India – World Guru of Mathematics (Part – 7)

Posted on August 29, 2011

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TRIGNOMENTRY, GEOMETRY AND ALGORITHM

Even in the area of Geometry, Indian mathematicians had their contribution. The complex origins of trigonometry are embedded in the history of the simple word “sine,” a mistranslation of an Arabic transliteration of a Sanskrit mathematical term! The complex etymology of “sine” reveals trigonometry’s roots in Babylonian, Greek, Hellenistic, Indian, and Arabic mathematics and astronomy. The word geometry emerged from the Sanskrit word Giamiti which means measuring the earth, “Gia” meaning earth and “miti” meaning measure.

The word trigonometry emerged from the Sanskrit  word Trikonamiti meaning measuring triangular forms, “Tri” means three, “Konam” means angle and “miti” means to measure. Euclid is credited with the invention of Geometry in 300 BCE while the concept of Geometry in India emerged in 1000 BCE, from the practice of making fire altars in square and rectangular shapes. The treatise of Surya Siddhanta (4th century CE) describes amazing details of Trigonometry, which were introduced to Europe 1200 years later in the 16th century by Briggs.

There was an area of mathematical applications called Rekha Ganita (Line Computation). The Sulva Sutras, which literally mean ‘Rule of the Chord’, give geometrical methods of constructing altars and temples. The temple layouts were called Mandalas. Some of important works in this field are by Apastamba, Baudhayana, Hiranyakesin, Manava, Varaha and Vadhula.

The Arab scholar Mohammed Ibn Jubair al Battani studied Indian use of ratios from Retha Ganita and introduced them among the Arab scholars like Al Khwarazmi, Washiya and Abe Mashar who incorporated the newly acquired knowledge of algebra and other branches of Indian mathema into the Arab ideas about the subject.

The chief exponent of this Indo-Arab amalgam in mathematics was Al Khwarazmi who evolved a technique of calculation from Indian sources. This technique which was named by westerners after Al Khwarazmi as “Algorismi” gave us the modern term Algorithm, which is used in computer software.

Algorithm which is a process of calculation based on decimal notation numbers. This method was deduced by Khwarazmi from the Indian techniques geometric computation which he had studied. Al Khwarazmi’s work was translated into Latin under the title “De Numero Indico” which means ‘of Indian Numerals’ thus betraying its Indian origin. This translation which belongs to the 12th century A.D credited to one Adelard who lived in a town called Bath in Britian.

Thus Al Khwarazmi and Adelard could look upon as pioneers who transmit Indian numerals to the west. Incidents according to the Oxford Dictionary, word algorithm which we use in the English language is a corruption of the name Khwarazmi which literally means ‘(a person) from Khawarizm’, which was the name of the town where Al Khwarazmi lived. Today unfortunately’, the original Indian texts that Al Khwarazmi studied are lost to us, only the translations are available.

“The Arabs borrowed so much from India, the field of mathematics that even the subject of mathematics in Arabic came to known as “Hindsa” which means ‘from India’ and a mathematician or engineer in Arabic is called “Muhandis” which means ‘an expert in Mathematics’. The word “Muhandis” possibly derived from the Arabic term mathematics viz. Hindsa.”

TRIGONOMETRY

Although trigonometry now is usually taught beginning with plane triangles, its origins lie in the world of astronomy and spherical triangles. Before the sixteenth century, astronomy was based on the notion that the earth stood at the center of a series of nested spheres. To calculate the positions of stars or planets, one needed to use concepts we now refer to as trigonometry.

The earliest uses of trigonometric functions were related to the chords of a circle, and the recognition that the length of the chord subtended by a given angle x was (in modern terms) 2sin(x/2). The Greek astronomer and mathematician Hipparchus produced the first known table of chords in 140 BC. His work was further developed by astronomers Menelaus (ca. AD 100) and Ptolemy (ca. AD 100), who relied on Babylonian observations and traditions.

Babylonian and Greek influences mingled with rich native mathematical developments in India around AD 500 to produce a trigonometry closer to its modern form. Hindu mathematical works such as that of Aryabhata give tables of half chords, known by the term jya-ardha or simply jya, which bears the following relationship to our modern concept of sine: jya x = rsinx.

Jya here represents the half chord AM.

In Ganitapada 6, Aryabhata gives the area of a triangle as

“tribhujasya phalashariram samadalakoti bhujardhasamvargah”

That translates to: “for a triangle, the result of a perpendicular with the half-side is the area.”

Golden Age Indian mathematicians made fundamental advances in the theory of trigonometry. They used ideas like the sine, cosine and tangent functions (which relate the angles of a triangle to the relative lengths of its sides) to survey the land around them, navigate the seas and even chart the heavens. For instance, Indian astronomers used trigonometry to calculate the relative distances between the Earth and the Moon and the Earth and the Sun. They realized that, when the Moon is half full and directly opposite the Sun, then the Sun, Moon and Earth form a right angled triangle, and were able to accurately measure the angle as 17°. Their sine tables gave a ratio for the sides of such a triangle as 400:1, indicating that the Sun is 400 times further away from the Earth than the Moon.

Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 AD, contains the roots of modern trigonometry, including the first real use of sine’s, cosines, inverse sine’s, tangents and secants.

As early as the 6th Century AD, the great Indian mathematician and astronomer Aryabhata produced categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. Aryabhata also demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value. But, perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any calculation can only ever be an approximation, something not proved in Europe until 1761.

Another set of trigonometric functions, tangent and cotangent, developed from the study of the lengths of shadows cast by objects of various heights. Thales of Miletus used shadow lenghts to calculate the heights of the pyramids in around 600 BC. Both Indian and Arabic mathematics developed a trigonometric tradition based on shadow lengths, a tradition that, in turn, influenced European mathematics.

As for the word “trigonometry,” it first appeared as the title of a book Trigonometria (a translation of the Sanskrit term as sited earlier), published by Bartholomeo Pitiscus in 1595.

From India the sine function (jya) was introduced to the Arab world in the 8th century, where the term jya was transliterated into jiba or jyb. Early Latin translations of Arabic mathematical treatises mistook jibafor the Arabic word jiab, which can mean the opening of a woman’s garment at the neck. Accordingly, jiab was translated into the Latin sinus, which can mean “fold” (in a garment), “bosom,” “bay,” or even “curve.” And after that, the sinus became sine in English.

The functions secant and cosecant derive from tables first used by navigators in the fifteenth century.

The Buddhist Pagodas borrowed their plan of construction from the geometric grid of the Mandala used for constructing temples in India

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