婆什迦罗第一Bhaskara-I

关于婆什迦罗第一的生平，除了能从他的著作中推断出的内容外，我们知之甚少。Shukla从婆什迦罗第一经常引用Asmakatantra而非Aryabhatiya Ⓣ（阿耶波多的著作）这一事实推断，他一定曾在阿斯马卡的一个数学家学派中工作，该学派很可能位于安得拉邦的尼扎马巴德县。如果这是正确的，而且看起来确实很可能，那么阿斯马卡的学派将是一批追随阿耶波多的学者，当然这与婆什迦罗第一本人确实是阿耶波多的追随者这一事实非常吻合。

在婆什迦罗第一的著作中还有其他关于印度地点的提及。例如，他提到Valabhi（今Vala），7世纪Maitraka王朝的首都，以及Sivarajapura，两者都位于Saurastra，即今天印度大陆西海岸的古吉拉特邦。还提到古吉拉特邦南部的Bharuch（或Broach）和旁遮普东部的Thanesar，后者从606年起被Harsa统治了41年。在婆什迦罗第一生命的前半段，Harsa是北印度最杰出的统治者。一个合理的猜测是婆什迦罗第一出生在Saurastra，后来移居到阿斯马卡。

婆什迦罗第一是两部论著以及阿耶波多著作注释的作者。他的作品有Mahabhaskariya Ⓣ（婆什迦罗第一的大书）、Laghubhaskariya Ⓣ（婆什迦罗第一的小书）和Aryabhatiyabhasya Ⓣ（对阿耶波多著作的注释）。Mahabhaskariya Ⓣ（婆什迦罗第一的大书）是一部关于印度数学天文学的八章著作，包含当时此类著作相当标准的主题。它讨论了诸如：行星的经度；行星彼此以及与亮星的合；日食和月食；升落；以及新月。

婆什迦罗第一在其论著Mahabhaskariya Ⓣ（婆什迦罗第一的大书）中收录了三节内容，通过一个有理的分数给出了三角函数正弦函数的近似值。这些内容出现在该著作第7章。婆什迦罗第一给出的公式惊人地准确，使用该公式导致的最大误差小于百分之一。该公式是

$\sin x = 16x (\pi - x)/[5\pi^{2} - 4x (\pi - x)]$

并且婆什迦罗第一将这项工作归于阿耶波多。我们计算了该公式给出的值，并将其与$\sin x$在$x$从0到$\pi/_{2}$、步长为$\pi/_{20}$时的正确值进行了比较。

在629年，婆什迦罗第一写了一部评注，即AryabhatiyabhasyaⓉ（对阿耶波多著作的评注），评注的是阿耶波多的AryabhatiyaⓉ（阿耶波多的著作）。AryabhatiyaⓉ（阿耶波多的著作）包含33节讨论数学的诗句，其余部分则涉及数学天文学。婆什迦罗第一的评注只针对这33节数学诗句。他考虑了一次不定方程和三角公式的问题。在讨论AryabhatiyaⓉ（阿耶波多的著作）的过程中，婆什迦罗第一表达了他关于如何将某一特定矩形视为圆内接四边形的想法。他是第一个开启关于四边均不相等且没有对边平行的四边形讨论的人。

许多世纪以来用于π的近似值之一是√10。婆什迦罗第一批评了这个近似值。他遗憾于无法获得用直径精确测量圆周的方法，并且他清楚地相信π不是有理数。

在[11]、[12]、[13]和[14]中，Shukla讨论了婆什迦罗第一数学的一些特征，例如：数字和符号表示、数学的分类、第一阶方程的名称和解法、二次方程、三次方程和多个未知数的方程、符号代数、婆什迦罗第一著作中不寻常和特殊的术语、权重和度量、求解线性不定方程的欧几里得算法方法、婆什迦罗第一给出的说明阿耶波多规则的例子、用于求解天文学中出现的某些方程的表格，以及婆什迦罗第一对早期印度数学家著作的引用。

We have very little information about Bhaskara I's life except what can be deduced from his writings. Shukla deduces from the fact that Bhaskara I often refers to the Asmakatantra instead of the Aryabhatiya Ⓣ that he must have been working in a school of mathematicians in Asmaka which was probably in the Nizamabad District of Andhra Pradesh. If this is correct, and it does seem quite likely, then the school in Asmaka would have been a collection of scholars who were followers of Aryabhata I and of course this fits in well with the fact that Bhaskara I himself was certainly a follower of Aryabhata I.

There are other references to places in India in Bhaskara's writings. For example he mentions Valabhi (today Vala), the capital of the Maitraka dynasty in the 7th century, and Sivarajapura, which were both in Saurastra which today is the Gujarat state of India on the west coast of the continent. Also mentioned are Bharuch (or Broach) in southern Gujarat and Thanesar in the eastern Punjab which was ruled by Harsa for 41 years from 606. Harsa was the pre-eminent ruler in north India through the first half of Bhaskara I's life. A reasonable guess would be that Bhaskara was born in Saurastra and later moved to Asmaka.

Bhaskara I was an author of two treatises and commentaries to the work of Aryabhata I. His works are the Mahabhaskariya Ⓣ, the Laghubhaskariya Ⓣ and the Aryabhatiyabhasya Ⓣ. The Mahabhaskariya Ⓣ is an eight chapter work on Indian mathematical astronomy and includes topics which were fairly standard for such works at this time. It discusses topics such as: the longitudes of the planets; conjunctions of the planets with each other and with bright stars; eclipses of the sun and the moon; risings and settings; and the lunar crescent.

Bhaskara I included in his treatise the Mahabhaskariya Ⓣ three verses which give an approximation to the trigonometric sine function by means of a rational fraction. These occur in Chapter 7 of the work. The formula which Bhaskara gives is amazingly accurate and use of the formula leads to a maximum error of less than one percent. The formula is

$\sin x = 16x (\pi - x)/[5\pi^{2} - 4x (\pi - x)]$

and Bhaskara attributes the work as that of Aryabhata I. We have computed the values given by the formula and compared it with the correct value for $\sin x$ for $x$ from 0 to $\pi/_{2}$ in steps of $\pi/_{20}$.

In 629 Bhaskara I wrote a commentary, the Aryabhatiyabhasya Ⓣ, on the Aryabhatiya Ⓣ by Aryabhata I. The Aryabhatiya Ⓣ contains 33 verses dealing with mathematics, the remainder of the work being concerned with mathematical astronomy. The commentary by Bhaskara I is only on the 33 verses of mathematics. He considers problems of indeterminate equations of the first degree and trigonometric formulae. In the course of discussions of the Aryabhatiya Ⓣ, Bhaskara I expressed his idea on how one particular rectangle can be treated as a cyclic quadrilateral. He was the first to open discussion on quadrilaterals with all the four sides unequal and none of the opposite sides parallel.

One of the approximations used for π for many centuries was √10. Bhaskara I criticised this approximation. He regretted that an exact measure of the circumference of a circle in terms of diameter was not available and he clearly believed that π was not rational.

In [11], [12], [13] and [14] Shukla discusses some features of Bhaskara's mathematics such as: numbers and symbolism, the classification of mathematics, the names and solution methods of equations of the first degree, quadratic equations, cubic equations and equations with more than one unknown, symbolic algebra, unusual and special terms in Bhaskara's work, weights and measures, the Euclidean algorithm method of solving linear indeterminate equations, examples given by Bhaskara I illustrating Aryabhata I's rules, certain tables for solving an equation occurring in astronomy, and reference made by Bhaskara I to the works of earlier Indian mathematicians.