印度数学

印度數學在公元前1200年^[1] 于印度次大陆^[2]出现，到18世纪结束。在印度数学的古典时期（公元400年至1200年），阿耶波多、婆羅摩笈多、婆什迦羅第二和伐罗诃密希罗等学者做出了重要的贡献。印度数学首先记录了今天使用的十进制^[3]。^[4]印度数学家早期的贡献包括对0作为数字的概念的研究^[5]，负数，^[6]算数，以及代数。^[7]另外，三角学^[8] 在印度更加先进，特别是发展出了正弦和余弦的现代定义。^[9]这些数学概念被传播到中东，中国和欧洲^[7]，从而导致了进一步的发展，形成了现在许多数学领域的基础。

古代和中世纪的印度数学作品，都是用梵语写成，通常由称作契经的一部分组成，在其中为了帮助学生记忆，用极少的字词陈述了一些规则和问题。在这之后是第二部分，包括一篇散文评论（有时是来自不同学者的多篇评论），其更详细地解释了问题并为解决方案提供了更多的理由。在散文部分，形式（和记忆）比起其涉及的思想来说并不是很重要。^[2][10]在公元前500年之前，所有的数学作品都是由口头传播，之后同时以口头和手稿的形式传播。现存的印度次大陆上最古老的数学文献是写在桦树皮上的巴赫沙利手稿，它在1881年于巴赫沙利村被发现，靠近白沙瓦（现位于巴基斯坦）并可能来自公元7世纪。^[11][12]

印度数学的后期里程碑是公元15世纪喀拉拉邦学派的数学家对三角函数（正弦，余弦和反正切）的级数展开的发展。 他们的卓越工作，在欧洲发明微积分之前两个世纪完成，提供了现在被称为幂级数的第一个例子（除了等比数列）^[13] 。 然而，他们没有制定出系统的微分和积分理论，也没有任何直接证据证明他们的结果是在喀拉拉邦以外传播的^{[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（英语：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（英语：Shulba Sutras）
• Kerala school of astronomy and mathematics（英语：Kerala school of astronomy and mathematics）
• Surya Siddhanta（英语：Surya Siddhanta）
• 婆羅摩笈多
• 斯里尼瓦瑟·拉马努金
• 巴赫沙利手稿
• 印度数学家列表（英语：List of Indian mathematicians）
• 印度科学和技术（英语：Indian science and technology）
• 印度逻辑（英语：Indian logic）
• 印度天文学（英语：Indian astronomy）
• 数学史
• List of numbers in Hindu scriptures（英语：List of numbers in Hindu scriptures）

引用

1. （Hayashi 2005，pp.360–361）
2. Encyclopaedia Britannica (Kim Plofker) 2007，第1頁 harvnb error: no target: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (help)
3. Ifrah 2000，第346頁 harvnb error: no target: CITEREFIfrah2000 (help): "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 （页面存档备份，存于互联网档案馆）. 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，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."
9. （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 ${\displaystyle \theta <\pi }$  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 error: no target: CITEREFEncyclopaedia_Britannica_(Kim_Plofker)2007 (help)
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 （法语） 
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. (编), 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 （页面存档备份，存于互联网档案馆）

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延伸阅读

梵语文献

• 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 (编), The Pañcasiddhāntikā of Varāhamihira, New edition with translation and commentary, (2 Vols.), Copenhagen, 1970 .
• Pingree, David (编), The Yavanajātaka of Sphujidhvaja, edited, translated and commented by D. Pingree, Cambridge, MA: Harvard Oriental Series 48 (2 vols.), 1978 .
• Sarma, K. V. (编), 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. (编), The Śulbasūtras of Baudhāyana, Āpastamba, Kātyāyana and Mānava, with Text, English Translation and Commentary, New Delhi: Indian National Science Academy, 1983 .
• Shukla, K. S. (编), 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 .
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外部链接

• Science and Mathematics in India（页面存档备份，存于互联网档案馆）
• , MacTutor History of Mathematics archive, St Andrew University, 2000.
• Indian Mathematicians （页面存档备份，存于互联网档案馆）
• Index of Ancient Indian mathematics （页面存档备份，存于互联网档案馆）, MacTutor History of Mathematics Archive, St Andrews University, 2004.
• Indian Mathematics: Redressing the balance, Student Projects in the History of Mathematics （页面存档备份，存于互联网档案馆）. Ian Pearce. MacTutor History of Mathematics Archive, St Andrews University, 2002.
• Indian Mathematics，In Our Time (BBC Radio 4)（英语：BBC Radio 4）的《In Our Time》節目。
• InSIGHT 2009（页面存档备份，存于互联网档案馆）, 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（页面存档备份，存于互联网档案馆）
• Combinatorial methods in ancient India （页面存档备份，存于互联网档案馆）
• Mathematics before S. Ramanujan （页面存档备份，存于互联网档案馆）