User:Peacearth/sandbox 1

阿耶波多 आर्यभट
印度校際天文及天體物理學中心的阿耶波多雕像，來自藝術家的想像，阿耶波多的真實樣貌不明。
出生	476 CE 可能是Ashmaka
逝世	550 CE
居住地	印度
學術工作
年代	笈多王朝
主要領域	數學、天文學
著名作品	阿里亚哈塔历书
著名思想	解釋日食與月食的成因、地球自轉、月球反射光、正弦函數、一元二次方程解、圆周率小數點後4位近似、99.8%精確度的地球周長、恒星年的長度
施影响于	婆羅摩笈多、伐罗诃密希罗

阿耶波多（梵語：आर्यभट; IAST: Āryabhaṭa，或譯阿里亚哈塔，阿耶波多一世，公元476年－550年）是5世纪末印度的著名数学家及天文学家。他的作品包括阿里亚哈塔历书（公元499年，他23岁的时候写成），分四部分。書中提供圓周率的一個近似值：((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416。此外，他還根據天文觀測，提出日心說，並發現日月食的成因。另外，印度在1975年發射的第一顆人造衛星以他的名字命名。

簡歷

出生地點與時間

阿里亚哈塔历书寫成時，阿耶波多Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476.

Aryabhata provides no information about his place of birth. The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India; Aryabhata is believed to have been born there.

其他假說

It has been claimed that the aśmaka (Sanskrit for "stone") where Aryabhata originated may be the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala. This is based on the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala has been used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala. K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence.

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.

教育

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.

著作

阿耶波多is the author of several treatises on mathematics and astronomy, some of which are lost.

His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known.

Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.

阿里亚哈塔历书

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuṭṭaka).
3. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.
4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE).

數學

Place value system and zero

The place-value system, first seen in the 3rd-century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients.

However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.

圓周率之估計

阿耶波多曾進行圓周率之估計，並有可能得出圓周率值是無理數的結論。在《阿里亞哈塔曆書》的第二部分，阿耶波多寫道：

原文：

“

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

”

翻譯：

“

4加上100，再乘以8，再加上62,000，按此規則可逼近直徑為20,000的圓之周長值。

”

這表示圓周率的值，也就是圓周與直徑的比，是${\frac {(4+100)\times 8+62000}{20000}}={\frac {62832}{20000}}=3.1416$，其精確度達5個有效数字。有學者推斷，阿耶波多使用「逼近」（āsanna）一詞，可能不僅為了表示這是估計值，而是有意表示他是個無理數。相較之下，在歐洲，圓周率是無理數的證明由约翰·海因里希·朗伯在1761年發表。在《阿里亞哈塔曆書》於公元820年被翻譯成阿拉伯文之後，這個估計值被花拉子米所著之代數書籍中被提及。

三角學

在《阿里亞哈塔曆書》中的〈算數〉（Ganitapada）篇第6回，阿耶波多描述三角形的面積計算方式：

原文：

“ tribhujasya phalashariram samadalakoti bhujardhasamvargah ”

翻譯：

“ 對於一個三角形，底的一半與高的乘積即為其面積 ”

阿耶波多也在該書中討論正弦函數的概念，稱之為「ardha-jya」，字面上為半弦之意。爾後，人們逐漸將其簡稱為「jya」。爾後其著作從梵文被譯為阿拉伯文，該函數名則被譯成「jiba」。然而，在阿拉伯文書寫體系中母因被省略，於是該詞變成了「jb」。之後人們將其母音重新替代為「jaib」，意為「口袋」或「折（衣服）」，因為「jiba」在阿拉伯文中沒有對應的意義。到了12世紀，克雷莫納的杰拉德將該署從阿拉伯文翻譯成拉丁文，於是「jaib」便成了拉丁文的對應詞「sinus」，意為「海灣」，最後便演變為如今正弦函數的英文名「sine」。

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to Diophantine equations that have the form ax + by = c. (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese remainder theorem.) This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems, elaborated by Bhaskara in 621 CE, is called the kuṭṭaka (कुट्टक) method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.

代數

在《阿里亚哈塔历书》中，阿耶波多推導出以下平方與立方级数求和之結果：

$1^{2}+2^{2}+\cdots +n^{2}={n(n+1)(2n+1) \over 6}$

and

$1^{3}+2^{3}+\cdots +n^{3}=(1+2+\cdots +n)^{2}$

天文學

Aryabhata's system of astronomy was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later writings on astronomy, which apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the Earth's rotation. He may have believed that the planet's orbits as elliptical rather than circular.

太陽系之運行

Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view, that the sky rotated. This is indicated in the first chapter of the Aryabhatiya, where he gives the number of rotations of the earth in a yuga, and made more explicit in his gola chapter:

In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by the cosmic wind.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles. They in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (c. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger śīghra (fast). The order of the planets in terms of distance from earth is taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms."

The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.

日食

阿耶波多以科學的方式解釋了日月食的成因。他表示，月球與行星並不會自行發光，而是反射太陽光。他也說明日食是月球落在地球上的陰影、月食是地球落在月球上的陰影造成的。他也討論了地球陰影的大小和範圍，並計算出日月食的規模。後事的印度天文學家更基於阿耶波多的基礎上，將預測做得更加精確。他們的計算是如此的精確已至於18世紀科學家紀堯姆·勒讓蒂爾造訪印度本地治里時發現印度人對1765年8月30日月食的持續時間只比實際短了41秒，而他手上的預測表則長了68秒。

地球自轉與公轉週期

若使用現今公制單位，阿耶波多求得之恆星日長度為23小時56分4.1秒，而現代測得之精確值為23小時56分4.091秒。另外，阿耶波多求得之恆星年長度為365天6小時12分30秒（365.25858天），只與現代測量值（365.25636天）相差3分20秒。

日心說

如前述，阿耶波多認為地球會繞著自身的自轉軸自轉。他的天文學模型As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the śīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlying heliocentric model, in which the planets orbit the Sun, though this has been rebutted. It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-Ptolemaic Greek, heliocentric model of which Indian astronomers were unaware, though the evidence is scant. The general consensus is that a synodic anomaly (depending on the position of the sun) does not imply a physically heliocentric orbit (such corrections being also present in late Babylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric.

影響

印度的第一顆人造衛星，以阿耶波多命名

Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (c. 820 CE), was particularly influential. Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trigonometry. He was also the first to specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, modern names "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced by Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misunderstood by Gerard of Cremona while translating an Arabic geometry text to Latin. He assumed that jiba was the Arabic word jaib, which means "fold in a garment", L. sinus (c. 1150).

Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world and used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) and remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the Panchangam (the Hindu calendar). In the Islamic world, they formed the basis of the Jalali calendar introduced in 1073 CE by a group of astronomers including Omar Khayyam, versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier Siddhanta calendars. This type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the Gregorian calendar.

以阿耶波多命名的事物

• 印度的第一顆人造衛星：阿耶波多人造衛星。
• 月球上的隕石坑：阿耶波多隕石坑。
• 鄰近印度奈尼塔爾的阿耶波多觀測科學研究學院（ARIES） 。
• 印度空间研究组织在平流层發現的細菌：阿耶波多芽孢桿菌（學名：Bacillus aryabhata）。
• 阿耶波多知識大學（AKU）：印度比哈尔邦的一所大學。

參見

• Āryabhaṭa numeration
• Āryabhaṭa's sine table
• Indian mathematics
• List of Indian mathematicians

参考文献

• Cooke, Roger. The History of Mathematics: A Brief Course. Wiley-Interscience. 1997. ISBN 0-471-18082-3. 
• Clark, Walter Eugene. The Āryabhaṭīya of Āryabhaṭa: An Ancient Indian Work on Mathematics and Astronomy. University of Chicago Press; reprint: Kessinger Publishing (2006). 1930. ISBN 978-1-4254-8599-3. 
• Kak, Subhash C. (2000). 'Birth and Early Development of Indian Astronomy'. In Selin, Helaine (编). Astronomy Across Cultures: The History of Non-Western Astronomy. Boston: Kluwer. 2000. ISBN 0-7923-6363-9. 
• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.
• Thurston, H. Early Astronomy. Springer-Verlag, New York. 1994. ISBN 0-387-94107-X. 

外部連結

• Eugene C. Clark's 1930 English translation of The Aryabhatiya in various formats at the Internet Archive.
• 約翰·J·奧康納; 埃德蒙·F·羅伯遜, Aryabhata_I, MacTutor数学史档案 （英语） 
• Achar, Narahari. Āryabhaṭa I. Thomas Hockey; et al (编). The Biographical Encyclopedia of Astronomers. New York: Springer: 63. 2007. ISBN 978-0-387-31022-0.  (PDF version)
• Aryabhata and Diophantus' son, Hindustan Times Storytelling Science column, Nov 2004
• Surya Siddhanta translations

{{Authority control}} [[Category:印度数学家]] [[Category:印度天文学家]]