Introduction
Bijaganita (IAST: Bījagaṇita) is a treatise on algebra and trigonometry written by the Indian astronomer Bhaskara II in the 12th century AD. It is the second volume of his main work Siddhānta Shiromani (“Crown of treatises”) alongside Lilāvati, Grahaganita and Golādhyāya.
The Bijaganita is divided into six parts:
- Introduction
- On Simple Equations
- On Quadratic Equations
- On Equations involving indeterminate Questions of the 1st Degree
- On Equations involving indeterminate Questions of the 2nd Degree
- On Equations involving Rectangles
The first two parts of the book cover basic algebraic topics such as solving linear and quadratic equations. The third and fourth parts deal with indeterminate equations, which are equations with more than one unknown. The fifth and sixth parts cover more advanced topics such as the construction of geometric shapes and the calculation of their areas and volumes.
Algebra
The Bijaganita is one of the earliest works on algebra to use symbolic notation. Bhaskara II used symbols to represent unknowns, coefficients, and operations such as addition, subtraction, multiplication, and division. He also developed a number of new algebraic methods, including the chakravala method for solving indeterminate quadratic equations.
Trigonometry
The Bijaganita also contains a section on trigonometry, which is the branch of mathematics that deals with the relationships between the sides and angles of triangles. Bhaskara II defined the six trigonometric functions (sine, cosine, tangent, cotangent, secant, and cosecant) and derived formulas for calculating them. He also developed a number of new trigonometric methods, including the sine addition and subtraction formulas.
Influence
The Bijaganita had a profound influence on the development of mathematics in India and beyond. It was translated into several languages, including Arabic and Persian, and was widely studied by mathematicians throughout the Islamic world. The book also had a significant influence on European mathematics, and its methods were later used by Renaissance mathematicians such as Fibonacci and Leonardo da Vinci.
Conclusion
The Bijaganita is a landmark work in the history of mathematics. It is one of the earliest works on algebra to use symbolic notation, and it contains a number of new and important algebraic and trigonometric methods. The book had a profound influence on the development of mathematics in India and beyond, and it continues to be studied by mathematicians today.
Example of a problem from the Bijaganita
One of the most famous problems from the Bijaganita is the following:
Two friends say to each other, “Give me 100, friend, and I shall be twice as rich as you.” The other replies, “If you give me 10, I shall be six times as rich as you.” What is the amount of their (respectively) capital?
To solve this problem, Bhaskara II used the following method:
- Let the capital of the first friend be x and the capital of the second friend be y.
- According to the first friend, if he receives 100 from the second friend, he will be twice as rich as the second friend. This can be expressed in the following equation:
x + 100 = 2y
- According to the second friend, if he receives 10 from the first friend, he will be six times as rich as the first friend. This can be expressed in the following equation:
y + 10 = 6x
- Solving these two equations simultaneously, we get the following solution:
x = 50 and y = 30
Therefore, the capital of the first friend is 50 and the capital of the second friend is 30.
This is just one example of the many problems that are solved in the Bijaganita. The book is a rich source of mathematical knowledge, and it continues to be studied by mathematicians today.
