**Surya Siddhanta**

Though its authorship is unknown, the Surya Siddhanta (c. 400) contains the roots of modern trigonometry. Some^{ }authors consider that it was written under the influence of Mesopotamia and Greece.

This ancient text uses the following as trigonometric functions for the first time:

**Sine (***Jya*).**Cosine (***Kojya*).**Inverse sine (***Otkram jya*).

It also contains the earliest uses of:

**Tangent.****Secant.**

The Hindu cosmological time cycles explained in the text, which was copied from an earlier work, gives:

- The average length of the sidereal year as
**365.2563795**days, which is only 1.4 seconds longer than the modern value of**365.2563627**days. - The average length of the tropical year as
**365.2421756**days, which is only 2 seconds shorter than the modern value of**365.2421988**days.

**Aryabhata I**

Aryabhata (476-550) wrote the *Aryabhatiya.* He described the important fundamental principles of mathematics in 332 shlokas. The treatise contained:

- Quadratic equations
- Trigonometry
- The value of π, correct to 4 decimal places.

Aryabhata also wrote the *Arya Siddhanta*, which is now lost. Aryabhata’s contributions include:

**Trigonometry:**

- Introduced the trigonometric functions.
- Defined the sine (
*jya*) as the modern relationship between half an angle and half a chord. - Defined the cosine (
*kojya*). - Defined the versine (
*utkrama-jya*). - Defined the inverse sine (
*otkram jya*). - Gave methods of calculating their approximate numerical values.
- Contains the earliest tables of sine, cosine and versine values, in 3.75° intervals from 0° to 90°, to 4 decimal places of accuracy.
- Contains the trigonometric formula
*sin (n + 1) x – sin nx = sin nx – sin (n – 1) x – (1/225)sin nx*. - Spherical trigonometry.

**Arithmetic:**

- Continued fractions.

**Algebra:**

- Solutions of simultaneous quadratic equations.
- Whole number solutions of linear equations by a method equivalent to the modern method.
- General solution of the indeterminate linear equation .

**Mathematical astronomy:**

- Proposed for the first time, a heliocentric solar system with the planets spinning on their axes and following an elliptical orbit around the Sun.
**Accurate calculations for astronomical constants, such as the:**- Solar eclipse.
- Lunar eclipse.
- The formula for the sum of the cubes, which was an important step in the development of integral calculus.

**Calculus:**

**Infinitesimals:**- In the course of developing a precise mapping of the lunar eclipse, Aryabhata was obliged to introduce the concept of infinitesimals (
*tatkalika gati*) to designate the near instantaneous motion of the moon.

- In the course of developing a precise mapping of the lunar eclipse, Aryabhata was obliged to introduce the concept of infinitesimals (
**Differential equations:**- He expressed the near instantaneous motion of the moon in the form of a basic differential equation.

**Exponential function:**- He used the exponential function
*e*in his differential equation of the near instantaneous motion of the moon.

- He used the exponential function

**Bhaskara II**

Bhāskara II (1114–1185) was a mathematician-astronomer who wrote a number of important treatises, namely the Siddhanta Shiromani, Lilavati, Bijaganita, Gola Addhaya, Griha Ganitam and Karan Kautoohal. A number of his contributions were later transmitted to the Middle East and Europe. His contributions include:

**Arithmetic:**

- Interest computation
- Arithmetical and geometrical progressions
- Plane geometry
- Solid geometry
- The shadow of the gnomon
- Solutions of combinations
- Gave a proof for division by zero being infinity.

**Algebra:**

- The recognition of a positive number having two square roots.
- Surds.
- Operations with products of several unknowns.
- The solutions of:
- Quadratic equations.
- Cubic equations.
- Quartic equations.
- Equations with more than one unknown.
- Quadratic equations with more than one unknown.
- The general form of Pell’s equation using the chakravala method.
- The general indeterminate quadratic equation using the chakravala method.
- Indeterminate cubic equations.
- Indeterminate quartic equations.
- Indeterminate higher-order polynomial equations.

**Geometry:**

- Gave a proof of the Pythagorean theorem.

**Calculus:**

- Conceived of differential calculus.
- Discovered the derivative.
- Discovered the differential coefficient.
- Developed differentiation.
- Stated Rolle’s theorem, a special case of the mean value theorem (one of the most important theorems of calculus and analysis).
- Derived the differential of the sine function.
- Computed π, correct to 5 decimal places.
- Calculated the length of the Earth’s revolution around the Sun to 9 decimal places.

**Trigonometry:**

- Developments of spherical trigonometry
- The trigonometric formulas: