Bhaskaracharya's Bijaganitham

Introduction

The most ancient and advanced among the world civilizations viz., The Hindu Civilization had contributed substantially for the development of scientific knowledge of humanity. The sacred Vedas integrated the spiritual and physical sciences and our sages revealed the scientific truth through realization. Among the various branches of Physical Sciences, Mathematics and Astronomy occupy prominent places in Indian contribution to the world. The concept of zero is acknowledge as of Indian origin. The decimal place value system with numerals 1 to 9 was in use in Bharat even from Vedic period. These numerals which were introduced to the western World through Arabs came to be known as Arab Numerals. Number Systems with letters of alphabet and familiar material objects helped presentation of mathematical problems through poetry. Our forefathers had made the process of learning Mathematics a pleasant experience by the amalgamation of Poetry and Mathematics.

Algebra or Bija Ganitha was recognized as the seed or Bija of Arithmetic. Calculations with assigned symbols for the unknown quantities was called Bija Ganitha by Pradhudaka Swamy (860 AD). Brahma Gupta named it as Kuttaka Ganitha. Some others called it as 'Avyaktha Ganitha' to distinguish it from Vyaktha Ganitha or Arithmetic of real number's Bhaskaracharya (1150 AD) defined Bija Ganitha as Mathematics of unknown quantities which would enhance the intellectual capacity of ordinary persons. He declared that this branch of Mathematics was invented by Mathematicians of earlier period and he had only explained it for the benefit of the common man. Hence it is not possible to trace the origin of Bija Ganitha in India. Narayana (1350 AD) had however stated that its source is Brahma himself. Bhaskara's Bija Ganitha is the second part of his comprehensive mathematical work called Sidhanta Siromany; its first part dealing with Arithmetic is known as 'Lilavathi'. The other two parts are 'Goladhyaya' and 'Graha Ganitha' which deal with Spherical Astronomy and motion of planets.

The subject matter of the text Bija Ganitham includes two parts; Basic Operations and Analysis. The basic mathematical operations of addition, subtraction, multiplication and division using positive and negative unknown quantities, operations with zero and surds are explained in the first part. It also includes the solution of indeterminate algebriaic equations of the first degree. Some of these topics appear in 'Lilavathi' also, as they are of interest in arithmetic.

The second part of the book is devoted to the solution of algebraic equations of the second degree involving one or more variables. Integer solution of indeterminate equation of the second degree calls for application of intelligence of higher level. Aryabhata, the pioneer of India Mathematics and Astronomy has dealt with the first degree which he called as 'Kuttaka Vyavahara'. It was Brahma Gupta (628 AD) who invented the method of solution of second degree indeterminate equation known as 'Bhavana'. This was extended and elaborated by Baskara II. He invented the general iterative method of solution of such equation as 'Chakravala' which cab be applied for solution of indeterminate equations in general.

European Mathematicians however designated the second degree indeterminate equation as Diophantus equation in honour of the Greek Mathematician of ancient period. Euler called it Pellian equation after John Pell, another Mathematician. The exact contribution of these two European Mathematicians is still unknown. The great Indian Mathematicians Brahma Gupta and Bhaskara who actually devised the method of solution of the equation have not been given the credit and recognition rightfully due to them. It is known that Fermat, challenged Fernicle to solve one of the equations solved by Bhaskara in a communication. This confirms that Bhaskara's works were not known to European Mathematicians even after a period of at least 500 years.

The general solution of the second degree indeterminate equation of the form Dx²+-y² in integer roots often necessitates the use of Chakravala method of Bhaskara and Bhavana method of Brahma Gupta. Hence the equation should be designated as Brahma Gupta-Bhaskara equation. The terminology used to identify the unknown variables by Bhaskara includes 'Yavat-thavat', Kalaka, Neelaka, Peetha etc. representing various colours. Hence the unknown quantities are classified as 'Varna' or colours. In this text, we will be using symbols such as x, y, z, etc. to represent various unknowns in order to make it familiar to the present generation.

The terms such as Squares, Cubes, Square roots, Cube roots etc. used in arithmetic bear the same meaning in the case of unknown quantities. Equations containing numerical coefficients and constant terms along with the powers and products of unknown quantities are solved by appropriate methods. The practice of identifying negative quantities by dot above was in vogue at the time. Spread over 11 chapters containing 65 sets of rules and 103 examples, the book deals extensively with the integer solution of indeterminate equations having multiple for imagination. It will be interesting to understand the level of knowledge existed in Bharat about 850 years ago while the rest of the world was not advanced to that level. We shall offer our respects at the feet of our great, great ancestors.

This translation includes the Sanskrit text with translation of the stanzas and detailed explanation. All the problems are fully solved to make the contents understandable. I hope that this will be well received by authorities for assimilation in School curricula of this country.

About the Author

Dr. V. Balakrishna Panicker is M. Tech in Mechanical Engineering from University of Kerala and Ph. D. from Indian Institute of Science, Bangalore. He was the Principal of N. S. S. College of Engineering Kasaragod and K M C T College of Engineering Kozhikode. He was the Joint Director of Fluid Control Research Institute, Palakkad. He has translated several ancient Sanskrit books on Mathematics into Malayalam which include Aryabhatiyam, Baudhayana Sulbasutram, Lilavati, Bijaganitham Vedanga Jyothisham etc. He is associated with several non governmental organizations like Chinmaya Mission, Bharatiya Vicharakendram Swadeshi Science Movement etc. He is one of the Vice Presidents of Bharatiya Vidya Bhavan, Palakkad Kendra and the President of Bharatiya Vidya Niketan Kerala State.