Kerala school of astronomy and mathematics

The Kerala school of astronomy and mathematics was a school of mathematics and astronomy founded by Madhava of Sangamagrama in Kerala, South India, which included among its members: Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri and Achyuta Panikkar. The school 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 independently created a number of important mathematics concepts. Their most important results—series expansion for trigonometric functions—were described in Sanskrit verse in a book by Neelakanta called "Tantrasangraha", and again in a commentary on this work, called "Tantrasangraha-vakhya", of unknown authorship. The theorems were stated without proof, but proofs for the series for sine, cosine, and inverse tangent were provided a century later in the work "Yuktibhasa" (c.1500-c.1610), written in Malayalam, by Jyesthadeva, and also in a commentary on "Tantrasangraha".Roy, Ranjan. 1990. "Discovery of the Series Formula for $pi$ by Leibniz, Gregory, and Nilakantha." "Mathematics Magazine" (Mathematical Association of America) 63(5):291-306.]

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). [Harv|Stillwell|2004|p=173] 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. [ Harv|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."] [ Harvnb|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 generalized to arbitrary functions—in fact, the explicit notion of an arbitrary function, not to mention that of itsderivative or an algorithm for taking the derivative, is irrelevant here" ] [Harvnb|Pingree|1992|p=562 Quote:"One example I can give you relates to the Indian Mādhava's demonstration, in about 1400 A.D., of the infinite power series of trigonometrical functions using geometrical and algebraic arguments. When this was first described in English by Charles 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."] [Harvnb|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. Thy 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."]

Contributions

Infinite Series and Calculus

The Kerala school has made a number of contributions to the fields of infinite series and calculus. These include the following (infinite) geometric series:

:$frac\{1\}\{1-x\}\; =\; 1\; +\; x\; +\; x^2\; +\; x^3\; +\; dots$ for Singh, A. N. Singh. 1936. "On the Use of Series in Hindu Mathematics." "Osiris" 1:606-628.]

This formula, however, was already known in the work of the 10th century Iraqi mathematician Alhazen (the Latinized form of the name Ibn al-Haytham) (965-1039). [Edwards, C. H., Jr. 1979. "The Historical Development of the Calculus". New York: Springer-Verlag.]

The Kerala school made intuitive use of mathematical induction, though the inductive hypothesis was not yet formulated or employed in proofs. They used this to discover a semi-rigorous proof of the result:

:$1^p+\; 2^p\; +\; cdots\; +\; n^p\; approx\; frac\{n^\{p+1\{p+1\}$ for large "n". This result was also known to Alhazen.

They applied ideas from (what was to become) differential and integral calculus to obtain (Taylor-Maclaurin) infinite series for $sin\; x$, $cos\; x$, and $arctan\; x$.Bressoud, David. 2002. "Was Calculus Invented in India?" "The College Mathematics Journal" (Mathematical Association of America). 33(1):2-13.] The "Tantrasangraha-vakhya" gives the series in verse, which when translated to mathematical notation, can be written as::$rarctan(frac\{y\}\{x\})\; =\; frac\{1\}\{1\}cdotfrac\{ry\}\{x\}\; -frac\{1\}\{3\}cdotfrac\{ry^3\}\{x^3\}\; +\; frac\{1\}\{5\}cdotfrac\{ry^5\}\{x^t\}\; -\; cdots\; ,$ where $y/x\; leq\; 1.$:$rsin\; frac\{x\}\{r\}\; =\; x\; -\; xcdotfrac\{x^2\}\{(2^2+2)r^2\}\; +\; xcdot\; frac\{x^2\}\{(2^2+2)r^2\}cdotfrac\{x^2\}\{(4^2+4)r^2\}\; -\; cdot$:$r\; -\; cos\; x\; =\; rcdot\; frac\{x^2\}\{(2^2-2)r^2\}\; -\; rcdot\; frac\{x^2\}\{(2^2-2)r^2\}cdot\; frac\{x^2\}\{(4^2-4)r^2\}\; +\; cdots\; ,$ where, for $r\; =\; 1$, the series reduce to the standard power series for these trigonometric functions, for example: ::$sin\; x\; =\; x\; -\; frac\{x^3\}\{3!\}\; +\; frac\{x^5\}\{5!\}\; -\; frac\{x^7\}\{7!\}\; +\; cdots$ and ::$cos\; x\; =\; 1\; -\; frac\{x^2\}\{2!\}\; +\; frac\{x^4\}\{4!\}\; -\; frac\{x^6\}\{6!\}\; +\; cdots$ (The Kerala school themselves did not use the "factorial" symbolism.)

The Kerala school made use of the 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 yet developed.) They also made use of the series expansion of $arctan\; x$ to obtain an infinite series expression (later known as Gregory series) for $pi$::$frac\{pi\}\{4\}\; =\; 1\; -\; frac\{1\}\{3\}\; +\; frac\{1\}\{5\}\; -\; frac\{1\}\{7\}\; +\; ldots$

Their rational approximation of the "error" for the finite sum of their series are of particular interest. For example, the error, $f\_i(n+1)$, (for "n" odd, and "i = 1, 2, 3") for the series::$frac\{pi\}\{4\}\; approx\; 1\; -\; frac\{1\}\{3\}+\; frac\{1\}\{5\}\; -\; cdots\; (-1)^\{(n-1)/2\}frac\{1\}\{n\}\; +\; (-1)^\{(n+1)/2\}f\_i(n+1)$::where $f\_1(n)\; =\; frac\{1\}\{2n\},\; f\_2(n)\; =\; frac\{n/2\}\{n^2+1\},\; f\_3(n)\; =\; frac\{(n/2)^2+1\}\{(n^2+5)n/2\}.$

They manipulated the error term to derive a faster converging series for $pi$::$frac\{pi\}\{4\}\; =\; frac\{3\}\{4\}\; +\; frac\{1\}\{3^3-3\}\; -\; frac\{1\}\{5^3-5\}\; +\; frac\{1\}\{7^3-7\}\; -\; cdots$

They used the improved series to derive a rational expression, $104348/33215$ for $pi$ correct up to nine decimal places, "i.e." $3.141592653$. They made use of an intuitive notion of a limit to compute these results. The Kerala school mathematicians also gave a semi-rigorous method of differentiation of some trigonometric functions,Katz, V. J. 1995. "Ideas of Calculus in Islam and India." "Mathematics Magazine" (Mathematical Association of America), 68(3):163-174.] though the notion of a function, or of exponential or logarithmic functions, was not yet formulated.

The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835, though there exists some other works, namely Kala Sankalita by J.Warren in 1825 [ [http://www.physics.iitm.ac.in/~labs/amp/kerala-astronomy.pdf Current Science] ] which briefly mentions the discovery of infinite series by Kerala astronomers . 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."cite book
author =Charles Whish
year = 1835
title = Transactions of the Royal Asiatic Society of Great Britain and Ireland
publisher = ] However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. Rajagopal and his associates. Their work includes commentaries on the proofs of the arctan series in "Yuktibhasa" given in two papers, [Rajagopal, C. and M. S. Rangachari. 1949. "A Neglected Chapter of Hindu Mathematics." "Scripta Mathematica". 15:201-209.] [Rajagopal, C. and M. S. Rangachari. 1951. "On the Hindu proof of Gregory's series." "Ibid." 17:65-74.] a commentary on the "Yuktibhasa"'s proof of the sine and cosine series [Rajagopal, C. and A. Venkataraman. 1949. "The sine and cosine power series in Hindu mathematics." "Journal of the Royal Asiatic Society of Bengal (Science)". 15:1-13.] and two papers that provide the Sanskrit verses of the "Tantrasangrahavakhya" for the series for arctan, sin, and cosine (with English translation and commentary). [Rajagopal, C. and M. S. Rangachari. 1977. "On an untapped source of medieval Keralese mathematics." "Archive for the History of Exact Sciences". 18:89-102.] [Rajagopal, C. and M. S. Rangachari. 1986. "On Medieval Kerala Mathematics." "Archive for the History of Exact Sciences". 35:91-99.]

Geometry, Arithmetic, and Algebra

In the fields of geometry, arithmetic, and algebra, the Kerala school discovered a formula for the ecliptic,Fact|date=February 2007 Lhuilier's formula for the circumradius of a cyclic quadrilateral by Parameshvara,J. J. O'Connor and E. F. Robertson (2000). [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Paramesvara.html Paramesvara] , "MacTutor History of Mathematics archive".] Ian G. Pearce (2002). [http://www-gap.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_2.html Mathematicians of Kerala] . "MacTutor History of Mathematics archive". University of St Andrews.] decimal floating point numbers, [D. F. Almeida, G. G. Joseph (2004). "Eurocentrism in the History of Mathematics: The Case of the Kerala School", "Race and Class".] the secant method and iterative methods for solution of non-linear equations by Parameshvara, [K. Plofker (1996). "An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text", "Historia Mathematica" 23 (3), p. 246-256.] and the Newton-Gauss interpolation formula by Govindaswami.Fact|date=February 2007

Astronomy

In astronomy, Madhava discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets.S. Kak (2002). [http://www.ece.lsu.edu/kak/grolier.pdf History of Indian Science] , p. 6. Louisiana State University.] Late Kerala school astronomers gave a formulation for the equation of the center of the planets, [Joseph (2000), p. 298-300.] and a heliocentric model of the solar system.

In 1500, Nilakanthan Somayaji (1444-1544) of the Kerala school of astronomy and mathematics, in his "Tantrasangraha", revised Aryabhata's model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century.George G. Joseph (2000), p. 408.]

Nilakanthan Somayaji, in his 'Aryabhatiyabhasya', a commentary on Aryabhata's 'Aryabhatiya', developed his own computational system for a partially heliocentric planetary model, in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun, which in turn orbits the Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Nilakantha's system, however, was mathematically more efficient than the Tychonic system, due to correctly taking into account the equation of the centre and latitudinal motion of Mercury and Venus. Most astronomers of the Kerala school of astronomy and mathematics who followed him accepted his planetary model. [K. Ramasubramanian, M. D. Srinivas, M. S. Sriram (1994). "Modification of the earlier Indian planetary theory by the Kerala astronomers (c. 1500 AD) and the implied heliocentric picture of planetary motion", "Current Science" 66, p. 784-790.]

Linguistics

In linguistics, the ayurvedic and poetic traditions of Kerala were founded by this school, and the famous poem, "Narayaneeyam", was composed by Narayana Bhattathiri.Fact|date=February 2007

Prominent mathematicians

Madhavan of Sangamagrama

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.

Little is known about Madhava, who lived at Irinjalakuda,Thrissur district between the years 1340 and 1425.the famous Bharatha temple is Koodalmaanikyam is at Irinjalakuda.'Koodal'in Sanskrit means Sangamam. Nilkantha attributes the series for "sine" to Madhava. It is not known if Madhava discovered the other series as well, or whether they were discovered later by others in the Kerala school.

Madhava's discoveries include the Taylor series for the sine, cosine, tangent and arctangen functions;MacTutor Biography|id=Madhava St Andrews University, 2000.] the second-order Taylor series approximations of the sine and cosine functions and the third-order Taylor series approximation of the sine function; the power series of π, usually attributed to LeibnizIan G. Pearce (2002). [http://www-gap.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch9_3.html Madhava of Sangamagramma] . "MacTutor History of Mathematics archive". University of St Andrews.] but now known as the Madhava-Leibniz series; [citation|title=Special Functions|last=George E. Andrews, Richard Askey|first=Ranjan Roy|publisher=Cambridge University Press|year=1999|isbn=0521789885|page=58] [citation|first=R. C.|last=Gupta|title=On the remainder term in the Madhava-Leibniz's series|journal=Ganita Bharati|volume=14|issue=1-4|year=1992|pages=68-71] the solution of transcendental equations by iteration;Fact|date=February 2007 and the approximation of transcendental numbers by continued fractions. Madhava correctly computed the value of $pi$ to 9 decimal places and 13 decimal places, and produced sine and cosine tables to 9 decimal places of accuracy. [Joseph (2000), p. 293] He also extended some results found in earlier works, including those of Bhaskara.

Narayanan Pandit

Narayana Pandit (1340-1400), had written two works, an arithmetical treatise called "Ganita Kaumudi" and an algebraic treatise called "Bijganita Vatamsa". Narayanan is also thought to be the author of an elaborate commentary of Bhaskara II's Lilavathi, titled "Karmapradipika" (or "Karma-Paddhati").J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Narayana.html Narayana] , "MacTutor History of Mathematics archive".]

Although the "Karmapradipika" contains little original work, it contains seven different methods for squaring numbers, a contribution that is wholly original to the author, as well as contributions to algebra and magic squares.

Narayanan's other major works contain a variety of mathematical developments, including a rule to calculate approximate values of square roots, investigations into the second order indeterminate equation "nq"² + 1 = "p"² (Pell's equation), solutions of indeterminate higher-order equations, mathematical operations with zero, several geometrical rules, and a discussion of magic squares and similar figures. Evidence also exists that Narayana made minor contributions to the ideas of differential calculus found in Bhaskara II's work. Narayana has also made contributions to the topic of cyclic quadrilaterals.

Parameshvaran

Parameshvara (1370-1460), the founder of the Drigganita system of Astronomy, was a prolific author of several important works. He belonged to the Alathur village situated on the bank of Bharathappuzha.He is stated to have made direct astronomical observations for fifty-five years before writing his famous work, Drigganita. He also wrote commentaries on the works of Bhaskara I, Aryabhata and Bhaskara II. His "Lilavathi Bhasya", a commentary on Bhaskara II's "Lilavathi", contains one of his most important discoveries: an early version of the mean value theorem. This is considered one of the most important results in differential calculus and one of the most important theorems in mathematical analysis, and was later essential in proving the fundamental theorem of calculus.

The "Siddhanta-Deepika" by Paramesvara is a commentary on the commentary of Govindsvamin on Bhaskara I's "Maha-bhaskareeya". This work contains some of his eclipse observations, including one made at Navakshethra in 1422 and two made at Gokarna in 1425 and 1430. It also presents a mean value type formula for inverse interpolation of the sine function, a one-point iterative technique for calculating the sine of a given angle, and a more efficient approximation that works using a two-point iterative algorithm, which is essentially the same as the modern secant method.

Parameshvaran was also the first mathematician to give the radius of a circle with an inscribed cyclic quadrilateral, an expression that is normally attributed to L'Huilier (1782).

Nilakanthan Somayaji

Nilakantha (1444-1544) was a disciple of Govinda, son of Parameshvara. He was a brahmin from Trkkantiyur in Ponnani taluk. His younger brother Sankara was also a scholar in astronomy. Nilakantha's most notable work "Tantra Samgraha" (which 'spawned' a later anonymous commentary "Tantrasangraha-vyakhya" and a further commentary by the name "Yukthideepika", written in 1501) he elaborates and extends the contributions of Madhava.J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Nilakantha.html Nilakantha] , "MacTutor History of Mathematics archive".]

Nilakantha was also the author of "Aryabhatiya-bhashya", a commentary of the "Aryabhatiya". Of great significance in Nilakantha's work includes the presence of inductive mathematical proofs, a derivation and proof of the Madhava-Gregory series of the arctangent trigonometric function, improvements and proofs of other infinite series expansions by Madhava, an improved series expansion of π that converges more rapidly, and the relationship between the power series of π and arctangent. He also gave sophisticated explanations of the irrationality of π, the correct formulation for the equation of the center of the planets, and a heliocentric model of the solar system.

Chitrabhanu

Citrabhanu (c. 1530) was a 16th century mathematician from Kerala who gave integer solutions to 21 types of systems of two simultaneous Diophantine equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Indian_mathematics.html An overview of Indian mathematics] , "MacTutor History of Mathematics archive".]

:$x\; +\; y\; =\; a,\; x\; -\; y\; =\; b,\; xy\; =\; c,\; x^2\; +\; y^2\; =\; d,\; x^2\; -\; y^2\; =\; e,\; x^3\; +\; y^3\; =\; f,\; x^3\; -\; y^3\; =\; g.$

For each case, Chitrabhanu gave an explanation and justification of his rule as well as an example. Some of his explanations are algebraic, while others are geometric.

Jyesthadevan

Jyesthadeva (c. 1500-1600) was another member of the Kerala School. His key work was the "Yuktibhasa" (written in Malayalam, a regional language of the Indian state of Kerala), the world's first Calculus text. It contained most of the developments of earlier Kerala School mathematicians, particularly from Madhava. Similar to the work of Nilakantha, it is unique in the history of Indian mathematics, in that it contains proofs of theorems, derivations of rules and series, a derivation and proof of the Madhava-Gregory series of the arctangent function, proofs of most mathematical theorems and infinite series earlier discovered by Madhava and other mathematicians of the Kerala School. It also contains a proof of the series expansion of the arctangent function (equivalent to Gregory's proof), and the sine and cosine functions.J. J. O'Connor and E. F. Robertson (2000). [http://www-gap.dcs.st-and.ac.uk/~history/Biographies/Nilakantha.html Nilakantha] , "MacTutor History of Mathematics archive".]

He also studied various topics found in many previous Indian works, including integer solutions of systems of first degree equations solved using "kuttaka" method, and rules of finding the sines and the cosines of the sum and difference of two angles. Jyesthadevan also gave the earliest statement of Wallis' theorem, and geometrical derivations of infinite series.

ankaran Varma

There remains a final Kerala work worthy of a brief mention, "Sadratnamala" an astronomical treatise written by Sankara Varma (1800-1838) that serves as a summary of most of the results of the Kerala School. What is of most interest is that it was composed in the early 19th century and the author stands out as the last notable name in Keralan mathematics. A remarkable contribution was his compution of π correct to 17 decimal places.

Possibility of transmission of Kerala School results to Europe

A. K. Bag suggested in 1979 that knowledge of these results might have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. [A. K. Bag (1979) "Mathematics in ancient and medieval India". Varanasi/Delhi: Chaukhambha Orientalia. page 285.] Kerala was in continuous contact with China and Arabia, and Europe. The suggestion of some communication routes and a chronology by some scholars [ C. K. Raju (2001). "Computers, Mathematics Education, and the Alternative Epistemology of the Calculus in the Yuktibhasa", "Philosophy East and West" 51 (3), p. 325-362.] could make such a transmission a possibility; however, there is no direct evidence by way of relevant manuscripts that such a transmission took place.Almeida, D. F., J. K. John, and A. Zadorozhnyy. 2001. "Keralese Mathematics: Its Possible Transmission to Europe and the Consequential Educational Implications." "Journal of Natural Geometry", 20:77-104.] According to David Bressoud, "there is no evidence that the Indian work of series was known beyond India, or even outside of Kerala, until the nineteenth century." Gold, D. and D. Pingree. 1991. "A hitherto unknown Sanskrit work concerning Madhava's derivation of the power series for sine and cosine." "Historia Scientiarum". 42:49-65.]

Both Arab and Indian scholars made discoveries before the 17th century that are now considered a part of calculus. However, they were not able to, as Newton and Leibniz were, to "combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today." The intellectual careers of both Newton and Leibniz are well-documented and there is no indication of their work not being their own; however, it is not known with certainty whether the immediate "predecessors" of Newton and Leibniz, "including, in particular, Fermat and Roberval, learned of some of the ideas of the Islamic and Indian mathematicians through sources of which we are not now aware." This is an active area of current research, especially in the manuscript collections of Spain and Maghreb, research that is now being pursued, among other places, at the Centre national de la recherche scientifique in Paris.

Notes

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first1=John
authorlink1=
year=2004
edition=2
title=Mathematics and its History
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