Indian Mathematics
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Indian Mathematics
May 5, 2005
Introduction:
Indian, in particular, Hindu, mathematics has not been given the credit or recognition that it deserves. Many of the foundational concepts used in all mathematics were first discovered by the Hindu Indians. This paper will discuss many of these concepts and how they were used in the fifth through the eighth centuries. Apart from direct testimony on the point, the literature of the Hindus furnishes unmistakable evidence to prove that the ancient Hindus possessed astonishing power of memory and concentration of thought. The science of mathematics, the most abstract of all sciences, must have an irresistible fascination for the minds of the Hindus. Mathematics is the science to which Indians have contributed the most. Our decimal system, place notation, numbers one through nine, and the ubiquitous 0, are all major Indian contributions to world science. Without them, our modern world of computer sciences, satellites, microchips, and artificial intelligence would all have been impossible. The majority of my writing will focus on a specific area of math called the shulba sutras, which consists of the majority of the discoveries made in geometry. This geometry fascinates me because of their purpose and meaning that is connected with everything they do. Math although seemingly very concrete, right and wrong, can be explained in a spiritual sense as well. The meanings behind all the numerical calculations are the actual significant part according the Vedic literature.
The Sulba Sutras
The Sulba Sutras, is an important part of the Vedic literature, which consists of a detailed analysis explaining the importance and interrelation between various branches of Vedic texts. Mr. Maharihsi Mahesh Yogi, has completely restored the thousands of years-old scattered Vedic Literature for the total significance of its theory and practice, and has organized it in the form of a complete science of consciousness. The Vedic literature is compiled into forty parts, including the four Vedas plus six sections each with six parts. The four Vedas, the Brahamanas, the Vendangas, the Upa-Vedas, and the Pratishakhya each “express a specific quality of consciousness,”(1) which means that we need to look beyond the surface to find the deeper meanings. There are four main Sulba Sutras, the Baudhayana, the Apastamba, the Manava, and the Katyayna.
One of the meanings of the Vedic Sulba Sutras is “string, cord or rope,”(1) which shows that the earliest geometrical and mathematical investigations among the Indians rose from the requirements of their religious rituals. “This could be a reference to the fact that measurements for the geometrical constructions are performed by drawing arcs with different radii and centers using a cord or sulba”.(1) The Sulba SÑ‹tras describes many geometrical properties and constructions such as the classical “Pythagorean” relationship between the sides of a right-angle triangle and arithmetical formulas such as calculating the square root of two accurate to thirteen decimal places. Beyond these constructions, there are deep and elegant symmetries such as a pair of formulas for converting a square to a circle and vice versa. Also there are beautiful blueprints for constructing citis or ceremonial platforms in images such as falcons, wheels and tortoises. At this level the Sutras seem to be a finely crafted manual for expert artists.
The general formats of the main Sulba Sutras are the same; each starts with sections on geometrical and arithmetical constructions and ends with detail s of how to build citis, which are known as the religious altars used in various ceremonies. There was a ritual which took place at an altar where foods, also sometimes animals, were sacrificed. Each of the citis are low platforms consisting of layers of carefully shaped and arranged bricks. The word citi, means consciousness and the word vedi means knowledge, and putting them together means pure, complete knowledge. Some are quite simple shapes such as a square or a rhombus while others are much more involved such as a falcon in flight with curved wings, a chariot wheel complete with spokes, or a tortoise with extended head and legs. These designs are particularly beautiful and elegant depictions of powerful and archetypal symbols, “the falcon as the great bird that can soar to heaven, the wheel as the wheel of life, and the tortoise as the representative of stability and perseverance”.(3)
The construction of a square into a circle with the exact same area is outlined in detail in the Vedic literature. In the ancient years from the fifth through the eighth century B.C. there were no sophisticated calculation devices so all calculations were done by hand. The way that this was done is to start with a square ABCD with center O. Draw an arc DG with center O so that OG is parallel to AD. Suppose that OG intersects DC at the point F. Let H be a point 1/3 of the distance from F to G. Then OH is the radius of the required circle.(1) See figure 1 below to get a visual understanding.
Figure 1.
r = OH = OF + FH
= OF + 1/3(OG – OF)
= a + 1/3(a*sqrt(2) – a)
r = a/3 (2 + sqrt(2)).
Then if we substitute pi = 3.141593, which is the ratio of the diameter to the circumference of the circle, and the sqrt (2) = 1.414214 we get the area of the constructed circle is
Area = pi(r) ^2 = 4.069011 . . . * a^2 which is 1.7% accurate of the correct value.
Next I will describe the procedure used to go in the opposite direction. To convert a circle into a square you must first divide the diameter into eight parts and one of the eight parts into