sign in sign up
<<
Home , Indian mathematics, Khandakhadyaka, Mahavira (mathematician)

VI Indian

A large and ancient civilization grew in the Indus River Valley in 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 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 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, and 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 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 . They were later written in , 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, 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 are prose technical treatises on performing sacrifices, together with the meanings associated to the sacrifices. Each 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 . The fifth and sixth vedangas are the Jyotis () and the Kalpa (rituals). One instance of is the construction of a fire-altar in the shape of a falcon. The construction involves parallelograms, triangles, squares and and had to be done with extreme care. Estimates for were done here. Equivalences of in differing shapes led to the ideas of squaring and, conversely, circling squares.

The Sulba- are further works expanding upon the Vedangas, possibly supplementing the Kalpa. Sulba refers to a chord of rope used to construct altars. A 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 . A third such author is Katyayana, who worked about 200 B.C.E.

In the sixth century B.C.E. religious reformers appeared in . The first we know of is , 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 . 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 root approximations, solutions of indeterminate 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 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 , which were astronomical treatises. We know of 18 of these which were written, but we only have parts of 5 of them surviving today. ( 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 . The latter book has not survived, but the Aryabhatia has and contains , plane and spherical , continued fractions, quadratic equations, sums of power and tables of . He considered indeterminate equations of the form ax + by = c, and sought Diophantine solutions which are integers. He used a method which is equivalent to the Euclidean algorithm, but is related to what is now termed continued fractions. An example of the use of the two methods, side by side, to find the greatest common divisor of 32 and 154 follows.

154 = 4(32) + 26 154/32 = 4 + 26/32 = 4 + 1/(32/26) 32 = 1(26) + 6 = 4 + 1/(1 + 6/26) = 4 + 1/(1 + 1/(26/6)) 26 = 4(6) + 2 = 4 + 1/(1 + 1/(4 + 2/6)) 2 is the g.c.d. 6 = 3(2) so 2 is the g.c.d of 154 and 32 = 4 + 1/(1 + 1/(4 + 1/3))

Aryabhata now uses this to find a solution to the equation 154x + 32 y = 6, say. He takes the last fractional expression and omits the last fraction, (1/3), in the lower right of the expression, obtaining 4 + 1/(1 + 1/4) = 4 + 1/(5/4) = 4 + 4/5 = 24/5. He then cross multiplies the 154/32 times the 24/5 and changes one sign to a negative to get 5(154)- 24(32) = 2. Since 6 = 2(3) we have the solution when we multiply by 3, 15(154) – 72(32) = 6 so x = 15 and y = -72 is one solution. This is the same as back substituting in the Euclidean algorithm to solve for 2 = 154s + 32t. Though the positional decimal system of writing numbers predates him, it was the work of Aryabhata that put the full system in place. This was further spread in Indian intellectual circles by the work of the later mathematician, in 857 A.D.

Another example of Jain mathematical ingenuity is the Jain Magic Square. Not only do all rows and all columns and the main diagonals add up to the magic number of 34, but the broken diagonals like 7 + 11 + 10 + 6 do also. Here it is:

7 12 1 14 2 13 8 11

16 3 10 5 9 6 15 4

Before considering the next great Indian mathematician, , it is convenient to take a look at an excellent philosopher, Nagarjuna, who lived from about 150 to 250 A.D. He !w as a profound influence on Indian intellectual life. Nagarjuna was a Buddhist and a skeptical philosopher who criticized logic and any simple answers to the questions of life. His central idea of the emptiness of all things and the ever present change in reality ties into the inadequacy of language as a perfect model of the world. He poked holes in any system which described the world in terms of fixed substances and concepts. He taught that things can have existence only because they lack a fixed substance, and so can change. This is the emptiness in everything. He referred to this emptiness as Sunya, which had a traditional Buddhist meaning of a lack of fixed essence in persons and which had already also been in use as the idea of the number zero. He taught that a disciplined skepticism leads to true wisdom. He was no nihilist. He thought that logical constructs were useful, if inaccurate, models of reality which people needed to have in order to function. But he cautioned against believing that they are perfect models of the real. As a result of the deep influence of these teachings on Indian intellectuals, we find some of their best mathematicians preferring insight into the truth of mathematical facts more than logical proofs.

We now come to the great Indian mathematician, Brahmagupta. (598 to 665 A.D.) who perhaps came from , a city in the state of Madhya Pradesh in north central India. He certainly became the director of the astronomical observatory there. This was the foremost center of mathematical activity at that time in India. He wrote the Brahmasphuta Siddhanta ( which means “The Opening of the Universe” ) in 628 A.D. He also wrote the , a treatise mostly on astronomy, in 665 at the age of 67. He was an orthodox Hindu. His writings on the solutions to the linear indeterminate equations in two variables go beyond the continued fractions methods of Aryabhata to classify all solutions to such equations and to give necessary and sufficient conditions for a solution to exist in the first place. We thus have:

Theorem ( Brahmagupta) (A) The equation ax + by = c for a, b and c integers has solutions if and only if the g.c.d of a and b divides c. (B) If (x1, y1) is a given solution then all solutions are givenby x = x1 + bt/d, y = y1 – at/d where t is any integer and d is the g.c.d. of a and b.

Here is a proof: (A) Suppose that (x1, y1) is an integral solution to ax + by = c and that d = g.c.d(a,b). Then ax1 + by1 = dmx1 + dny1 = d(mx1 + ny1) = c, so c is an integer multiple of d, or d divides c. On the other hand, suppose d divides c, c = dr, where r is an integer. Since d = g.c.d(a,b) there are integers s and t so that d = as + bt, by Aryabhata’s method or by back substitution in the Euclidean algorithm. So c = dr = r(as + bt) = a(rs) + b(rt) and x = rs, y = rb gives the required integral solution to the equation.

(B) Now how about all solutions? Suppose (x1, y1) is one solution. Any other solution (x, y) has the property that a(x1 – x) + b(y1 – y) = 0, since both ax1 + by1 = c and ax + by = c, so one subtracts the two. Say also that a = dm and b = dn So a(x1 – x) = -b(y1 – y) and we get m(x1 – x) = -n(y1 – y)

y1 " y x1 " z So = " and we denote this quantity by the letter t. m n

From ax1 – ax + by1 – by = 0 we get ax1 + by1 – by = ax and a(x1 – x) + by1 = by

! b # y1 " y & So, dividing the first by a we get x1 + % ( = x, or x = x1 + bt/d d $ m '

a # x1 " x & And dividing the second by b gives us y1 + % ( = y or y = y1 – at/d. d $ n ' ! This finishes Brahmagupta’s proof of his theorem, in our symbolism and language.

Brahmagupta prefers !to have the insight to see the truth of mathematical propositions. He sometimes apologizes to the reader for supplying logical proofs as crutches when insight cannot do the job. He states with no reasoning that the following laws for calculations with positive and negative numbers hold true:

A debt subtracted from a fortune is a larger fortune. A fortune subtracted from a debt is a larger debt. The product or quotient of two fortunes is a fortune. The product or quotient of two debts is a fortune. The product or quotient of a fortune and a debt, in either order, is a debt.

Unfortunately he also “sees” that 0/0 = 0 and N/0 = a fraction with zero denominator, a number.

Brahmagupta states without proof that n(n + 1)(2n + 1) 1 + 4 + 9 + 16 + """ + n 2 = 6 2 # n(n + 1)& and that 1 + 8 + 27 + 64 + """ + n 3 = % ( . $ 2 ' ! He also stated the Theorem on cyclic quadrilaterals: If ABCD is a quadrilateral inscribed in a , that the area A = (s " a)(s " b)(s " c)(s " d) where a = AB, b = ! BC, c = CD, d = DA an s is the s = (a +b+c+d)/2, a la Heron of Alexandria. ! B

A

C

D

Brahmagupta also worked on Diophantine, indeterminate, quadratic equations such as y 2 = 80x 2 + 1 or y 2 = Ax 2 + 1 where A is not a perfect square, or one is finding Pythagorean triples, which were by then fully understood. He had to find one solution by an ad hoc method and then he created a remarkable cross-multiplication which would give him an infinity of other integral solutions. So let’s look at the equation ! above, y 2 = 80x 2! + 1. We quickly see that x = 1 y = 9 will work. One writes the x’s

x1 y1 and y’s down in a table: Here we have x1 = x2 = 1 and y1 = y2 = 9. x2 y2

! To find the new x cross-multiply in the table and add: x3 = x1 " y2 + y1 " x2. To find the new y you multiply vertically! a nd by A: y = A " x " x + y " y . ! 3 1 2 1 2

! ! If you start with our values twice you get new (x, y) = ( 18, 161). If you then perform this multiplication using the two pairs of values (1, 9) and (18, 161) you get a new pair (323, 2889) which works. Brahmagupta is multiplying numbers like (y + x√3) as follows:

( 2 + 7√3)( 6 + 5√3) = [(2)(6) + 3(5)(7)] + [(2)(5) + (7)(6)]√3

A later towering mathematical figure is Bhaskara II of the Ujjain School of Mathematics. He lived from 1115 to 1185 A.D. and came from the city of Bijapore in Mysore State, which is now the State of in southern India. Bhaskara found a way to finish the program of Brahmagupta in solving equations of the form y 2 = Ax 2 + 1. He found a way to produce the first solution in a reliable way. He wrote the Siddhanta Siromain based on Brahmagupta’s Brahmasphuta Siddhanta. The Siddhanta Siromain contains cosmology, geography, new of the motion of the moon, new measurements of the heliacal risings of the planets, new ! astronomical equipment and through an advanced use of trigonometry presents a more theoretical view of astronomy. Bhaskara also wrote a small mathematical work called the Lilavati, which means “The Beautiful” and was dedicated to his daughter. In the Lilavati we find a more complete understanding of the rules for calculating with zero, and the remark that 5/0 is infinite. But there is still some problem remaining when he writes that 5(0)/0 = 5. In another written work of Bhaskara, the Bijagnita, he works on solving indeterminate equations of the form ax + by + cz = d. He gives a paper-folding proof of the Pythagorean Theorem, inspired by the Chinese classic, the Jiuzhang Suanshu. He also writes on five different styles of performing multiplication, among them the grid, or gelosia, method. To multiply 37 times 859 one draws a 2 by 3 grid with diagonals, as shown below. One arranges the numbers to be multiplied along the edges of the grid. The individual digits are then multiplied into the corresponding cross-grid spaces, with the answer having the tens digit above the diagonal and the units digit blow the diagonal. One then begins at the bottom and adds diagonally along slanted rows, carrying to the next row above, as shown in the figure to the right. 31, 783 is the answer, reading around.

3 7 3 7 2 5 8 2 5 8 3 4 6 4 6 3 1 1 5 1 3 5 5 5 5 5 2 6 7 9 2 6 7 9 3 7 3 8 3

To multiply 357,398 by 25,276, one creates a 6 by 5 grid as shown below. Then the inividual digits are multiplied into the associated grid blocks as done above.

3 5 7 3 9 8

2

5

2

7

6

Our final mathematical talent in ancient Madhava of Sangamagramma, in the southern Sate of , one of the locations with a majority of the population of Dravidian descent. He lived from 1340 to 1425 A.D. In his work on astronomy he created very accurate tables of sines and cosines and was led to see that, for Δx very small, 2 sin(x + "x) # sin(x) + cos(x)$ "x % sin(x)$ ("x) /2 which we can recognize as equivalent to the fact that the of the is the cosine. He also somehow came to see the equivalent of the Taylor’s series for ! x 3 x 5 x 7 sin(x) = x " + " + ### and so on, as well as that for the cosine, 3! 5! 7! x 2 x 4 x 6 cos(x) = 1 " + " + ###. 2! 4! 6! He discovered the equivalent of the series for the inverse tangent function ! x 3 x 5 x 7 tan"1(x) = x " + " + ### ! 3 5 7 # " 1 1 1 and so, since tan"1(1) = , he knew that = 1 # + # + $$$. 4 4 3 5 7 All of this was before the analogous discoveries in the creation of the took place in Europe, 300! years later.

! !