马达瓦Madhava of Sangamagrama

马达瓦出生在印度西南部喀拉拉邦海岸附近的科钦附近。只有通过对喀拉拉数学过去二十五年的研究，马达瓦的杰出贡献才得以曝光。在[10]中，拉贾戈帕尔和兰加查里将他的成就置于背景中，他们写道：-

马达瓦迈出了决定性的一步，从古代数学的有限程序前进到处理它们向无穷的极限过渡，这正是现代经典分析的核心。

马达瓦的所有数学著作都已失传，尽管他的一些天文学文本保存了下来。然而，他在数学上的杰出工作主要是通过大约100年后生活的其他喀拉拉数学家的报告而被发现的，例如尼拉卡莎·萨默亚士。

马达瓦在1400年左右发现了与sin $x$、cos $x$和$\arctan x$的科林·麦克劳林展开等价的级数，这比它们在欧洲被重新发现早了二百多年。细节出现在他的追随者所写的许多著作中，例如Mahajyanayana prakara，其意为Method of computing the great sines。事实上，一些历史学家如Sarma（例如见[2]）曾声称这部著作是马达瓦本人所写，但这似乎极不可能，现在大多数历史学家已接受它是马达瓦的一位追随者在16世纪的作品。这在[4]中有详细讨论。

耶斯特迪瓦大约在1550年用喀拉拉邦的地区语言马拉雅拉姆语写了Yukti-Bhasa。在[9]中，Gupta给出了该文本的翻译，这在[2]和其他许多来源中也有给出。耶斯特迪瓦对马达瓦的级数描述如下：-

第一项是给定正弦与所需弧的半径的乘积除以该弧的余弦。后续项通过迭代过程获得，即第一项反复乘以正弦的平方并除以余弦的平方。然后所有项都除以奇数1, 3, 5, ....。弧通过分别加减奇数秩和偶数秩的项来获得。规定这里应取弧的正弦或其补角的正弦中较小的那个作为给定正弦。否则，通过上述迭代获得的项将不会趋于消失的量级。

这是一段描述马达瓦级数的非凡段落，但请记住，即使耶斯特迪瓦的这段文字也是在詹姆斯·格雷果里重新发现这个级数展开之前100多年写的。也许我们应该用现代符号写下马达瓦所发现的级数究竟是什么。首先要注意的是，印度语中θ的正弦在我们的记号中会写成$r \sin \theta$，而印度语的余弦在我们的记号中会是$r \cos \theta$，其中$r$是半径。因此级数是

$r \theta = r\Large\frac{r \sin \theta}{r \cos \theta}\normalsize - r\Large\frac{r \sin \theta)^{3}}{3r(r \cos \theta)^{3}}\normalsize + r\Large\frac{r \sin \theta)^{5}}{5r(r \cos \theta)^{5}}\normalsize- r\Large\frac{r \sin \theta)^{7}}{7r(r \cos \theta)^{7}}\normalsize + ...$

代入$\tan = \Large \frac {\sin}{\cos}$并消去$r$得到

$\theta = \tan \theta - \large\frac{1}{3}\normalsize \tan^{3}\theta + \large\frac{1}{5}\normalsize \tan^{5}\theta - ...$

这等价于詹姆斯·格雷果里的级数

$\tan^{-1}\theta = \theta - \large\frac{1}{3}\normalsize \theta^{3} + \large\frac{1}{5}\normalsize \theta^{5} - ...$

现在马达瓦将$q = \large\frac{\pi}{4}\normalsize$代入他的级数得到

$\Large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - ...$

并且他也将$\theta = \Large\frac{\pi}{6}\normalsize$代入他的级数得到

$\pi = \sqrt{12}(1 - \large\frac{1}{3\times 3}\normalsize + \large\frac{1}{5\times 3^{2}}\normalsize - \large\frac{1}{7\times 3^{3}}\normalsize + ...)$

我们知道马达瓦得到了π的精确到小数点后11位的近似值，当他给出

$\pi = 3.14159265359$

这可以通过取上述马达瓦级数的最后一项的21项得到。在[5]中，Gupta给出了梵文文本的翻译，其中给出了马达瓦精确到11位的π的近似值。

也许更令人印象深刻的是，马达瓦给出了他的级数的余项，从而改进了近似值。他通过添加修正项$R_{n}$改进了$\large\frac{\pi}{4}\normalsize$级数的近似值，得到

$\Large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - ... \large\frac{1}{2n-1}\normalsize ± R_{n}$

马达瓦给出了$R_{n}$的三种形式，改进了近似值，即

$R_{n} = \Large\frac{1}{4n}$或
$R_{n} = \Large\frac{n}{4n^{2} + 1}$或
$R_{n} = \Large\frac{n^{2} + 1}{4n^{3} + 5n}$。

人们做了大量工作，试图重建马达瓦可能是如何找到他的修正项的。最有说服力的是，它们来自一个连分数的前三个渐近分数，而这个连分数本身可以从对π的标准印度近似值，即$\large\frac{62832}{20000}\normalsize$，推导出来。

马达瓦还给出了一张表，列出了在给定圆的四分之一内以等间隔画出的二十四条弧的半正弦弦的几乎准确的值。人们认为，他找到这些高度精确的表格的方法是使用了级数展开的等价形式

$\sin \theta = \theta - \large\frac{1}{3!}\normalsize \theta^{3} + \large\frac{1}{5!}\normalsize \theta^{5} - ...$

$\cos \theta = 1 - \large\frac{1}{2!}\normalsize \theta^{2} + \large\frac{1}{4!}\normalsize \theta^{4} - ...$

Yukti-Bhasa中的耶斯特迪瓦解释了马达瓦如何在1400年左右找到他的级数展开，这些展开等价于艾萨克·牛顿在1676年左右重新发现的现代版本。历史学家声称，马达瓦使用的方法相当于逐项积分。

Rajagopal声称马达瓦迈出了通向现代经典分析的决定性一步，鉴于他的非凡成就，这似乎非常公平。同样，Joseph在[1]中写道：-

我们可以认为马达瓦是数学分析的创始人。他在这一领域的一些发现表明他具有非凡的直觉，几乎可以与近代的直觉天才拉马努金相媲美，后者在靠近马达瓦出生地的贡伯戈讷姆度过了童年和青年时代。

Madhava of Sangamagrama was born near Cochin on the coast in the Kerala state in southwestern India. It is only due to research into Keralese mathematics over the last twenty-five years that the remarkable contributions of Madhava have come to light. In [10] Rajagopal and Rangachari put his achievement into context when they write:-

[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis.

All the mathematical writings of Madhava have been lost, although some of his texts on astronomy have survived. However his brilliant work in mathematics has been largely discovered by the reports of other Keralese mathematicians such as Nilakantha who lived about 100 years later.

Madhava discovered the series equivalent to the Maclaurin expansions of sin $x$, cos $x$, and $\arctan x$ around 1400, which is over two hundred years before they were rediscovered in Europe. Details appear in a number of works written by his followers such as Mahajyanayana prakara which means Method of computing the great sines. In fact this work had been claimed by some historians such as Sarma (see for example [2]) to be by Madhava himself but this seems highly unlikely and it is now accepted by most historians to be a 16th century work by a follower of Madhava. This is discussed in detail in [4].

Jyesthadeva wrote Yukti-Bhasa in Malayalam, the regional language of Kerala, around 1550. In [9] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:-

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. Perhaps we should write down in modern symbols exactly what the series is that Madhava has found. The first thing to note is that the Indian meaning for sine of θ would be written in our notation as $r \sin \theta$ and the Indian cosine of would be $r \cos \theta$ in our notation, where $r$ is the radius. Thus the series is

$r \theta = r\Large\frac{r \sin \theta}{r \cos \theta}\normalsize - r\Large\frac{r \sin \theta)^{3}}{3r(r \cos \theta)^{3}}\normalsize + r\Large\frac{r \sin \theta)^{5}}{5r(r \cos \theta)^{5}}\normalsize- r\Large\frac{r \sin \theta)^{7}}{7r(r \cos \theta)^{7}}\normalsize + ...$

putting $\tan = \Large \frac {\sin}{\cos}$ and cancelling $r$ gives

$\theta = \tan \theta - \large\frac{1}{3}\normalsize \tan^{3}\theta + \large\frac{1}{5}\normalsize \tan^{5}\theta - ...$

which is equivalent to Gregory's series

$\tan^{-1}\theta = \theta - \large\frac{1}{3}\normalsize \theta^{3} + \large\frac{1}{5}\normalsize \theta^{5} - ...$

Now Madhava put $q = \large\frac{\pi}{4}\normalsize$ into his series to obtain

$\Large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - ...$

and he also put $\theta = \Large\frac{\pi}{6}\normalsize$ into his series to obtain

$\pi = \sqrt{12}(1 - \large\frac{1}{3\times 3}\normalsize + \large\frac{1}{5\times 3^{2}}\normalsize - \large\frac{1}{7\times 3^{3}}\normalsize + ...)$

We know that Madhava obtained an approximation for π correct to 11 decimal places when he gave

$\pi = 3.14159265359$

which can be obtained from the last of Madhava's series above by taking 21 terms. In [5] Gupta gives a translation of the Sanskrit text giving Madhava's approximation of π correct to 11 places.

Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation. He improved the approximation of the series for $\large\frac{\pi}{4}\normalsize$ by adding a correction term $R_{n}$ to obtain

$\Large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - ... \large\frac{1}{2n-1}\normalsize ± R_{n}$

Madhava gave three forms of $R_{n}$ which improved the approximation, namely

$R_{n} = \Large\frac{1}{4n}$ or
$R_{n} = \Large\frac{n}{4n^{2} + 1}$ or
$R_{n} = \Large\frac{n^{2} + 1}{4n^{3} + 5n}$.

There has been a lot of work done in trying to reconstruct how Madhava might have found his correction terms. The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to π namely $\large\frac{62832}{20000}\normalsize$.

Madhava also gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions

$\sin \theta = \theta - \large\frac{1}{3!}\normalsize \theta^{3} + \large\frac{1}{5!}\normalsize \theta^{5} - ...$

$\cos \theta = 1 - \large\frac{1}{2!}\normalsize \theta^{2} + \large\frac{1}{4!}\normalsize \theta^{4} - ...$

Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676. Historians have claimed that the method used by Madhava amounts to term by term integration.

Rajagopal's claim that Madhava took the decisive step towards modern classical analysis seems very fair given his remarkable achievements. In the same vein Joseph writes in [1]:-

We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition, making him almost the equal of the more recent intuitive genius Srinivasa Ramanujan, who spent his childhood and youth at Kumbakonam, not far from Madhava's birthplace.