伐罗诃密希罗Varāhamihira

我们对伐罗诃密希罗的了解确实非常有限。根据他的一部著作，他在Kapitthaka接受教育。然而，这远未解决问题，只是引发了关于这个地方可能在哪里的各种解释的讨论。Dhavale在[3]中讨论了这个问题。我们不知道他是否出生在Kapitthaka，无论那可能在哪里，尽管我们将其作为最可能的猜测。然而，我们确实知道他在Ujjain工作，那里自公元400年左右以来一直是重要的数学中心。Ujjain的数学学派因伐罗诃密希罗在那里工作而重要性增加，并在很长一段时间内继续是印度两个领先的数学中心之一，特别是以婆罗摩笈多作为其下一个主要人物。

伐罗诃密希罗最著名的著作是Pancasiddhantika（《五部天文法典》），年代为公元575年。这部著作本身很重要，也为我们提供了关于现已失传的更古老的印度文本的信息。这部著作是一篇关于数学天文学的论文，它总结了五部更早的天文学论著，即Surya, Romaka, Paulisa, Vasistha和Paitamaha悉檀多。Shukla在[11]中写道：-

伐罗诃密希罗的《五部悉檀多》是阿耶波多一世之前印度天文学史最重要的资料来源之一。

伐罗诃密希罗 所总结的一部论著是 Romaka-Siddhanta，它本身基于公元1世纪希腊人给出的太阳和月亮运动的本轮理论。Romaka-Siddhanta 基于 喜帕恰斯 的回归年和19年的默冬章。伐罗诃密希罗 所总结的其他著作也基于希腊人关于天体运动的本轮理论。他修订了历法，通过更新这些早期著作以考虑自它们写成以来的岁差。Pancasiddhantika 还包含许多使用位值制数系的例子。

然而，关于解释伐罗诃密希罗的天文学文本和其他类似著作的数据存在相当多的争论。一些人认为天文学理论起源于巴比伦，而另一些人则认为印度人通过自己的观测改进了巴比伦模型。在这个领域还有很多工作要做，以澄清其中一些有趣的理论。

在 [1] 中，Ifrah 指出 伐罗诃密希罗 是印度历史上最著名的占星家之一。他的著作 Brihatsamhita（《大汇编》）讨论了诸如 [1] 等主题：-

……天体的描述、它们的运动和合、气象现象、这些运动、合和现象所代表的预兆的指示、应采取什么行动和完成什么操作、在人类、动物、宝石等中寻找的迹象。

伐罗诃密希罗 做出了一些重要的数学发现。其中包括某些三角公式，翻译成我们现在的记号对应于

$\sin x = \cos(\large\frac{\pi}{2}\normalsize - x)$，

$\sin^{2}x + \cos^{2}x = 1$，和

$\large\frac{1}{2}\normalsize (1 - \cos 2x) = \sin^{2}x$。

对三角学的另一个重要贡献是他的正弦表，他改进了 阿耶波多 的正弦表，给出了更准确的值。应该强调，准确性对这些印度数学家非常重要，因为他们计算正弦表是为了应用于天文学和占星术。这促使他们通过发展新的插值方法来实现更高的准确性。

耆那教数学学派在数百年的时间里研究了计算从 $n$ 个对象中选取 $r$ 个对象的方法数的规则。他们给出了计算二项式系数 $_{n}C_{r}$ 的规则，这些规则相当于

$_{n}C_{r} = \large\frac{1}{r!}\normalsize n(n-1)(n-2)...(n-r+1)$

然而，伐罗诃密希罗以一种相当不同的方式处理了计算$_{n}C_{r}$的问题。他把数$n$写成一列，底部是$n = 1$。然后他把数$r$排成行，左侧是$r = 1$。从数组的左下角开始，对应于值$n = 1, r = 1$，$_{n}C_{r}$的值通过求和两个条目得到，即紧接在$(n, r)$位置下方的那个和紧接其左边的那个。当然，这个表正是用于求二项式系数的Pascal's triangle，尽管是从与我们今天构建它的方式不同的角度来看。伐罗诃密希罗这项工作的完整细节见[5]。

Hayashi在[6]中考察了伐罗诃密希罗关于幻方的工作。他特别考察了出现在伐罗诃密希罗工作中的四阶泛对角幻方。

Our knowledge of Varahamihira is very limited indeed. According to one of his works, he was educated in Kapitthaka. However, far from settling the question this only gives rise to discussions of possible interpretations of where this place was. Dhavale in [3] discusses this problem. We do not know whether he was born in Kapitthaka, wherever that may be, although we have given this as the most likely guess. We do know, however, that he worked at Ujjain which had been an important centre for mathematics since around 400 AD. The school of mathematics at Ujjain was increased in importance due to Varahamihira working there and it continued for a long period to be one of the two leading mathematical centres in India, in particular having Brahmagupta as its next major figure.

The most famous work by Varahamihira is the Pancasiddhantika (The Five Astronomical Canons) dated 575 AD. This work is important in itself and also in giving us information about older Indian texts which are now lost. The work is a treatise on mathematical astronomy and it summarises five earlier astronomical treatises, namely the Surya, Romaka, Paulisa, Vasistha and Paitamaha siddhantas. Shukla states in [11]:-

The Pancasiddhantika of Varahamihira is one of the most important sources for the history of Hindu astronomy before the time of Aryabhata I I.

One treatise which Varahamihira summarises was the Romaka-Siddhanta which itself was based on the epicycle theory of the motions of the Sun and the Moon given by the Greeks in the 1st century AD. The Romaka-Siddhanta was based on the tropical year of Hipparchus and on the Metonic cycle of 19 years. Other works which Varahamihira summarises are also based on the Greek epicycle theory of the motions of the heavenly bodies. He revised the calendar by updating these earlier works to take into account precession since they were written. The Pancasiddhantika also contains many examples of the use of a place-value number system.

There is, however, quite a debate about interpreting data from Varahamihira's astronomical texts and from other similar works. Some believe that the astronomical theories are Babylonian in origin, while others argue that the Indians refined the Babylonian models by making observations of their own. Much needs to be done in this area to clarify some of these interesting theories.

In [1] Ifrah notes that Varahamihira was one of the most famous astrologers in Indian history. His work Brihatsamhita (The Great Compilation) discusses topics such as [1]:-

... descriptions of heavenly bodies, their movements and conjunctions, meteorological phenomena, indications of the omens these movements, conjunctions and phenomena represent, what action to take and operations to accomplish, sign to look for in humans, animals, precious stones, etc.

Varahamihira made some important mathematical discoveries. Among these are certain trigonometric formulae which translated into our present day notation correspond to

$\sin x = \cos(\large\frac{\pi}{2}\normalsize - x)$,

$\sin^{2}x + \cos^{2}x = 1$, and

$\large\frac{1}{2}\normalsize (1 - \cos 2x) = \sin^{2}x$.

Another important contribution to trigonometry was his sine tables where he improved those of Aryabhata I giving more accurate values. It should be emphasised that accuracy was very important for these Indian mathematicians since they were computing sine tables for applications to astronomy and astrology. This motivated much of the improved accuracy they achieved by developing new interpolation methods.

The Jaina school of mathematics investigated rules for computing the number of ways in which $r$ objects can be selected from $n$ objects over the course of many hundreds of years. They gave rules to compute the binomial coefficients $_{n}C_{r}$ which amount to

$_{n}C_{r} = \large\frac{1}{r!}\normalsize n(n-1)(n-2)...(n-r+1)$

However, Varahamihira attacked the problem of computing $_{n}C_{r}$ in a rather different way. He wrote the numbers $n$ in a column with $n = 1$ at the bottom. He then put the numbers $r$ in rows with $r = 1$ at the left-hand side. Starting at the bottom left side of the array which corresponds to the values $n = 1, r = 1$, the values of $_{n}C_{r}$ are found by summing two entries, namely the one directly below the $(n, r)$ position and the one immediately to the left of it. Of course this table is none other than Pascal's triangle for finding the binomial coefficients despite being viewed from a different angle from the way we build it up today. Full details of this work by Varahamihira is given in [5].

Hayashi, in [6], examines Varahamihira's work on magic squares. In particular he examines a pandiagonal magic square of order four which occurs in Varahamihira's work.