印度數學

印度數學在公元前1200年^[1] 於印度次大陸^[2]出現，到18世紀結束。在印度數學的古典時期（公元400年至1200年），阿耶波多、婆羅摩笈多、婆什迦羅第二和伐羅訶密希羅等學者做出了重要的貢獻。印度數學首先記錄了今天使用的十進制^[3]。^[4]印度數學家早期的貢獻包括對0作為數字的概念的研究^[5]，負數，^[6]算數，以及代數。^[7]另外，三角學^[8] 在印度更加先進，特別是發展出了正弦和餘弦的現代定義。^[9]這些數學概念被傳播到中東，中國和歐洲^[7]，從而導致了進一步的發展，形成了現在許多數學領域的基礎。

印度河流域文明中重量標準化的立方

古代和中世紀的印度數學作品，都是用梵語寫成，通常由稱作契經的一部分組成，在其中為了幫助學生記憶，用極少的字詞陳述了一些規則和問題。在這之後是第二部分，包括一篇散文評論（有時是來自不同學者的多篇評論），其更詳細地解釋了問題並為解決方案提供了更多的理由。在散文部分，形式（和記憶）比起其涉及的思想來說並不是很重要。^[2][10]在公元前500年之前，所有的數學作品都是由口頭傳播，之後同時以口頭和手稿的形式傳播。現存的印度次大陸上最古老的數學文獻是寫在樺樹皮上的巴赫沙利手稿，它在1881年於巴赫沙利村被發現，靠近白沙瓦（現位於巴基斯坦）並可能來自公元7世紀。^[11][12]

印度數學的後期里程碑是公元15世紀喀拉拉邦學派的數學家對三角函數（正弦，餘弦和反正切）的級數展開的發展。 他們的卓越工作，在歐洲發明微積分之前兩個世紀完成，提供了現在被稱為冪級數的第一個例子（除了等比數列）^[13] 。 然而，他們沒有制定出系統的微分和積分理論，也沒有任何直接證據證明他們的結果是在喀拉拉邦以外傳播的^{[14][15][16][17]}。

史前時期

在摩亨佐-達羅、哈拉帕與其他幾個印度河流域文明的遺址中，都發現了使用"實用數學"的證據。他們使用4:2:1的比例製造磚頭，此作法被認為有利於磚頭結構的穩定性。他們使用了標準化的砝碼，以比例:1/20, 1/10, 1/5, 1/2, 1, 2, 5, 10, 20, 50, 100, 200與500進行製作，其中的一單位的重量約等於28克。他們大量生產有規則形狀的砝碼，有六面體、桶形、錐形與柱形，從而展示出對於基本幾何的理解。^[18]

印度文明的居民曾試圖對於長度量測進行高精準度的標準化。他們設計出了摩亨佐達羅尺，尺上的單位長度約為3.4公分，並且再分割成十等分。古代摩亨若達羅的磚頭尺寸通常是此單位長度的整數倍。^[19][20]

在洛沙爾(Lothal,2200 BCE)與卓拉維拉(Dholavira)發現的貝殼製中空柱狀物體上面有8個裂縫，被認為可用來製作羅盤，證明了能在平面上測量角度的能力，並可藉此確定星星的位置進行導航。^[21]

吠陀時期

吠陀時期的宗教文本提供了使用大數的證據。在《夜柔吠陀》（公元前1200-900年）文本中，包含的數字高達10¹²。^[1]

耆那學者（前400年-200年）

耆那學者認為世界是永恆的，只有形式上的變化，與婆羅門教創造萬物的理論不同。耆那教由筏陀摩那在前6世紀創立，但耆那數學著作大部分是在前6世紀後撰寫的。耆那學者將數字分為三類。

耆那重要數學家包括賢臂（英語：Bhadrabahu）(Bhadrabahu，卒於公元前298年），他是兩部天文著作的作者。

書寫形式

巴赫沙利手稿

古典時期（400-1600年）

參見

• Shulba Sutras（英語：Shulba Sutras）
• Kerala school of astronomy and mathematics（英語：Kerala school of astronomy and mathematics）
• Surya Siddhanta（英語：Surya Siddhanta）
• 婆羅摩笈多
• 斯里尼瓦瑟·拉馬努金
• 巴赫沙利手稿
• 印度數學家列表（英語：List of Indian mathematicians）
• 印度科學和技術（英語：Indian science and technology）
• 印度邏輯（英語：Indian logic）
• 印度天文學（英語：Indian astronomy）
• 數學史
• List of numbers in Hindu scriptures（英語：List of numbers in Hindu scriptures）

引用

1. (Hayashi 2005，第360–361頁)
2. Encyclopaedia Britannica (Kim Plofker) 2007，第1頁 harvnb模板錯誤: 無指向目標: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (幫助)
3. Ifrah 2000，第346頁 harvnb模板錯誤: 無指向目標: CITEREFIfrah2000 (幫助): "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，第44–47頁
5. Bourbaki 1998，第46頁: "...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."
6. Bourbaki 1998，第49頁: 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."
7. "algebra" 2007. Britannica Concise Encyclopedia （頁面存檔備份，存於網際網路檔案館）. 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，第45頁) 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，第126頁): "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 ${\displaystyle \theta <\pi }$ on a circle of radius r, in other words the number ${\displaystyle 2r\sin \left(\theta /2\right)}$; the introduction of the sine, more easily handled, is due to Hindu mathematicians of the Middle Ages)."
10. Filliozat 2004，第140–143頁
11. Hayashi 1995
12. Encyclopaedia Britannica (Kim Plofker) 2007，第6頁 harvnb模板錯誤: 無指向目標: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (幫助)
13. Stillwell 2004，第173頁
14. Bressoud 2002，第12頁 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，第293頁 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"
16. Pingree 1992，第562頁 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，第173–174頁 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."
18. Sergent, Bernard, Genèse de l'Inde, Paris: Payot: 113, 1997, ISBN 978-2-228-89116-5 （法語）
19. Coppa, A.; et al, Early Neolithic tradition of dentistry: Flint tips were surprisingly effective for drilling tooth enamel in a prehistoric population, Nature, 6 April 2006, 440 (7085): 755–6, Bibcode:2006Natur.440..755C, PMID 16598247, S2CID 6787162, doi:10.1038/440755a.
20. Bisht, R. S., Excavations at Banawali: 1974–77, Possehl, Gregory L. (編), Harappan Civilisation: A Contemporary Perspective, New Delhi: Oxford and IBH Publishing Co.: 113–124, 1982
21. S. R. Rao (1992). Marine Archaeology, Vol. 3,. pp. 61-62. Link http://drs.nio.org/drs/bitstream/handle/2264/3082/J_Mar_Archaeol_3_61.pdf?sequence=2 （頁面存檔備份，存於網際網路檔案館）

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延伸閱讀

梵語文獻

• Keller, Agathe, 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, 2006, ISBN 978-3-7643-7291-0.
• Keller, Agathe, 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, 2006, ISBN 978-3-7643-7292-7.
• Neugebauer, Otto; Pingree, David (編), The Pañcasiddhāntikā of Varāhamihira, New edition with translation and commentary, (2 Vols.), Copenhagen, 1970.
• Pingree, David (編), The Yavanajātaka of Sphujidhvaja, edited, translated and commented by D. Pingree, Cambridge, MA: Harvard Oriental Series 48 (2 vols.), 1978.
• Sarma, K. V. (編), Āryabhaṭīya of Āryabhaṭa with the commentary of Sūryadeva Yajvan, critically edited with Introduction and Appendices, New Delhi: Indian National Science Academy, 1976.
• Sen, S. N.; Bag, A. K. (編), The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, with Text, English Translation and Commentary, New Delhi: Indian National Science Academy, 1983.
• Shukla, K. S. (編), 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, 1976.
• Shukla, K. S. (編), 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, 1988.

外部連結

• Science and Mathematics in India（頁面存檔備份，存於網際網路檔案館）
• An overview of Indian mathematics, MacTutor History of Mathematics archive, St Andrew University, 2000.
• Indian Mathematicians （頁面存檔備份，存於網際網路檔案館）
• Index of Ancient Indian mathematics （頁面存檔備份，存於網際網路檔案館）, MacTutor History of Mathematics Archive, St Andrews University, 2004.
• Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics （頁面存檔備份，存於網際網路檔案館）. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
• Indian Mathematics，In Our Time (BBC Radio 4)（英語：BBC Radio 4）的《In Our Time》節目。
• InSIGHT 2009（頁面存檔備份，存於網際網路檔案館）, 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（頁面存檔備份，存於網際網路檔案館）
• Combinatorial methods in ancient India （頁面存檔備份，存於網際網路檔案館）
• Mathematics before S. Ramanujan （頁面存檔備份，存於網際網路檔案館）