阿耶波多Aryabhata

阿耶波多也被称为阿耶波多，以区别于大约400年后同名的后来数学家。比鲁尼无助于理解阿耶波多的生平，因为他似乎相信有两位名叫阿耶波多的不同数学家生活在同一时代。因此，他制造了两位不同Aryabhata的混淆，直到1926年B Datta表明比鲁尼的两位Aryabhata是同一人，这一混淆才得以澄清。

我们知道阿耶波多出生的年份，因为他告诉我们，他在499年完成AryabhatiyaⓉ（阿耶波多的著作）时二十三岁。我们把Kusumapura作为阿耶波多的出生地，认为它靠近Pataliputra（后者于1541年在比哈尔邦被重新建立为Patna），但这远不确定，甚至Kusumapura本身的位置也不确定。正如Parameswaran在[26]中所写：-

……关于Asmakajanapada和Kusumapura的位置，无法给出最终结论。

我们确实知道阿耶波多在Kusumapura撰写了AryabhatiyaⓉ（阿耶波多的著作），当时Pataliputra是笈多帝国的首都和主要学术中心，但历史学家提出了许多其他地点作为他的出生地。一些人推测他出生在印度南部，也许是喀拉拉邦、泰米尔纳德邦或安得拉邦，而另一些人则推测他出生在印度东北部，也许在孟加拉。在[8]中，有人声称阿耶波多出生在南印度伐迦陀迦王朝的Asmaka地区，尽管作者承认他一生大部分时间生活在北方笈多帝国的Kusumapura。然而，将Asmaka作为阿耶波多的出生地，依据的是尼拉卡莎·萨默亚士尼拉卡莎·萨默亚士在15世纪晚期所作的一条评论。现在大多数历史学家认为，尼拉卡莎·萨默亚士将阿耶波多与婆什迦罗第一混淆了，后者是AryabhatiyaⓉ（阿耶波多的著作）的后期注释者。

我们应该注意到，Kusumapura成为印度两个主要数学中心之一，另一个是Ujjain。两者都在北部，但Kusumapura（假设它靠近Pataliputra）位于恒河畔，位置更靠北。Pataliputra作为阿耶波多时代笈多帝国的首都，是一个通信网络的中心，使来自世界其他地区的学问能够轻易到达这里，也使阿耶波多及其学派所取得的数学和天文学进展能够传遍印度，并最终传入伊斯兰世界。

至于阿耶波多撰写的文本，只有一部留存下来。然而Jha在[21]中声称：-

……阿耶波多至少撰写了三部天文学文本，还写了一些自由诗节。

留存下来的文本是阿耶波多的杰作AryabhatiyaⓉ（阿耶波多的著作），这是一部小型天文学论著，以118节诗写成，总结了截至当时的印度数学。其数学部分包含33节诗，给出了66条数学规则，没有证明。AryabhatiyaⓉ（阿耶波多的著作）包含10节诗的引言，接着是数学部分，正如我们刚才提到的，有33节诗，然后是25节诗关于时间推算和行星模型的部分，最后50节诗关于球面和食。

这种编排存在一个困难，巴特尔·伦德特·范德瓦尔登在[35]中对此进行了详细讨论。巴特尔·伦德特·范德瓦尔登认为，实际上10节Introduction的成文时间晚于其他三个部分。认为这两部分并非有意作为一个整体的一项理由是，第一部分与其余三个部分的韵律不同。然而，问题不止于此。我们说过第一部分有十节，阿耶波多确实将该部分命名为Set of ten giti stanzas。但它实际上包含十一个giti诗节和两个arya诗节。巴特尔·伦德特·范德瓦尔登认为有三节是后来添加的，并指出其余部分中也有少量诗节，他认为这些也是由阿耶波多在Kusumapura的学派成员添加的。

AryabhatiyaⓉ（阿耶波多的著作）的数学部分涵盖算术、代数、平面三角学和球面三角学。它还包含连分数、二次方程、幂级数求和以及一张正弦表。让我们更详细地考察其中一些内容。

首先我们来看阿耶波多发明并在AryabhatiyaⓉ（阿耶波多的著作）中使用和阐述的表示数字的系统。它包括给印度字母表中的33个辅音赋予数值，以表示1、2、3、…、25、30、40、50、60、70、80、90、100。更大的数字由这些辅音后接一个元音来表示，从而得到100、10000、…。事实上，该系统允许用字母记法表示高达$10^{18}$的数字。Ifrah在[3]中论证说，阿耶波多也熟悉数字符号和位值制。他在[3]中写道：-

……极有可能阿耶波多知道零的符号和位值制的数字。这一推测基于以下两个事实：第一，如果没有零或位值制，他的字母计数系统的发明是不可能的；第二，他进行了平方根和立方根的计算，而如果所讨论的数字不是按照位值制和零来书写的，这些计算是不可能的。

接下来我们简要看一下AryabhatiyaⓉ（阿耶波多的著作）中包含的一些代数内容。据我们所知，这是第一部考察形如$by = ax + c$和$by = ax - c$的方程的整数解的著作，其中$a, b, c$是整数。这个问题源于天文学中确定行星周期的问题。阿耶波多使用kuttaka方法来解决这类问题。kuttaka一词意为“粉碎”，该方法包括将问题分解为新的问题，使得每一步的系数都变得越来越小。这里的方法本质上是使用欧几里得算法来求$a$和$b$的最大公因数，但也与连分数有关。

阿耶波多给出了π的一个精确近似值。他在AryabhatiyaⓉ（阿耶波多的著作）中写道：-

将四加到一百，乘以八，然后加上六万二千。结果大约是一个直径为两万的圆的周长。根据这一法则，给出了周长与直径的关系。

这给出$\pi = \large\frac{62832}{20000}\normalsize = 3.1416$，这是一个惊人地精确的值。事实上，π = 3.14159265，精确到8位。如果获得如此精确的值令人惊讶，那么也许更令人惊讶的是，阿耶波多没有使用他精确的π值，而是在实践中更倾向于使用√10 = 3.1622。阿耶波多没有解释他是如何找到这个精确值的，但例如，Ahmad [5]将这个值视为单位圆内接256边形的周长一半的近似值。然而，在[9]中，Bruins表明这一结果不能通过边数加倍来获得。另一篇讨论阿耶波多这个精确π值的有趣论文是[22]，其中Jha写道：-

阿耶波多的π值非常接近现代值，并且在古代人中最准确。有理由相信阿耶波多设计了一种特殊方法来求得这个值。有充分根据表明阿耶波多本人使用了它，并且几位后来的印度数学家乃至阿拉伯人也采用了它。关于阿耶波多的π值源自希腊的猜想被批判性地考察，并被发现没有根据。阿耶波多独立发现了这个值，并且也意识到π是一个无理数数。他无疑有印度背景，但在计算π方面胜过了他所有的前辈。因此，发现这个精确π值的功劳可以归于著名数学家阿耶波多。

我们现在来看阿耶波多论著中包含的三角学。他给出了一张正弦表，按$\large\frac{90°}{24}\normalsize$ = 3° 45'的间隔计算近似值。为了做到这一点，他使用了一个用$\sin nx$和$\sin (n - 1)x$表示$\sin(n + 1)x - \sin nx$的公式。他还把正矢（versin = 1 - cosine）引入三角学。

阿耶波多给出的其他规则包括求前$n$个整数之和、这些整数的平方和以及立方和的规则。阿耶波多给出了三角形和圆面积的正确公式，但大多数历史学家认为球体和棱锥体积的公式是错误的。例如，Ganitanand在[15]中将阿耶波多给出高为h、底面积为$A$的棱锥体积的错误公式$V = Ah/2$这一事实描述为“数学失误”。他似乎还给出了球体体积的错误表达式。然而，正如常见的情况一样，事情并不像表面看起来那么简单，Elfering（例如参见[13]）认为这不是错误，而是错误翻译的结果。

这与AryabhatiyaⓉ（阿耶波多的著作）第二部分的第6、7和10节有关，在[13]中，Elfering给出了一种翻译，该翻译对棱锥体积和球体体积都给出了正确答案。然而，在他的翻译中，Elfering以与这些术语通常含义不同的方式翻译了两个专业术语。如果没有一些支持性证据表明这些专业术语在其他地方被用于这些不同含义，那么仍然会显得阿耶波多确实给出了这些体积的错误公式。

我们已经考察了AryabhatiyaⓉ（阿耶波多的著作）中包含的数学内容，但这是一部天文学著作，因此我们应该对其包含的天文学内容稍作说明。阿耶波多对行星在空间中的位置进行了系统处理。他给出地球周长为4 967由旬，直径为$1 581\large\frac{1}{24}\normalsize$由旬。由于1由旬=5英里，这给出周长为24 835英里，这与目前公认的24 902英里非常接近。他认为天体的表观旋转是由于地球的轴向旋转造成的。这是对太阳系本质的一个相当非凡的观点，后来的注释者无法接受，大多数人修改了文本，以使阿耶波多免于他们认为的愚蠢错误！

阿耶波多以地球/太阳轨道半径来表示行星轨道的半径，本质上就是它们绕太阳的旋转周期。他认为月亮和行星通过反射阳光而发光，令人难以置信的是，他认为行星的轨道是ellipses。他正确地解释了日食和月食的原因。直到那时，印度人的信仰是日食和月食是由一个叫做Rahu的恶魔引起的。他给出的年长为365天6小时12分30秒，这是一个高估值，因为真实值小于365天6小时。

婆什迦罗第一在大约100年后为Aryabhatiya Ⓣ（阿耶波多的著作）写了一篇评注，他这样写到阿耶波多：-

阿耶波多是大师，他在到达数学、运动学和球面学终极知识海洋的最远海岸并探究其最深处之后，把这三门科学交给了博学的世界。

Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to believe that there were two different mathematicians called Aryabhata living at the same time. He therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age when he wrote Aryabhatiya Ⓣ which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.

We do know that Aryabhata wrote Aryabhatiya Ⓣ in Kusumapura at the time when Pataliputra was the capital of the Gupta empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya Ⓣ.

We should note that Kusumapura became one of the two major mathematical centres of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to reach it easily, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to reach across India and also eventually into the Islamic world.

As to the texts written by Aryabhata only one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least three astronomical texts and wrote some free stanzas as well.

The surviving text is Aryabhata's masterpiece the Aryabhatiya Ⓣ which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. Its mathematical section contains 33 verses giving 66 mathematical rules without proof. The Aryabhatiya Ⓣ contains an introduction of 10 verses, followed by a section on mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses being on the sphere and eclipses.

There is a difficulty with this layout which is discussed in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 verse Introduction was written later than the other three sections. One reason for believing that the two parts were not intended as a whole is that the first section has a different meter to the remaining three sections. However, the problems do not stop there. We said that the first section had ten verses and indeed Aryabhata titles the section Set of ten giti stanzas. But it in fact contains eleven giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have been added and he identifies a small number of verses in the remaining sections which he argues have also been added by a member of Aryabhata's school at Kusumapura.

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. Let us examine some of these in a little more detail.

First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya Ⓣ. It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system allows numbers up to $10^{18}$ to be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and the place-value system. He writes in [3]:-

... it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.

Next we look briefly at some algebra contained in the Aryabhatiya Ⓣ. This work is the first we are aware of which examines integer solutions to equations of the form $by = ax + c$ and $by = ax - c$, where $a, b, c$ are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to solve problems of this type. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. The method here is essentially the use of the Euclidean algorithm to find the highest common factor of $a$ and $b$ but is also related to continued fractions.

Aryabhata gave an accurate approximation for π. He wrote in the Aryabhatiya Ⓣ the following:-

Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.

This gives $\pi = \large\frac{62832}{20000}\normalsize = 3.1416$ which is a surprisingly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realised that π is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to the celebrated mathematician, Aryabhata I.

We now look at the trigonometry contained in Aryabhata's treatise. He gave a table of sines calculating the approximate values at intervals of $\large\frac{90°}{24}\normalsize$ = 3° 45'. In order to do this he used a formula for $\sin(n + 1)x - \sin nx$ in terms of $\sin nx$ and $\sin (n - 1)x$. He also introduced the versine (versin = 1 - cosine) into trigonometry.

Other rules given by Aryabhata include that for summing the first $n$ integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are correct, but the formulae for the volumes of a sphere and of a pyramid are claimed to be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula $V = Ah/2$ for the volume of a pyramid with height h and triangular base of area $A$. He also appears to give an incorrect expression for the volume of a sphere. However, as is often the case, nothing is as straightforward as it appears and Elfering (see for example [13]) argues that this is not an error but rather the result of an incorrect translation.

This relates to verses 6, 7, and 10 of the second section of the Aryabhatiya Ⓣ and in [13] Elfering produces a translation which yields the correct answer for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical terms in a different way to the meaning which they usually have. Without some supporting evidence that these technical terms have been used with these different meanings in other places it would still appear that Aryabhata did indeed give the incorrect formulae for these volumes.

We have looked at the mathematics contained in the Aryabhatiya Ⓣ but this is an astronomy text so we should say a little regarding the astronomy which it contains. Aryabhata gives a systematic treatment of the position of the planets in space. He gave the circumference of the earth as 4 967 yojanas and its diameter as $1 581\large\frac{1}{24}\normalsize$ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24 835 miles, which is an excellent approximation to the currently accepted value of 24 902 miles. He believed that the apparent rotation of the heavens was due to the axial rotation of the Earth. This is a quite remarkable view of the nature of the solar system which later commentators could not bring themselves to follow and most changed the text to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of the planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Sun. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses of the Sun and the Moon. The Indian belief up to that time was that eclipses were caused by a demon called Rahu. His value for the length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since the true value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the Aryabhatiya Ⓣ about 100 years later wrote of Aryabhata:-

Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost depths of the sea of ultimate knowledge of mathematics, kinematics and spherics, handed over the three sciences to the learned world.