Mazhab astronomi dan matematika Kerala

Mazhab astronomi dan matematika Kerala adalah mazhab matematika dan astronomi yang didirikan oleh Madhava dari Sangamagrama di Kerala, India Selatan; yang para anggotanya termasuk antara lain Parameshvara, Neelakanta Somayaji, Jyeshtadeva, Achyuta Pisharati, Melpathur Narayana Bhattathiri, dan Achyuta Panikkar. Mazhab ini berkembang antara abad ke-14 dan ke-16 masehi, dan penemuan orisinil terakhir mazhab ini tampaknya terdapat pada Narayana Bhattathiri (1559-1632). Dalam upaya untuk memecahkan masalah-masalah astronomi, mazhab Kerala secara mandiri menciptakan beberapa konsep-konsep penting matematika. Hasil terpenting mereka, yaitu ekspansi deret untuk fungsi trigonometri, digambarkan dalam syair bahasa Sanskerta dalam sebuah buku karangan Neelakanta yang bernama Tantrasangraha, serta dalam komentar tentang karya ini yang disebut Tantrasangraha-vakhya, yang tidak diketahui penulisnya. Teorema-teorema yang ada dinyatakan tanpa bukti, tetapi bukti-bukti untuk deret sinus, kosinus, dan tangen terbalik diberikan satu abad kemudian dalam karya Yuktibhasa (k. 1500 - k. 1610), yang ditulis dalam bahasa Malayalam oleh Jyesthadeva, dan juga dalam komentar tentang Tantrasangraha.^[1]

Karya mereka, yang diselesaikan dua abad sebelum penemuan kalkulus di Eropa, memberikan gambaran tentang apa yang sekarang disebut sebagai contoh pertama suatu deret pangkat (berbeda dengan deret geometri).^[2] Namun mereka tidak merumuskan teori diferensial dan integral secara sistematis, serta tidak terdapat bukti yang jelas bahwa hasil karya mereka pernah dikabarkan di luar wilayah Kerala.^[3]^[4]^[5]^[6]

Lihat pula

Astronomi India

Catatan kaki

^ 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.
^ (Stillwell 2004, hlm. 173)
^ (Bressoud 2002, hlm. 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, hlm. 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 its derivative or an algorithm for taking the derivative, is irrelevant here"
^ Pingree 1992, hlm. 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."
^ Katz 1995, hlm. 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."

Pranala luar

(Inggris)Sekolah Kerala, Matematika Eropa dan Navigasi, 2001.
(Inggris)Sebuah tinjauan atas matematika India Diarsipkan 2002-10-15 di Wayback Machine., Arsip Sejarah Matematika MacTutor, 2002.
(Inggris)Matematika India: Menyesuaikan keseimbangan, Arsip Sejarah Matematika MacTutor, 2002.
(Inggris)Matematika Kerala Diarsipkan 2018-05-29 di Wayback Machine., Arsip Sejarah Matematika MacTutor, 2002.
(Inggris)Kemungkiran penyampaian matematika Kerala ke Eropa Diarsipkan 2018-01-03 di Wayback Machine., Arsip Sejarah Matematika MacTutor, 2002.
(Inggris)Bukan Newton maupun Leibnitz - Pra-Sejarah Kalkulus dan Ilmu Perbintangan pada Kerala Abad Pertengahan Diarsipkan 2006-08-06 di Wayback Machine., 2005.
(Inggris)"Bangsa India mendahului 'penemuan' Newton sejauh 250 tahun" phys.org, 2007

[roy-1] 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.

[2] (Stillwell 2004, hlm. 173)

[3] (Bressoud 2002, hlm. 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."

[4] Plofker 2001, hlm. 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 its derivative or an algorithm for taking the derivative, is irrelevant here"

[5] Pingree 1992, hlm. 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."

[6] Katz 1995, hlm. 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."

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