尼拉卡莎·萨默亚士Nilakantha Somayaji

尼拉卡莎·萨默亚士出生于一个来自喀拉拉邦南马拉巴尔的南布蒂里婆罗门家庭。南布蒂里是喀拉拉邦的主要种姓。这是一个正统种姓，其成员认为自己是古代吠陀宗教的后裔。

他出生在一座名为凯拉卢尔的房子中，据称该房子与现今印度南部蒂鲁尔附近特尔卡蒂尤尔村的埃塔马纳相符。他的父亲是贾塔维达斯，家庭属于加尔吉亚氏族，这是一个禁止与氏族外任何人通婚的印度种姓。该家庭遵循阿什瓦拉亚纳经，这是《梨俱吠陀》中祭祀仪式的手册，《梨俱吠陀》是一部吠陀赞歌集。他崇拜人格化的神苏摩，苏摩是“植物之主”和疾病的治疗者。这解释了萨默亚士这个名字的含义，即他来自一个有资格举行苏摩仪式的家庭。

萨默亚士在老师拉维的指导下学习天文学和吠檀多，吠檀多是印度教哲学六个正统体系之一。他还受教于达摩德拉，后者是Paramesvara的儿子。Paramesvara是一位著名的印度天文学家，达摩德拉遵循其父的教导。这也使萨默亚士成为Paramesvara的追随者。萨默亚士撰写的若干数学天文学文本流传了下来。他总共撰写了大约十部天文学论著。

Tantrasamgraha Ⓣ（策略集）是他于1501年写的主要天文学论著。它由432首梵文偈颂组成，分为8章，涵盖了印度天文学的各个方面。它基于行星运动的均轮-本轮和偏心模型。前两章讨论行星的运动和黄经。第三章Treatise on shadow讨论与太阳在天球上的位置有关的各种问题，包括其在三种坐标系（即黄道坐标、赤道坐标和地平坐标）中表达的关系。

第四和第五章是Treatise on the lunar eclipse和On the solar eclipse，这两章讨论日食和月食的各个方面。第六章是On vyatipata，讨论太阳和月亮黄经的完全偏差。第七章On visibility computation讨论月亮和行星的升落。最后一章On elevation of the lunar cusps考察月亮被太阳照亮部分的大小，并给出其图形表示。

Tantrasamgraha Ⓣ（策略集）在萨默亚士所使用的数学方面非常重要。特别是他使用了马达瓦发现的结果，并且它是他所发现的卓越数学结果的重要来源。然而，萨默亚士不仅仅使用马达瓦的结果，他还扩展并改进了它们。出现了一部题为Tantrasamgraha-vakhya Ⓣ（对Ⓣ的阐释）的匿名评注，稍后在大约1550年，耶斯特迪瓦发表了一部题为Yuktibhasa Ⓣ（逻辑）的评注，其中包含对马达瓦和萨默亚士早期结果的证明。对于一部印度文本来说，给出数学证明是相当不寻常的。

级数$\large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - \large\frac{1}{7}\normalsize + ...$是arctan的级数表示的一个特例，即$\tan^{-1}x = x - \large\frac{1}{3}\normalsize x^{3} + \large\frac{1}{5}\normalsize x^{5} - ...$。众所周知，只需令$x = 1$即可得到$\large\frac{\pi}{4}\normalsize$的级数。[4]的作者报告了这些级数在哥特弗里德·威廉·莱布尼茨和詹姆斯·格雷果里17世纪70年代著作中的出现。这两位欧洲数学家对该级数的贡献是众所周知的，但在[4]中也讨论了马达瓦近三百年前著作中关于该级数的结果，这些结果由Nilakantha在TantrasamgrahaⓉ（策略集）中呈现。

萨默亚士通过获得圆弧的近似表达式然后取极限，推导出了级数展开$\tan^{-1}x = x - \large\frac{1}{3}\normalsize x^{3} + \large\frac{1}{5}\normalsize x^{5} - ...$。他工作的一个有趣特点是引入了几个额外的$\large\frac{\pi}{4}\normalsize$级数，这些级数比$\large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - \large\frac{1}{7}\normalsize + ...$收敛得更快。[4]的作者基于他拥有尾级数$\large\frac{1}{(n+2)}\normalsize - \large\frac{1}{n+4}\normalsize + \large\frac{1}{n+6}\normalsize - \large\frac{1}{n+8}\normalsize + ....$的某种连分数表示的假设，重建了他可能如何得出这些结果。TantrasamgrahaⓉ（策略集）并不是我们拥有其文本的萨默亚士的唯一著作。他还写了GolasaraⓉ（词汇表），用五十六首梵文诗写成，展示了如何用数学计算来计算天文数据。Siddhanta DarpanaⓉ（原理镜）用三十二首梵文诗写成，描述了一个行星模型。CandracchayaganitaⓉ（算术）用三十一首梵文诗写成，解释了用于计算月亮天顶距的计算方法。

在萨默亚士时代，Nambudiri种姓的首领是Netranarayana，他成为萨默亚士另一部主要著作的赞助人，即Aryabhatiyabhasya Ⓣ（对阿耶波多著作的评注），这是对阿耶波多的Aryabhatiya Ⓣ（阿耶波多的著作）的评注。在这部著作中，萨默亚士提到了他观测到的两次食，第一次在1467年3月6日，第二次在1501年7月28日，地点在Anantaksetra。萨默亚士还在Aryabhatiyabhasya Ⓣ（对阿耶波多著作的评注）中提到了他写的其他著作，例如关于食的Grahanirnaya Ⓣ（裁决），这些著作没有留存下来。

Nilakantha was born into a Namputiri Brahmin family which came from South Malabar in Kerala. The Nambudiri is the main caste of Kerala. It is an orthodox caste whose members consider themselves descendants of the ancient Vedic religion.

He was born in a house called Kelallur which it is claimed coincides with the present Etamana in the village of Trkkantiyur near Tirur in south India. His father was Jatavedas and the family belonged to the Gargya gotra, which was a Indian caste that prohibits marriage to anyone outside the caste. The family followed the Ashvalayana sutra which was a manual of sacrificial ceremonies in the Rigveda, a collection of Vedic hymns. He worshipped the personified deity Soma who was the "master of plants" and the healer of disease. This explains the name Somayaji which means he was from a family qualified to conduct the Soma ritual.

Nilakantha studied astronomy and Vedanta, one of the six orthodox systems of Indian Hindu philosophy, under the teacher Ravi. He was also taught by Damodra who was the son of Paramesvara. Paramesvara was a famous Indian astronomer and Damodra followed his father's teachings. This led Nilakantha also to become a follower of Paramesvara. A number of texts on mathematical astronomy written by Nilakantha have survived. In all he wrote about ten treatises on astronomy.

The Tantrasamgraha Ⓣ is his major astronomy treatise written in 1501. It consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates.

The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipata and deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.

The Tantrasamgraha Ⓣ is very important in terms of the mathematics Nilakantha uses. In particular he uses results discovered by Madhava and it is an important source of the remarkable mathematical results which he discovered. However, Nilakantha does not just use Madhava's results, he extends them and improves them. An anonymous commentary entitled Tantrasamgraha-vakhya Ⓣ appeared and, somewhat later in about 1550, Jyesthadeva published a commentary entitled Yuktibhasa Ⓣ that contained proofs of the earlier results by Madhava and Nilakantha. This is quite unusual for an Indian text in giving mathematical proofs.

The series $\large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - \large\frac{1}{7}\normalsize + ...$ is a special case of the series representation for arctan, namely

Nilakantha derived the series expansion

The head of the Nambudiri caste in Nilakantha's time was Netranarayana and he became Nilakantha's patron for another of his major works, namely the Aryabhatiyabhasya Ⓣ which is a commentary on the Aryabhatiya Ⓣ of Aryabhata I. In this work Nilakantha refers to two eclipses which he observed, the first on 6 March 1467 and the second on 28 July 1501 at Anantaksetra. Nilakantha also refers in the Aryabhatiyabhasya Ⓣ to other works which he wrote such as the Grahanirnaya Ⓣ on eclipses which have not survived.