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


* Bourbaki, Nicolas, Elements of the History of Mathematics, Berlin, Heidelberg, and New York: Springer-Verlag, 301 pages, 1998, ISBN 978-3-540-64767-6 .

* Boyer, C. B.; Merzback (fwd. by Isaac Asimov), U. C., History of Mathematics, New York: John Wiley and Sons, 736 pages, 1991, ISBN 978-0-471-54397-8 .

* Bressoud, David, Was Calculus Invented in India?, The College Mathematics Journal (Math. Assoc. Amer.), 2002, 33 (1): 2–13, JSTOR 1558972, doi:10.2307/1558972 .

* Bronkhorst, Johannes, Panini and Euclid: Reflections on Indian Geometry, Journal of Indian Philosophy (Springer Netherlands), 2001, 29 (1–2): 43–80, doi:10.1023/A:1017506118885 .

* Burnett, Charles, The Semantics of Indian Numerals in Arabic, Greek and Latin, Journal of Indian Philosophy (Springer-Netherlands), 2006, 34 (1–2): 15–30, doi:10.1007/s10781-005-8153-z .

* Burton, David M., The History of Mathematics: An Introduction, The McGraw-Hill Companies, Inc.: 193–220, 1997 .

* Cooke, Roger, The History of Mathematics: A Brief Course, New York: Wiley-Interscience, 632 pages, 2005, ISBN 978-0-471-44459-6 .

* Dani, S. G., On the Pythagorean triples in the Śulvasūtras (PDF), Current Science, 25 July 2003, 85 (2): 219–224 [2019-08-11], （原始內容存檔 (PDF) 於 2011-10-12） .

* Datta, Bibhutibhusan, Early Literary Evidence of the Use of the Zero in India, The American Mathematical Monthly, December 1931, 38 (10): 566–572, JSTOR 2301384, doi:10.2307/2301384 .

* Datta, Bibhutibhusan; Singh, Avadesh Narayan, History of Hindu Mathematics : A source book, Bombay: Asia Publishing House, 1962 .

* De Young, Gregg, Euclidean Geometry in the Mathematical Tradition of Islamic India, Historia Mathematica, 1995, 22 (2): 138–153, doi:10.1006/hmat.1995.1014 .

* Encyclopædia Britannica (Kim Plofker), mathematics, South Asian, Encyclopædia Britannica Online, 2007: 1–12 [18 May 2007] .

* Filliozat, Pierre-Sylvain, Ancient Sanskrit Mathematics: An Oral Tradition and a Written Literature, Chemla, Karine; Cohen, Robert S.; Renn, Jürgen; et al (編), History of Science, History of Text (Boston Series in the Philosophy of Science), Dordrecht: Springer Netherlands, 254 pages, pp. 137–157: 360–375, 2004, ISBN 978-1-4020-2320-0 .

* Fowler, David, Binomial Coefficient Function, The American Mathematical Monthly, 1996, 103 (1): 1–17, JSTOR 2975209, doi:10.2307/2975209 .

* Hayashi, Takao, The Bakhshali Manuscript, An ancient Indian mathematical treatise, Groningen: Egbert Forsten, 596 pages, 1995, ISBN 978-90-6980-087-5 .

* Hayashi, Takao, Aryabhata's Rule and Table of Sine-Differences, Historia Mathematica, 1997, 24 (4): 396–406, doi:10.1006/hmat.1997.2160 .

* Hayashi, Takao, Indian Mathematics, Grattan-Guinness, Ivor (編), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 1, pp. 118–130, Baltimore, MD: The Johns Hopkins University Press, 976 pages, 2003, ISBN 978-0-8018-7396-6 .

* Hayashi, Takao, Indian Mathematics, Flood, Gavin (編), The Blackwell Companion to Hinduism, Oxford: Basil Blackwell, 616 pages, pp. 360–375: 360–375, 2005, ISBN 978-1-4051-3251-0 .

* Henderson, David W., Square roots in the Sulba Sutras, Gorini, Catherine A. (編), Geometry at Work: Papers in Applied Geometry, 53, pp. 39–45, Washington DC: Mathematical Association of America Notes, 236 pages: 39–45, 2000 [2019-08-11], ISBN978-0-88385-164-7, （原始內容存檔於 2015-03-10） .

* Joseph, G. G., The Crest of the Peacock: The Non-European Roots of Mathematics, Princeton, NJ: Princeton University Press, 416 pages, 2000, ISBN 978-0-691-00659-8 .

* Katz, Victor J., Ideas of Calculus in Islam and India, Mathematics Magazine (Math. Assoc. Amer.), 1995, 68 (3): 163–174, JSTOR 2691411, doi:10.2307/2691411 .

* Katz, Victor J. (編), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages, pp 385–514, 2007, ISBN 978-0-691-11485-9 .

* Keller, Agathe, Making diagrams speak, in Bhāskara I's commentary on the _Aryabhaṭīya_, Historia Mathematica, 2005, 32 (3): 275–302, doi:10.1016/j.hm.2004.09.001 .

* Kichenassamy, Satynad, Baudhāyana's rule for the quadrature of the circle, Historia Mathematica, 2006, 33 (2): 149–183, doi:10.1016/j.hm.2005.05.001 .

* Pingree, David, On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle, Journal of Historical Astronomy, 1971, 2 (1): 80–85, doi:10.1177/002182867100200202 .

* Pingree, David, The Mesopotamian Origin of Early Indian Mathematical Astronomy, Journal of Historical Astronomy, 1973, 4 (1): 1–12, doi:10.1177/002182867300400102 .

* Pingree, David; Staal, Frits, Reviewed Work(s): The Fidelity of Oral Tradition and the Origins of Science by Frits Staal, Journal of the American Oriental Society, 1988, 108 (4): 637–638, JSTOR 603154, doi:10.2307/603154 .

* Pingree, David, Hellenophilia versus the History of Science, Isis, 1992, 83 (4): 554–563, Bibcode:1992Isis...83..554P, JSTOR 234257, doi:10.1086/356288

* Pingree, David, The logic of non-Western science: mathematical discoveries in medieval India, Daedalus, 2003, 132 (4): 45–54 [2019-08-11], doi:10.1162/001152603771338779, （原始內容存檔於 2011-01-15） .

* Plofker, Kim, An Example of the Secant Method of Iterative Approximation in a Fifteenth-Century Sanskrit Text, Historia Mathematica, 1996, 23 (3): 246–256, doi:10.1006/hmat.1996.0026 .

* Plofker, Kim, The "Error" in the Indian "Taylor Series Approximation" to the Sine, Historia Mathematica, 2001, 28 (4): 283–295, doi:10.1006/hmat.2001.2331 .

* Plofker, K., Mathematics of India, Katz, Victor J. (編), The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton, NJ: Princeton University Press, 685 pages, pp 385–514: 385–514, 2007, ISBN 978-0-691-11485-9 .

* Plofker, Kim, Mathematics in India: 500 BCE–1800 CE, Princeton, NJ: Princeton University Press. Pp. 384., 2009, ISBN 978-0-691-12067-6 .

* Price, John F., Applied geometry of the Sulba Sutras (PDF), Gorini, Catherine A. (編), Geometry at Work: Papers in Applied Geometry, 53, pp. 46–58, Washington DC: Mathematical Association of America Notes, 236 pages: 46–58, 2000 [2019-08-11], ISBN978-0-88385-164-7, （原始內容 (PDF) 存檔於 2007-09-27） .

* Roy, Ranjan, Discovery of the Series Formula for $\pi $ by Leibniz, Gregory, and Nilakantha, Mathematics Magazine (Math. Assoc. Amer.), 1990, 63 (5): 291–306, JSTOR 2690896, doi:10.2307/2690896 .

* Singh, A. N., On the Use of Series in Hindu Mathematics, Osiris, 1936, 1 (1): 606–628, JSTOR 301627, doi:10.1086/368443

* Staal, Frits, The Fidelity of Oral Tradition and the Origins of Science, Mededelingen der Koninklijke Nederlandse Akademie von Wetenschappen, Afd. Letterkunde, NS 49, 8. Amsterdam: North Holland Publishing Company, 40 pages, 1986 .

* Staal, Frits, The Sanskrit of science, Journal of Indian Philosophy (Springer Netherlands), 1995, 23 (1): 73–127, doi:10.1007/BF01062067 .

* Staal, Frits, Greek and Vedic Geometry, Journal of Indian Philosophy, 1999, 27 (1–2): 105–127, doi:10.1023/A:1004364417713 .

* Staal, Frits, Squares and oblongs in the Veda, Journal of Indian Philosophy (Springer Netherlands), 2001, 29 (1–2): 256–272, doi:10.1023/A:1017527129520 .

* Staal, Frits, Artificial Languages Across Sciences and Civilisations, Journal of Indian Philosophy (Springer Netherlands), 2006, 34 (1): 89–141, doi:10.1007/s10781-005-8189-0 .

* Stillwell, John, Mathematics and its History, Undergraduate Texts in Mathematics 2, Springer, Berlin and New York, 568 pages, 2004, ISBN 978-0-387-95336-6, doi:10.1007/978-1-4684-9281-1 .

* Thibaut, George, Mathematics in the Making in Ancient India: reprints of 'On the Sulvasutras' and 'Baudhyayana Sulva-sutra', Calcutta and Delhi: K. P. Bagchi and Company (orig. _Journal of Asiatic Society of Bengal_), 133 pages, 1984 [1875] .

* van der Waerden, B. L., Geometry and Algebra in Ancient Civilisations, Berlin and New York: Springer, 223 pages, 1983, ISBN 978-0-387-12159-8

* van der Waerden, B. L., On the Romaka-Siddhānta, Archive for History of Exact Sciences, 1988, 38 (1): 1–11, doi:10.1007/BF00329976

* van der Waerden, B. L., Reconstruction of a Greek table of chords, Archive for History of Exact Sciences, 1988, 38 (1): 23–38, doi:10.1007/BF00329978

* Van Nooten, B., Binary numbers in Indian antiquity, Journal of Indian Philosophy (Springer Netherlands), 1993, 21 (1): 31–50, doi:10.1007/BF01092744

* Whish, Charles, On the Hindú Quadrature of the Circle, and the infinite Series of the proportion of the circumference to the diameter exhibited in the four S'ástras, the Tantra Sangraham, Yucti Bháshá, Carana Padhati, and Sadratnamála, Transactions of the Royal Asiatic Society of Great Britain and Ireland, 1835, 3 (3): 509–523, JSTOR 25581775, doi:10.1017/S0950473700001221

* Yano, Michio, Oral and Written Transmission of the Exact Sciences in Sanskrit, Journal of Indian Philosophy (Springer Netherlands), 2006, 34 (1–2): 143–160, doi:10.1007/s10781-005-8175-6



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）