# Indian Mathematics - Redressing the balance

### Ian G Pearce

### The Classical period: III. Brahmagupta, and the influence on Arabia

Brahmagupta was born in 598 AD, possibly in Ujjain (possibly a native of Sind) and was the most influential and celebrated mathematician of the Ujjain school.

It is important here to note that one must not ignore contributions made by Varahamihira, who was an influential figure at the same Ujjain school during the 6

Returning to Brahmagupta, he not only elaborated the mathematical results of Aryabhata but also made notable contributions to many topics.

L Gurjar describes his results as:

I believe that the Indian influence on Arabic work is often ignored or played down and consider this to be unfortunate (at the least). This issue is definitely worthy of discussion as it is noticeable that much is made of the Greek influence on Arabic works but far less of the Indian influence, which in retrospect was quite significant.

His second work was written much later in his life in 665 AD and was titled

In the BSS among the major developments are those in the areas of:

Brahmagupta possessed a greater understanding of the number system (and place value system) than anyone to that point. Many rules are given and an advanced technique for multiplication exhibited.

Operations with zero, Brahmagupta was the first to attempt to divide by zero, and while his attempts; showing $\Large\frac n 0\normalsize = \infty$, were not ultimately successful they demonstrate an advanced understanding of an extremely abstract concept.

Operations with negative numbers.

Theory of Arithmetic progressions.

Algorithm for calculating square roots that is equivalent to the Newton-Raphson iterative formula, but clearly pre-dates it by many centuries. (See chapter 8.6)

Brahmagupta stands in high esteem for his contributions to this topic. A rule that he gives for finding the values of the diagonals of cyclic quadrilaterals is generally known as "Ptolemy's theorem". Ptolemy 'pre-dated' Brahmagupta by 500 years, so it is wholly reasonable to attribute the 'discovery' of these rules to him. However, Brahmagupta's independent discovery should still be considered a remarkable achievement.

Furthermore, some of his work (regarding right angled triangles, which was later developed by Mahavira, Bhaskara II, et al) is often attributed to Fibonacci (13

As L Gurjar quotes:

Solutions to $Nx^2 + 1 = y^2$,

He also made many other contributions to solving a variety of algebraic equations, including $ax + c = by$ (which is the focus of a paper by P Majumdar).

Brahmagupta may have been one of the first mathematicians to recognise that the quadratic equation has

In his other work, the content is far more 'pure' astronomy, but an interpolation formula used to calculate values of sines bears great similarity to the Newton-Stirling interpolation formula, which is clearly of great historical and mathematical interest.

Without a doubt, Brahmagupta made remarkable contributions to mathematics (and astronomy) and his work continued to be influential for many centuries. In 860 AD an extensive and important commentary on the BSS was written by Prthudakasvami (or Prithudaka Swami). His work was extremely elaborate and unlike many Indian works did not 'suffer' brevity of expression.

The spread of Buddhism (around 500 AD) into China resulted in a period of cultural and scientific exchanges lasting several centuries. Chinese scholars are known to have translated the work of Brahmagupta; this highlights not only the quality of the work but the influence it had on the world outside India. R Gupta mentions four '

This discussion helps to highlight the influence that Indian mathematics had on Arabic mathematics, and ultimately, through Latin translations, on European mathematics, an influence that is considerably neglected. It must be argued that sufficient credit has not been given.

There seems to have been relatively little further 'interaction' between Indian and Arab scholars, and thus Indian works had limited, if any, further influence on mathematical developments in other countries. However Indian mathematics continued to flourish independently throughout the subcontinent for another 400 years, and some of the most outstanding contributions to the history of world mathematics were in fact made during this time period.

It is important here to note that one must not ignore contributions made by Varahamihira, who was an influential figure at the same Ujjain school during the 6

^{th}century. He is thought to have lived from 505 AD till 587 AD and made only fairly small contributions to the field of mathematics, he is described by Ifrah as:However he increased the stature of the Ujjain school while working there, a legacy that was to last for a long period, and although his contributions to mathematics were small they were of some importance. They included several trigonometric formulas, improvement of Aryabhata's sine tables, and derivation of the...One of the most famous astrologers in Indian history.[EFR/JJO'C18, P 1]

*Pascal triangle*by investigating the problem of computing binomial coefficients.Returning to Brahmagupta, he not only elaborated the mathematical results of Aryabhata but also made notable contributions to many topics.

L Gurjar describes his results as:

.His contributions to mathematics are found in two works, the first of which..Unique in the history of world mathematics.[LG, P 91]

*Brahmasphutasiddhanta*(BSS) must be considered one of the important mathematical works from this early period, not only of India, but also of the world. Not only was its mathematical content of an exceptional quality, but the work also had a significant influence on the burgeoning scientific awakening in the Arab empire.I believe that the Indian influence on Arabic work is often ignored or played down and consider this to be unfortunate (at the least). This issue is definitely worthy of discussion as it is noticeable that much is made of the Greek influence on Arabic works but far less of the Indian influence, which in retrospect was quite significant.

His second work was written much later in his life in 665 AD and was titled

*Khandakhayaka*. Although the BSS contains 25 chapters it is generally considered that the first ten chapters make up the first work, and that at a later date Brahmagupta made revisions and additions.In the BSS among the major developments are those in the areas of:

**Arithmetic:**Brahmagupta possessed a greater understanding of the number system (and place value system) than anyone to that point. Many rules are given and an advanced technique for multiplication exhibited.

Operations with zero, Brahmagupta was the first to attempt to divide by zero, and while his attempts; showing $\Large\frac n 0\normalsize = \infty$, were not ultimately successful they demonstrate an advanced understanding of an extremely abstract concept.

Operations with negative numbers.

Theory of Arithmetic progressions.

Algorithm for calculating square roots that is equivalent to the Newton-Raphson iterative formula, but clearly pre-dates it by many centuries. (See chapter 8.6)

**Geometry:**Brahmagupta stands in high esteem for his contributions to this topic. A rule that he gives for finding the values of the diagonals of cyclic quadrilaterals is generally known as "Ptolemy's theorem". Ptolemy 'pre-dated' Brahmagupta by 500 years, so it is wholly reasonable to attribute the 'discovery' of these rules to him. However, Brahmagupta's independent discovery should still be considered a remarkable achievement.

Furthermore, some of his work (regarding right angled triangles, which was later developed by Mahavira, Bhaskara II, et al) is often attributed to Fibonacci (13

^{th}c.) and Vieta (16^{th}c.), highlighting the constant European bias.As L Gurjar quotes:

...(Brahmagupta) derived certain results, which were troubling the brains of Western mathematicians as late as the 17^{th}century. [LG, P 91]

**Algebra:**Solutions to $Nx^2 + 1 = y^2$,

*Pell's equation*, his most outstanding contribution to mathematics. (See chapter 8.6)He also made many other contributions to solving a variety of algebraic equations, including $ax + c = by$ (which is the focus of a paper by P Majumdar).

Brahmagupta may have been one of the first mathematicians to recognise that the quadratic equation has

*two*solutions.In his other work, the content is far more 'pure' astronomy, but an interpolation formula used to calculate values of sines bears great similarity to the Newton-Stirling interpolation formula, which is clearly of great historical and mathematical interest.

Without a doubt, Brahmagupta made remarkable contributions to mathematics (and astronomy) and his work continued to be influential for many centuries. In 860 AD an extensive and important commentary on the BSS was written by Prthudakasvami (or Prithudaka Swami). His work was extremely elaborate and unlike many Indian works did not 'suffer' brevity of expression.

The spread of Buddhism (around 500 AD) into China resulted in a period of cultural and scientific exchanges lasting several centuries. Chinese scholars are known to have translated the work of Brahmagupta; this highlights not only the quality of the work but the influence it had on the world outside India. R Gupta mentions four '

*Brahminical*' translations in a paper. During this time the decimal system and notation was adopted by Chinese scholars and as R Gupta states:... Indian mathematical astronomy exerted a great influence in China during the (glorious) Thang Period (618-907). [RG4, P 11]The lasting legacy of the BSS however was its translation by

*Arab*scholars and its contribution to the 'forward progress' of mathematics. These translations, along with translated work of Aryabhata and (possibly) the*Surya Siddhanta*were responsible for alerting the Arabs, and the West to Indian mathematics (and astronomy), as G Joseph states:...This was to have momentous consequences for the development of the two subjects. [GJ, P 267]Of particular interest is the well told story of the Indian scholar who traveled to Baghdad, at the behest of Caliph al-Mansur (early ruler of the Arab Empire). R Gupta reports the story as such:

... In the year 156 (772/773 AD) there came to Caliph al-Mansur a man (an Ujjain scholar by the name of Kanka) from India, an expert in hisab (computation) bringing with him a work called Sindhind (i.e. Siddhanta) concerning the motions of the planets. [RG, P 12]A translation of this work, thought to be Brahmagupta's BSS, was subsequently carried out by al-Fazari (and an Indian scholar) and had a far-reaching influence on subsequent Arabic works. The famous Arabic scholar al-Khwarizmi (credited with 'inventing' algebra) is known to have made use of the translation, called

*Zij al-Sindhind*. Al-Khwarizmi (c. 780-850 AD) is known to have written two subsequent works, one based on Indian astronomy (Zij) and the other on arithmetic (possibly*Kitab al-Adad al-Hindi*). Later Latin translations of this second work (*Algorithmi De Numero Indorum*), composed in Spain around the 11th century, are thought to have played a crucial role in introducing the Indian place-value system numerals and the corresponding computational methods into (wider) Europe. Both Indian astronomy and arithmetic had a huge impact in Spain.This discussion helps to highlight the influence that Indian mathematics had on Arabic mathematics, and ultimately, through Latin translations, on European mathematics, an influence that is considerably neglected. It must be argued that sufficient credit has not been given.

There seems to have been relatively little further 'interaction' between Indian and Arab scholars, and thus Indian works had limited, if any, further influence on mathematical developments in other countries. However Indian mathematics continued to flourish independently throughout the subcontinent for another 400 years, and some of the most outstanding contributions to the history of world mathematics were in fact made during this time period.