus; but many historians still find it impossible to conceive of the problem and its solution in terms of anything other than the calculus and proclaim that the calculus is what Mādhava found. In this case the elegance and brilliance of Mādhava's mathematics are being distorted as they are buried under the current mathematical solution to a problem to which he discovered an alternate and powerful solution."
- ^ (Katz 1995, pp. 173–174) Quote:"How close did Islamic and Indian scholars come to inventing the calculus? Islamic scholars nearly developed a general formula for finding integrals of polynomials by A.D. 1000—and evidently could find such a formula for any polynomial in which they were interested. But, it appears, they were not interested in any polynomial of degree higher than four, at least in any of the material that has come down to us. Indian scholars, on the other hand, were by 1600 able to use ibn al-Haytham's sum formula for arbitrary integral powers in calculating power series for the functions in which they were interested. By the same time, they also knew how to calculate the differentials of these functions. So some of the basic ideas of calculus were known in Egypt and India many centuries before Newton. It does not appear, however, that either Islamic or Indian mathematicians saw the necessity of connecting some of the disparate ideas that we include under the name calculus. They were apparently only interested in specific cases in which these ideas were needed. ... There is no danger, therefore, that we will have to rewrite the history texts to remove the statement that Newton and Leibniz invented calculus. They were certainly the ones who were able to combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between them, and turn the calculus into the great problem-solving tool we have today."
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- ^ Rao, S. R. (July 1992). "A Navigational Instrument of the Harappan Sailors" (PDF). Marine Archaeology. 3: 61–62. Archived from the original (PDF) on 8 August 2017.
- ^ A. Seidenberg, 1978. The origin of mathematics. Archive for History of Exact Sciences, vol 18.
- ^ (Staal 1999)
- ^ a b (Hayashi 2003, p. 118)
- ^ a b (Hayashi 2005, p. 363)
- ^ Pythagorean triples are triples of integers (a, b, c) with the property: a2+b2 = c2. Thus, 32+42 = 52, 82+152 = 172, 122+352 = 372, etc.
- ^ (Cooke 2005, p. 198): "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."
- ^ (Cooke 2005, pp. 199–200): "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."
- ^ a b c (Joseph 2000, p. 229)
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- Pell numbers. If x/y is one term in this sequence of approximations, the next is (x + 2y)/(x + y). These approximations may also be derived by truncating the continued fractionrepresentation of √2.
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- ^ Mathematics Department, University of British Columbia, The Babylonian tabled Plimpton 322 Archived 17 June 2020 at the Wayback Machine.
- ^ Three positive integers form a primitive Pythagorean triple if c2 = a2+b2 and if the highest common factor of a, b, c is 1. In the particular Plimpton322 example, this means that 135002+127092 = 185412 and that the three numbers do not have any common factors. However some scholars have disputed the Pythagorean interpretation of this tablet; see Plimpton 322 for details.
- ^ a b (Dani 2003)
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- ^ (Singh 1936, pp. 623–624)
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- ^ a b (Filliozat 2004, p. 137)
- ^ (Pingree 1988, p. 637)
- ^ (Staal 1986)
- ^ a b c d (Filliozat 2004, p. 139)
- ^ a b c d e (Filliozat 2004, pp. 140–141)
- ^ (Yano 2006, p. 146)
- ^ a b c (Filliozat 2004, pp. 143–144)
- ^ (Filliozat 2004, p. 144)
- ^ a b (Pingree 1988, p. 638)
- ^ a b c (Hayashi 2003, pp. 122–123)
- ^ (Hayashi 2003, p. 123)
- ^ a b c (Hayashi 2003, p. 119)
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- ^ (Plofker 2007, p. 395); (Plofker 2009, pp. 47–48)
- ^ (Hayashi 2005, p. 366)
- ^ a b c (Plofker 2009, p. 45)
- ^ a b c d (Plofker 2009, p. 46)
- ^ a b c d e (Plofker 2009, p. 47)
- ^ a b (Plofker 2009)
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- ^ "Illuminating India: Starring the oldest recorded origins of 'zero', the Bakhshali manuscript".
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- ^ "Carbon dating finds Bakhshali manuscript contains oldest recorded origins of the symbol 'zero'". Bodleian Library. 14 September 2017. Retrieved 14 September 2017.
- ^ (Neugebauer & Pingree 1970)
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The word Siddhanta means that which is proved or established. The Sulva Sutras are of Hindu origin, but the Siddhantas contain so many words of foreign origin that they undoubtedly have roots in Mesopotamia
- ^ a b c d e f g h (Katz 1995)
- ^ (Hayashi 2005, p. 369)
- ^ a b c d (Hayashi 2003, pp. 121–122)
- ^ (Stillwell 2004, p. 77)
- ^ (Stillwell 2004, p. 87)
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- ^ (Stillwell 2004, pp. 74–76)
- ^ Gupta, R. C. (2000), "History of Mathematics in India", in Hoiberg, Dale; Ramchandani, Indu (eds.), Students' Britannica India: Select essays, Popular Prakashan, p. 329
- ^ a b Singh, A. N., Mathematics of Dhavala, Lucknow University, archived from the original on 11 May 2011, retrieved 31 July 2010
- ^ a b Joseph (2000), p. 298–300.
- ISBN 978-0-471-18082-1.
- ^ a b c d e f g h i (Roy 1990)
- ^ a b c (Bressoud 2002)
- ^ (Singh 1936)
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- ^ Rajagopal, C.; Rangachari, M. S. (1949), "A Neglected Chapter of Hindu Mathematics", Scripta Mathematica, 15: 201–209.
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- ^ Cajori, Florian (1893), "The Hindoos", A History of Mathematics P 86, Macmillan & Co.,
In algebra, there was probably a mutual giving and receiving [between Greece and India]. We suspect that Diophantus got his first glimpse of algebraic knowledge from India
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- ^ a b Almeida, D. F.; John, J. K.; Zadorozhnyy, A. (2001), "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications", Journal of Natural Geometry, 20: 77–104.
- ^ Gold, D.; Pingree, D. (1991), "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine", Historia Scientiarum, 42: 49–65.
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Further reading
Source books in Sanskrit
- Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 1: The Translation: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 172 pages, ISBN 978-3-7643-7291-0.
- Keller, Agathe (2006), Expounding the Mathematical Seed. Vol. 2: The Supplements: A Translation of Bhaskara I on the Mathematical Chapter of the Aryabhatiya, Basel, Boston, and Berlin: Birkhäuser Verlag, 206 pages, ISBN 978-3-7643-7292-7.
- Sarma, K. V., ed. (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Sūryadeva Yajvan, critically edited with Introduction and Appendices, New Delhi: Indian National Science Academy.
- Sen, S. N.; Bag, A. K., eds. (1983), The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, with Text, English Translation and Commentary, New Delhi: Indian National Science Academy.
- Shukla, K. S., ed. (1976), Āryabhaṭīya of Āryabhaṭa with the commentary of Bhāskara I and Someśvara, critically edited with Introduction, English Translation, Notes, Comments and Indexes, New Delhi: Indian National Science Academy.
- Shukla, K. S., ed. (1988), Āryabhaṭīya of Āryabhaṭa, critically edited with Introduction, English Translation, Notes, Comments and Indexes, in collaboration with K.V. Sarma, New Delhi: Indian National Science Academy.
External links
- Science and Mathematics in India
- An overview of Indian mathematics, St Andrews University, 2000.
- Indian Mathematicians
- Index of Ancient Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2004.
- Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
- Indian Mathematics on In Our Time at the BBC
- InSIGHT 2009, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.
- Mathematics in ancient India by R. Sridharan
- Combinatorial methods in ancient India
- Mathematics before S. Ramanujan