Indijska matematika

Indijska matematika se razvila na Indijskom potkontinentu^[1] od 1200. pr. Kr.^[2] sve do kraja 18. stoljeća.

Povijest

U klasičnom razdoblju indijske matematike (400. – 1200.) su zabilježena značajna postignuća zahvaljujući učenjacima kao što su Aryabhata, Brahmagupta i Bhaskara II. Dekadski brojevni sustav koji se koristi danas^[3] se prvi put koristio u indijskoj matematici.^[4] Indijski matematičari su dali rani doprinos proučavanju koncepta nule kao broja,^[5] negativnih brojeva,^[6] aritmetike i algebre.^[7] Uz to se u Indiji razvila i trigonometrija,^[8] uključujući suvremene defincije sinusa i kosinusa.^[9] Ti matematički koncepti su se kasnije prenijeli na Bliski Istok, Kinu i Europu^[7] te bitno pridonijeli razvoju koncepata koji danas tvore temelj mnogih područja matematike.

Drevni i srednjovjekovni matematički tekstovi, svi napisani na sanskrtu, najčešće su se sastojali od sutra u kojima su načela ili problemi izneseni u ekonomičnim stihovima kako bi ih učenik mogao što lakše upamtiti. Njih je slijedila druga sekscija koja se sastojala od komentara u prozi (ponekad nekoliko komentara od različitih učenjaka) koji su detaljnije obrazložili problem ili izložili njegovo rješenja. U proznom dijelu, forma (i njena memorizacija) nisu bili tako važni kao same ideje.^[1]^[10] Sva matematička djela su se prenosila usmenom predajom sve do oko 500. pr. Kr. a nakon čega su se prenosili i usmeno i preko rukopisa. Najstariji sačuvani matematički dokument na Indijskom kontinentu je rukopis iz Bakhshalija, otkriven godine 1881. u Bakhshaliju kraj Peshawara (moderni Pakistan) a koji datira iz 7. stoljeća.^[11]^[12]

Važno poglavlje u povijesti indijske matematike bio je razvoj ekspanzije nizova za trigonometrijske funkcije (sinuse, kosinuse i obrnute trigonometrijske funkcije) u Keralskoj školi u 15. stoljeću. Njihovo djelo, napravljeno dva stoljeća prije otkrića infinitezimalnog računa u Europi, predstavljalo je prvi primjer reda potencija.^[13] Međutim, ona nije razvila koncepte diferencijala i integracije, niti ima neposrednih dokaza da su se ta dostignuća proširila izvan Kerale.^[14]^[15]^[16]^[17]

Izvori

↑ ^a ^b Encyclopaedia Britannica (Kim Plofker) 2007, str. 1 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (pomoć)
↑ Hayashi 2005, str. 360–361 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFHayashi2005 (pomoć)
↑ Ifrah 2000, str. 346 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFIfrah2000 (pomoć): "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, str. 44–47 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPlofker2009 (pomoć)
↑ Bourbaki 1998, str. 46 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBourbaki1998 (pomoć): "...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."
↑ Bourbaki 1998, str. 49 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBourbaki1998 (pomoć): Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. 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 Brahmagupta during 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. 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, str. 45 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPingree2003 (pomoć) 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, str. 126 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBourbaki1998 (pomoć): "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 $\theta <\pi$ on a circle of radius r, in other words the number $2r\sin \left(\theta /2\right)$ ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."
↑ Filliozat 2004, str. 140–143 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFFilliozat2004 (pomoć)
↑ Hayashi 1995 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFHayashi1995 (pomoć)
↑ Encyclopaedia Britannica (Kim Plofker) 2007, str. 6 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (pomoć)
↑ Stillwell 2004, str. 173 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFStillwell2004 (pomoć)
↑ Bressoud 2002, str. 12 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBressoud2002 (pomoć) 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, str. 293 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPlofker2001 (pomoć) 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, str. 562 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPingree1992 (pomoć) 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 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, str. 173–174 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFKatz1995 (pomoć) 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."

Vanjske poveznice

Science and Mathematics in India Arhivirana inačica izvorne stranice od 28. srpnja 2010. (Wayback Machine)
An overview of Indian mathematics, MacTutor History of Mathematics Archive, St Andrews University, 2000.
'Index of Ancient Indian mathematics' Arhivirana inačica izvorne stranice od 22. listopada 2019. (Wayback Machine), 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.

Online course material for InSIGHT, a workshop on traditional Indian sciences for school children conducted by the Computer Science department of Anna University, Chennai, India.

[plofker-1] Encyclopaedia Britannica (Kim Plofker) 2007, str. 1 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (pomoć)

[hayashi2005-p360-361-2] Hayashi 2005, str. 360–361 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFHayashi2005 (pomoć)

[irfah346-3] Ifrah 2000, str. 346 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFIfrah2000 (pomoć): "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."

[4] Plofker 2009, str. 44–47 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPlofker2009 (pomoć)

[bourbaki46-5] Bourbaki 1998, str. 46 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBourbaki1998 (pomoć): "...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."

[bourbaki49-6] Bourbaki 1998, str. 49 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBourbaki1998 (pomoć): Modern arithmetic was known during medieval times as "Modus Indorum" or method of the Indians. 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 Brahmagupta during 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."

[concise-britannica-7] "algebra" 2007. Britannica Concise Encyclopedia. Encyclopædia Britannica Online. 16 May 2007. Quote: "A full-fledged decimal, positional system certainly existed in India by the 9th century (AD), yet many of its central ideas had been transmitted well before that time to China and the Islamic world. Indian arithmetic, moreover, developed consistent and correct rules for operating with positive and negative numbers and for treating zero like any other number, even in problematic contexts such as division. Several hundred years passed before European mathematicians fully integrated such ideas into the developing discipline of algebra."

[8] Pingree 2003, str. 45 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPingree2003 (pomoć) 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."

[9] Bourbaki 1998, str. 126 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBourbaki1998 (pomoć): "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 $\theta <\pi$ on a circle of radius r, in other words the number $2r\sin \left(\theta /2\right)$ ; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."

[filliozat-p140to143-10] Filliozat 2004, str. 140–143 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFFilliozat2004 (pomoć)

[hayashi95-11] Hayashi 1995 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFHayashi1995 (pomoć)

[plofker-brit6-12] Encyclopaedia Britannica (Kim Plofker) 2007, str. 6 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (pomoć)

[13] Stillwell 2004, str. 173 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFStillwell2004 (pomoć)

[14] Bressoud 2002, str. 12 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFBressoud2002 (pomoć) 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."

[15] Plofker 2001, str. 293 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPlofker2001 (pomoć) 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"

[16] Pingree 1992, str. 562 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFPingree1992 (pomoć) 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 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."

[17] Katz 1995, str. 173–174 Pogreška u predlošku harvnb: ne postoji izvor s oznakom: CITEREFKatz1995 (pomoć) 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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