VI Indian Mathematics A large and ancient civilization grew in the Indus River Valley in Pakistan beginning at least by 7,000 B.C.E. One of the older and larger sites is at Mehrgarh. The remains of this site are on the Kachi Plain near the Bolan pass in Baluchistan, Pakistan. The earliest people inhabiting Mehrgarh were semi-nomadic, farmed wheat, barley and kept sheep, goats and cattle. There was no use of pottery in the early era, from 7,000 to 5500 B.C.E. The structures were simple mud buildings with four rooms. The burials of males included more material goods in this time. From 5500 to 3500 B.C.E. there was use of pottery and more elaborate female burials. Stone and copper drills were used to create beads, and the remains of two men have dental holes drilled in their teeth. Mehrgarh was abandoned between 2600 and 2000 B.C.E., when the Harrappan civilization grew nearer to the Indus River. From 2300 to 1750 B.C.E. there was an advanced and large civilization in the Indus and Ghaggar-Hakra River Valleys in Pakistan. The Ghaggar-Hakra River system is largely dried up, now. Many sites have been excavated over a large area stretching from the foothills of the Himalayas to the Arabian sea. The most famous cities are Mohenjo-Daro and Harrappa. These were both planned cities which were larger than any cities of the time in Mesopotamia or Egypt. We have thousands of stone and pottery objects with the as yet undeciphered script of these people on them. There were standard weights and measures in regular geometric shapes such as cubes, cones and cylinders with weights in the ratios of 1, 2 , 5, 10, 20, 50, 100, 200, 500 and 1/2, 1/10 and 1/20. There was a ruler with a standard unit of length which was subdivided into ten equal intervals. Their mud bricks were constructed in integer multiples of this length. We thus have evidence of a decimal number system. There were baths in individual houses with sewer systems serving the cities, and larger public baths. It is clear that mathematics played a role in such a large and organized, agriculturally based, society. After 1750 the Indus valley cities were abandoned quite possibly due to climactic conditions and disruptive changes in the landscape which led to the diverting of large river channels. By 1700 to 1000 B.C.E. there was oral transmission of knowledge called the Vedas. They were later written in Sanskrit, the root language of all Indo-European languages, by the second century B.C.E. The old writing is on birch bark and palm leaves, so it disintegrates over time. Veda means knowledge in Sanskrit. The Vedas are divided into four main categories. These are: • The Samhitas, which are collections of hymns, rituals for sacrifices and chants. The four Smahitas are the The Rig-veda, Sama-veda, Yajur-veda and the Atharva- veda. These are the oldest vedas, dating from around 1500 to 1000 B.C.E. • The Brahmanas are prose technical treatises on performing sacrifices, together with the meanings associated to the sacrifices. Each Brahmana is associated with one of the Samhitas. They were composed from 1000 to about 600 B.C.E. • The Aranyakas explain some more dangerous rituals and were composed about 700 B.C.E. • The Upanishads are the newest vedas from about 800 to 500 B.C.E. They are more philosophical in tone and discuss knowledge and meditation as paths to spiritual awakening. They are called the Vedantas, meaning the end of the Vedas. The mathematics in the Vedas occurs in appendices called Vedangas. The fifth and sixth vedangas are the Jyotis (astronomy) and the Kalpa (rituals). One instance of geometry is the construction of a fire-altar in the shape of a falcon. The construction involves parallelograms, triangles, squares and trapezoids and had to be done with extreme care. Estimates for Pi were done here. Equivalences of areas in differing shapes led to the ideas of squaring circles and, conversely, circling squares. The Sulba-Sutras are further works expanding upon the Vedangas, possibly supplementing the Kalpa. Sulba refers to a chord of rope used to construct altars. A sutra was a stylized form of vedic literature. They date from about 800 to 200 B.C.E. Three authors can be singled out for mathematical exposition in the Sulba-Sutras. Baudhayana who worked between 800 and 600 B.C.E had Pythagorean triples and a form of the Pythagorean Theorem in his teachings. The work of Apastamba (around 600 B.C.E.) contains the approximation to 2 which is given by 1 + 1/3 + 1/12 + 1/((12)(34)) = 1.4142156, which is correct to 5 decimal places. The early Indians were aware that only ! approximations could be found for irrational numbers. A third such author is Katyayana, who worked about 200 B.C.E. In the sixth century B.C.E. religious reformers appeared in India. The first we know of is Mahavira, who preached the Jain religion. The Jains developed a fair amount of mathematics due to their belief in the immortality of souls, and because they became involved in the commerce of the day. Mahavira preached Ahimsa (non-violence) and Satya (truthfulnesss) and that the soul is seduced by the influence of karmic, material particles stuck to it resulting from erroneous choices in life. The true destiny of the soul is spiritual liberation from the illusions of karma and reality. He denounced worship of gods, founded orders of monks and nuns and was carrying on the Jain religion which had preceded him. Gautama the Buddha lived about 550 to 480 B.C.E. he was a reformer who denounced organized religion. He believed that the root cause of human suffering is ignorance, and devised ways to eliminate this ignorance. The Jaina mathematics was very different from Mediterranean mathematics in its consideration of large numbers and of infinity. Since Jains believed that the world never began and never will end, and since souls ultimately become enlightened and leave this center earth of karmic illusion, there has to be an infinity of souls. They ended up with five different types of infinity, although not in a mathematically careful way. Their cosmology had a time period of 2588 years in it. This is an astoundingly large number, the like of which is not found in other ancient cultures. In the power vacuum left by the death of Alexander The Great, Chandragupta Maurya established the! Mauryan Empire which grew from the eastern side of India to encompass most of India and Pakistan. Chandragupta was a Jain. He defeated an invasion by Seleucus I, who had taken over the eastern portions of Alexander’s empire. He ruled from 322 to 298 B.C.E. He arranged for a Seleucid princess to marry a Mauryan noble. Thus there would have been transfers of knowledge between these two cultures. Babylonian and Greek philosophy, mathematics and astronomy would have passed to the east and analogous Indian knowledge would have passed to the west. During this period of time there came to be less use of altars in India and the need for mathematics in the society changed to more practical matters. At the end of his life, Chandragupta went to the south of India, became a Jain ascetic and ended his life by starving. The next great figure on the stage of Indian history is the legendary ruler Ashoka, (304 to 232 B.C.E.) a grandson of Chandragupta, who ruled India from 272 to 232 B.C.E. He expanded the Mauryan Empire to Persia in the west and Assam in the east. After witnessing many deaths in the battle of Kalinga which he himself fought, he embraced Buddhism and worked to spread it throughout southeast Asia. He had contacts with the Seleucids and with the Mediterranean cultures. The oldest extant written record of the Jaina mathematics is the Bakshali Manuscript which is on birch bark and is dated to be from between 300 and 600 A.D. It is written in sutra format. This work contains good square root approximations, solutions of indeterminate algebra problems in several unknowns (in our terminology) a decimal place value system, use of a dot for zero, quadratic equations and the quadratic formula. # # r & 2 & % % ( ( 2 r $ 2n' The square root algorithm is N = n + r = n + " % ( . 2n % # r & ( % 2% n + ( ( $ $ 2n' ' This is very accurate for several decimal places. The most prosperous, long lasting and large empire in Indian history is that of the Guptas, which lasted fr!om 320 to 720 A.D. The most prestigious ruler of the Guptas was Chandragupta II. The arts, literature and sciences flowered during this era. The epic Indian works of literature, the Ramayana and the Mahabbharatta, were completed during this time. There was internal peace and trade abroad. The Hindu Brahmin religion spread across India. A first known center of learning was established at Nalanda, near current-day Patna in Bihar state in northern India during the fifth century A.D. The mathematics of this classical period of Indian history is in the Siddhantas, which were astronomical treatises. We know of 18 of these which were written, but we only have parts of 5 of them surviving today. Aryabhata ( 476 to 550 A.D.), a Jain from Pataliputra in northern India, near current-day Patna, wrote a small book on astronomy, the Aryabhatia, around 499 A.D. and the Arya Siddhanta. The latter book has not survived, but the Aryabhatia has and contains arithmetic, plane and spherical trigonometry, continued fractions, quadratic equations, sums of power series and tables of sines.