印度数学

**印度數學**在公元前 1200 年 於印度次大陸出現，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位。印度數學家早期的貢獻包括對 0 作為數字的概念的研究，負數，算數，以及代數。另外，三角學
在印度更加先進，特別是發展出了正弦和餘弦的現代定義。這些數學概念被傳播到中東，中國和歐洲，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。

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。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。

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） 。 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。

## 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。

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。

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。

## 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。

## 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 (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

## 引用

## 參考文獻

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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) 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）