# Indian Sciences – Mathematics

Mathematics is the fundamental building block of everything in the Universe. Indians have been constantly contributing towards its development for at least 4 millennia.

Mathematics is the fundamental building block of everything in the Universe. It wouldn’t be wrong to call it a ‘language’, complex one at that, but nonetheless. The patterns, the sequences and such can be used to define any physical object or phenomena in existence. Mathematics helps us understand and do these. As for the history of it, in short, the concept was realized when prehistoric people started using tallying on bones to have a ‘count’ of their collection of physical objects (could be food items or days, seasons and such).

“The Indians have an extremely subtle and penetrating intellect, and when it comes to arithmetic, geometry and other such advanced disciplines, other ideas must make way for theirs. The best proof of this is the nine symbols with which they represent each number no matter how large.”

A manuscript found in a Spanish monastery records (976 AD).

Coming back to the topic at hand, mathematics in India, pieces of evidence of practical mathematics used by the Indus Valley Civilization have been uncovered. Bricks of a set proportion (4:2:1) were used as they were considered to be better suited for the stability of a brick structure. Weights in geometrical shapes were produced, which showed that the people possessed at least the basic knowledge of geometry. They even had a ruler which could be used for measurement to a high degree of accuracy. Hollow cylindrical objects made of shell and found are demonstrated to have the ability to measure angles in a plane, as well as to determine the position of stars for navigation.

Pieces of evidence for the use of large numbers were found in the religious texts from the Vedic period. Numbers as high as 1012 were included in the texts during 1200-900 BCE. In the 4th stanza of Purusha Sukta, we can see that the concept of fractions was known as well (mention of 3/4th and 1/4th).

Sulba Sutras, written by Baudhayana, is one of the oldest known mathematical texts in existence. The text lists rules for the construction of fire altars. In it, an explicit statement of the now-known Pythagoras’ theorem and its applications in various geometric constructions is seen. They reflect a blending of geometric and subtle algebraic thinking and insight which we associate with Euclid. In fact, the Sulba construction of a square equal in area to a given rectangle is exactly the same as that given by Euclid several centuries later.

Pythagoras’ theorem was known in other ancient civilizations like the Babylonian, but the emphasis there was on the numerical and not so much on the proper geometric aspect while in the Sulbasutras one sees depth in both aspects – especially the geometric. Baudhayana even gives a formula for the square root of 2, which is accurate up to five decimal places. Apart from these, he is also said to have been the first one to calculate the value of Pi, which is an integral part of geometry now.

Another important contribution was given by Panini, although a Sanskrit grammarian. His grammar includes early use of Boolean logic, the null operator, and of context-free grammar, and includes a precursor of the Backus–Naur form (used in the description programming languages).

During the post-Vedic period, Pingala is most notable among other mathematicians. He stumbled upon both Pascal’s triangle (as it is known now) and binomial coefficients as he worked on the enumeration of syllabic combinations. His work also contains the basic idea of Fibonacci Numbers.

Between 400 BCE – 200 CE, Jain mathematicians acted as the links between the mathematics of the Vedic period and the Classical period. They were fascinated with the enumeration of large numbers and infinite. Hence, they classified numbers into enumerable, innumerable and infinite. They further classified infinite into five; infinite in: one direction, two directions, area, everywhere and perpetually. Jains also devised notations for simple powers and exponents. This helped them in defining simple algebraic equations. They have also described fractions, series, set theory and logarithms.

The use of decimals was first recorded in India, which was then transmitted to Islamic countries and then on to Europe. The Earliest written evidence of decimals in India dates to 595 CE in Gujarat. However, there are textual sources from the 1st century CE where the author talks about decimal usage in word form.

Bakhshali Manuscript is the oldest existing mathematical manuscript in India. Written between the 8th – 12th century CE, it contains topics such as (fractions, square roots, profit and loss, simple interest, the rule of three, and regula falsi), algebra (simultaneous linear equations and quadratic equations), arithmetic progressions and geometric problems (including problems about volumes of irregular solids).

The period between 400 – 1600 CE is known as the golden age of mathematics. Mathematicians like Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava of Sangamagrama and Nilakantha Somayaji, all come in this period. Aryabhata is one of the most well-known Indian mathematicians to ever exist. His contributions: In trigonometry:

• Introduced trigonometric functions.
• Defined sine, cosine, inverse sine and so on.
• Tables of sine, cosine.
• Spherical trigonometry.

In Algebra:

• Gave solutions of simultaneous quadratic equations.
• Whole number solutions of linear equations by a method equivalent to the modern method.
• The general solution of the indeterminate linear equation.

And of course, zero. He showed that zero was not only a numeral but also a symbol and a concept.
Varahamira’s contributions were towards trigonometry, including sine and cosine tables and the formulas relating to them.

Brahmagupta worked with two sections: “basic operations” (including cube roots, fractions, ratio and proportion, and barter) and “practical mathematics” (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).

Bhaskara I developed Aryabhata’s work and gave solutions of indeterminate equations, a rational approximation of the sine function, and also a formula for calculating the sine of an acute angle without the use of a table, correct to two decimal places.

Mahavira Acharya wrote treatises about a wide range of mathematical topics, which include zero, squares, cubes, square and cube roots, plane and solid geometry, and areas of ellipses and quadrilaterals inside a circle. He further stated that the square roots of negative numbers do not exist.

Bhaskara II wrote a number of important treatises. The most known of these is Siddhanta Shiromani. His contributions were:Arithmetic:

• Interest computation.
• Arithmetical and geometrical progressions.
• Plane geometry.
• Solid geometry.
• Solutions of combinations.
• Gave proof for division by zero being infinity.

Algebra:

• The recognition of a positive number having two square roots.
• Operations with products of several unknowns.
• Solutions of quadratic, cubic, and quartic equations.
• Solutions of indeterminate cubic, quartic equations.

Calculus:

• Conceived differential calculus.
• Discovered the derivative.
• Discovered the differential coefficient.
• Developed differentiation.
• Derived the differential of the sine function.
• Calculated the length of the Earth’s revolution around the Sun to 9 decimal places.

Madhava of Sangamagrama founded the Kerala school of astronomy and mathematics. Among its members was Nilakantha Somayaji. The school flourished between the 14th and 16th centuries. The members developed a number of important mathematical concepts. The most important of these was the series expansion for trigonometric functions (series for sine, cosine and inverse tangent). The discovery of this series of expansions of the calculus was an achievement because these were done centuries before calculus was developed in Europe. According to Charles Whish, the Kerala mathematicians had “laid the foundation for a complete system of fluxions” and these works abounded “with fluxional forms and series to be found in no work of foreign countries.”

Florian Cajori, in his widely acclaimed text on the history of mathematics, remarks:

“… it is remarkable to what extent Indian mathematics enters into the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian. Think of our notation of numbers, brought to perfection by the Hindus, think of the Indian arithmetical operations nearly as perfect as our own, think of their elegant algebraical methods, and then judge whether the Brahmins on the banks of the Ganges are not entitled to some credit.”

And further says that

“Unfortunately, some of the most brilliant results in indeterminate analysis, found in the Hindu works, reached Europe too late to exert the influence they would have exerted, had they come two or three centuries earlier.”

So, it can be seen that many of the mathematical theorems we study now in the modern era, were known to ancient Indians. What shadowed them was the lack of proper documentation and dissemination.