how did geometry develop in india?

Answer 1

2b or not 2b

Answer 2

Hi,



LAVLESH
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Geometry in Ancient India

Aniruddha Avanipal
Hindu Geometry - Part 1

MUSIC

Introduction
In ancient India, geometry was known as 'Shulba' or 'Rajju'. The word 'shulba' itself is derived from the Sanskrit root 'shulb' meaning 'to measure' and hence its etymological significance is 'measuring' or 'act of measurement'. The word 'Rajju' literally means a 'rope' or an instrument of measuring. The rules of geometry itself were known as 'Shulba-Sutra', the Sanskrit word 'sutra' meaning 'aphorism' or a short rule.

Shulba-Sutras or the rules of geometry were instrumental for the Vedic Hindus to construct altars. Those altars used to vary in their shapes and sizes and acted as symbolic representations for higher spiritual conceptions invented by Vedic Sages. From Shulba-sutras we get a glimpse of advanced knowledge of geometry that the Vedic Hindus developed.

The root of the Shulba-Sutras The Vedas which were composed between 4000 BCE to 3100 BCE are not only ancient repository of spiritual Hindu knowledge but also contain in-depth analysis of physical sciences such as Mathematics, Physics, Astronomy, Architecture etc. The Vedas also have six supplementary texts known as the 'Vedangas' or limbs of the Vedas. One of these is 'Kalpa-Sutra-Vedanga' or set of rules related to rituals or ceremonies. The Kalpa-sutras are broadly divided in two classes, the 'Grhya-sutras' or the rules for ceremonies relating to family or domesti affairs such as marriage, birth etc and the 'Shrauta-sutras' or the rules for ceremonies ordained by Vedas. The shulba-sutras belong to the latter class.

Patanjali, the great writer of 'Yoga Sutra', stated that there are as many as 1131 to 1137 different schools of the Vedas. In his own words, "There were 21 different schools of the Rig-Veda; 101 schools of the Yajur-Veda; 1000 of the Sama-Veda; and 9 or 15 of the Atharva Veda". Each school of the Veda had its own 'Shrauta-sutra' and hence probably its own Shulba. It appears there were numerous manuals of geometry in ancient India. But many of them are lost and at present we know of only seven distinct Shulba-sutras compiled by Baudhayana, Apastamba, Katyayana, Manava, Varaha and Vadhula.

Dating of Hindu Geometry

Before providing the actual dates of Shulba-sutra, it is crucial to point out that many of the important development in India have been dated by the western historians based on the foundation laid by the erroneous and mythical "Aryan Invasion Theory". Max M�ller (1823-1903), who was largely responsible for this theory based it on pure conjectures rather than on scientific proofs. He concluded that northern India was invaded and conquered by nomadic, light-skinned RACE of a people called 'Aryans' who descended from Central Asia (or some unknown land ?) around 1500 BC, and destroyed an earlier and more advanced civilization of the people habitated in the Indus Valley and imposed upon them their culture and language. According to this erroneous theory, Rig Veda was composed around 1200 BCE.

However, Aryan Invasion theory has been proved to be entirely mythical and baseless by modern historians like David Frawley, N S Rajaram, N Jha, Dinesh Agrawal etc. A complete description on the myth of Aryan Invasion Theory is outside the scope of this article. Please refer to the 'reference section' of this article for a comprehensive description of the scientific, archaeological and logical proofs disproving the Aryan Invasion theory.

Through their ground-breaking work, historians like Frawley and Rajaram not only proved the baslessness of Aryan Invasion Theory, but also established correct dating of many important developments of Hindu civilization.

The correct dates of some of those land-mark incidents are listed below:

* 4000 BCE: Rig Veda
* 3700 BCE: Battle of Ten Kings (referred to in the Rig Veda) Beginning of Puranic dynastic lists: Agastya, the messenger of Vedic religion in the Dravida country. Vasistha, his younger brother, author of Vedic works. Rama and Ramayana 3600 BCE: Yajur-, Sama-, Atharvaveda: Completion of Vedic Canon.
* 3100 BCE: Age of Krishna and Vyasa. Mahabharata War. Early Mahabharata.
* 3000 BCE: Shatapathabrahmana, Shulbasutras, Yajnavalkyasutra, Panini, author of the Ashtadhyayi, Yaska, author of the Nirukta

As we can see, literature like Shulba sutras and Shatapatha Brahmana date back to as far as 3000 BCE. Hence Shulba of Hindus which includes both 2-dimensional and 3-dimensional geometry is almost 5000 years old.

Influence of Hindu Geometry on Greeks In his monumental work, The origin of mathematics, Archive for History of Exact Sciences. vol. 18, 301-342, A. Seidenberg remarks:

"By examining the evidence in the Shatapatha Brahmana, we now know that Indian geometry predates Greek geometry by centuries. For example, the earth was represented by a circular
altar and the heavens were represented by a square altar and the ritual consisted of converting the circle into a square of an identical area. There we see the beginnings of geometry!

Two aspects of the 'Pythagoras' theorem are described in the Vedic literature. One aspect is purely algebraic that presents numbers a,b,c for which the sum of the squares of the first two
equals the square of the third. The second is the geometric, according to which the sum of the areas of two square areas of different size is equal to another square. The Babylonians knew
the algebraic aspect of this theorem as early as 1700 BCE, but they did not seem to know the geometric aspect. The Shatapatha Brahmana, which precedes the age of Pythagoras,
knows both aspects. Therefore, the Indians could not have learnt it from the Old-Babylonians or the Greeks, who claim to have rediscovered the result only with Pythagoras. India is thus
the cradle of the knowledge of geometry and mathematics." So, contrary to the European belief that Hindus were influenced by the Greek geometry, the facts prove that it is the other way round. Most of the aspects of planar geometry described by Euclid and other Greek mathematicians were already known to Indians at least 2500 years before the Greeks. In fact, there are proofs which hint towards the fact Greeks were influenced by the ancient Hindu Mathematics and Geometry. Bibhuti Bhushan Datta in his book "Ancient Hindu Geometry" states:

"...One who was well versed in that science was called in ancient India as samkhyajna (the expert of numbers), parimanajna (the expert in measuring), sama-sutra-niranchaka
(Uinform-rope-stretcher), Shulba-vid (the expert in Shulba) and Shulba-pariprcchaka (the inquirer into the Shulba).

Of these term, viz, 'sama-sutra-niranchaka' perhaps deserves more particular notice. For we find an almost identical term, 'harpedonaptae' (rope-stretcher) appearing in the writings of
the Greek Democritos (c. 440 BC). It seems to be an instance of Hindu influence on Greek geometry. For the idea in that Greek term is neither of the Greeks nor of their acknowledged teachers in the science of geometry, the Egyptians, but it is characteristically of Hindu origin."

I will conclude this part of the article by pointing out the fact that the English word 'Geometry' has a Greek root which itself is derived from the Sanskrit word 'Jyamiti'. In Sanskrit 'Jya' means an arc or curve and 'Miti' means correct perception or measurement.

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Sulbasutra Geometry
For Prof. W. Casselman
By Susanna Kong

Math 309

April 2001
Introduction

The basis and inspiration for the whole of Indian mathematics is geometry. The beginnings of algebra can be traced to the constructional geometry of the Vedic priests preserved in the Sulbasutras, a manual of geometrical constructions from the 5th to the 8th centuries. Earlier remnants of geometrical knowledge of the Indus Valley Civilisation can be found in excavations at Harappa and Mohenjo-daro where there is evidence of circle-drawing instruments from as early as 2500 B.C.E. (Amma 1)

Early geometry of the Sulbasutras was based on religious needs with regards to the construction of altars such as agnicitis, vedis, mandapas etc. that are required for sacrificial ritual (Kulkarni 19, Amma 3).

The history of Indian geometry can be divided into three distinct periods:

§ pre-Aryan, such as excavated in Indus Valley.

§ Vedic or Sulbasutra

§ post-Christian.

Evidence of the pre-Aryan period includes well-planned towns and geometric designs including circles, squares and triangles. A link between this and the Vedic period can be found in the motif of a rectangle with the four sides curved inwards resembling a stretched hide; in the former period it can be seen as a decorative pattern while in the later period it is seen in the shape of the sacrificial altars or vedi. The vedi as well as the fireplaces or agni had such exact measurements and geometric shapes that they were codified and became the Sulbasutras. However, it is not known from how far back such knowledge originated as the sacrificial act is as old as the Vedas or older (Amma 5).

In the post-Christian era Aryabhata I (b. 475 C.E.) dominated Indian mathematics although what exactly his contributions were is somewhat unclear. Also important are Bhaskara I (c. 522 C.E.), Bramhagupta (628 C.E.) and Bhaskara II (b. 1114 C.E.). In this period, geometry took somewhat of a backseat to the pursuit of astronomy and algebra.
Sulbasutra Geometry

The Sulbasutras contain the geometry necessary for construction of the vedi and the agni for the obligatory and votive rites. These in turn are a part of the Kalpasutras , which are attached to the Vedas as one of the six V dangas or limbs of the Vedas. It is likely that the Sulbasutra sections were a part of the Srautasutras of the Yajur Veda, the Veda designed for the performance of sacrifices.

The three most primitive agni, Garhapatya, Aavaniya and Daksinagni, are older than the Rg Veda and the Mahavedi was likely known to the Indus Valley civilisation. It is generally recognised that the Pythagorean theorem was known in India at the latest in the 8th century B.C.E. (Amma 14-15).

The Sulbasutras deal with the correct construction of the vedi and agni including orientation, size, shape and areas and, as such, they are not meant as mathematical theorems or proofs. The geometry in the Sulbasutras can be categorised into that which expressly states theorems, constructions and implicit geometrical truths contained in constructions.
Theorem of the Square of the Diagonal or Pythagorean Theorem

Modern historical research has shown that what is known as Pythagoras’ Theorem was in fact known to the ancient Indians. Whereas Pythagoras (c. 540 B.C.E.) is given credit for this formula in modern western culture, the Vedic priests were using this principle to build their altars.

In the Black Yajur Veda (8th century B.C.E.) and the Satapath Brahmana, 36 units is the length of the east-west line or praci or line of symmetry or prsthya of the Mahavedi and 30 units as one of the north-south lines. The praci and half the side make the sides containing the right angle in the triangle with sides 36, 15, and 39 units. There is a possibility that this theorem was known even earlier by the construction of the three agni, Garhapatya, Ahavaniya and Daksina. Other rational right triangles as well as the irrational right triangle 1,1 √2 and approximate right triangles are also mentioned in the Sulbasutras (Amma 17-18).

There are several Vedic proofs for this theorem which are all fairly simple (Tirthaji 349-351).
Proof I

The square AE = the square KG and the four congruent right-angled triangles all around it.

The areas are c2, (b-a)2 and 4( ½ ab) respectively.

So c2 = a2 – 2ab + b2 + 4( ½ ab) = a2 + b2
Proof I


Proof II

Construction:

CD = AB = m; DE = BC = n

So, ABC and CDE are congruent and ACE is a right-angled isosceles.

The trapezium ABDE = ABC + CDE + ACE

So ½ mn + ½ h2 + ½ mn = ½ (m + n) = ½ m2 + mn + ½ n2

So, ½ h2 = ½ m2 + ½ n2

So, h2 = m2 + n2
Proof II


Proof III

AE = BF = CG = DH = m and EB = FC = GD = HA = n

The square AC = the square EG + the four congruent right-angled triangles around it.

So, h2 + 4 (½ mn) = (m + n)2 = m2 + 2mn + n2

So, h2 = m2 + n2
Proof III


Proof IV

BD is perpendicular to AC.

So, triangles ABC, ABD and BCD are similar.

So, AB2/AC2 = ADB/ABC and BC2/AC2 = BCD/ABC

So, (AB2 + BC2)/AC2 = (ADB + BCD)/ABC = ABC/ABC = 1

So, AB2 + BC2 = AC2
Proof IV


Proof V

Using coordinates, the distance between point A at (a,0) and point B at (0,b) is:

BA = √ [(a – 0)2 + (0 – b)2] = √ (a2 + b2).
Proof V


Square Root of 2

“The oldest value of √2 is obtained from one of the cuniform tablets from old Babylonian times (1600 B.C.) now in the Yale Babylonian collection. It shows a square with two diagonals and values in sexagesimal system from which √2 in that system worked out as √2 = 1; 24, 51, 10. The sexagesimal equivalent of the value of √2 as given in the Sulbasutra is 1; 24, 51, 10, 37…” (Kulkarni 99-106).

Three derivations of the value of √2 as interpreted from the Sulbasutrakaras are given below:



1. Muller (1930)

ABED is the square with the length of side equal to one unit.

Another square ACFG is to be drawn such that its area is twice the area of the square ABED, i.e. the magnitude of AC = √2

The value of x is obtained.

The area of square ABED = area of rectangle BCHE + GJED + square JFHE.

So, 12 = 2(x) + x2

Say x = ½, then ¼ + 1 > 12

Say x = ⅓, then 1/9 + ⅔ = 7/9 < 12

So, √2 = 1 + ⅓ + x1

So 2(4/3x1) + x12 = (√2)2 – (4/3)2 = 2/9

2(4/3x1) < 2/9

so, x1 < 1/3.4

so, √2 = 1 + 1/3 + 1/3.4 = 17/12

now, 2(17/12)x2 – x22 = (17/12)2 – (√2)2 = 289/144 – 2 = 1/144

2(17/12) > 1/144

so, x2 = (1/144)(12/17)(½) = (1/3)( ¼ )(1/3)( ¼ )

so, √2 = 1 + ⅓ + 1/3.4 - (⅓)( ¼ )(⅓)( ¼ )



In decimal form this final calculation equals 1.40972222222 whereas today’s value of √2 is 1.41421356237, a difference of .00449134015.
I Muller



2. Dutta B. B. (1932)

Take two squares whose sides are of unit length.

Divide the second square into three equal stripes I, II an III.

Sub-divide the last strip into three small squares III1, III2, III3 of sides ⅓rd each. Then on placing I, II and III1 about the first square S in the position I’, II’ and III1’ a new square is formed. Now divide each of the portions III2 and III3 into four equal stripes.

Placing four and four of them about the square just formed, on its east and south side, say, and introducing a small square at the south-east corner, a large square will be formed, each side of which is equal to 1 + ⅓ + 1/(3)(4).

Now this square is larger than the two original squares by an amount [1/(3)(4)]2, the areas of small square introduced at the corner.

In order to get the equivalent area remove two thin stripes say of width x from either side of the square.

2 x [1 + ⅓ + 1/(3)(4) ] – x2 = [1/(3)(4)]2

Neglecting x2 as too small 2x(17/12) = [1/(3)(4)]2

So, x = [1/(3)(4)]2 (12/17) ( ½ ) = 1/(3)(4)(34)

Thus we have √2 = 1 + ⅓ + 1/(3)(4) – 1/(3)(4)(34) nearly.



In decimal form this equals 1.41421568627 whereas today’s value of √2 is 1.41421356237, a difference of .0000021239.
II Dutta



3.

ABC is a circle such that AB = BC = 1 and AC = √2, and ÐABC = 90

OC = OB = 1/√2

Taking C as centre and CB as radius draw an arc which cuts AC at D.

CD = CB = 1

OD = 1 – 1/√2 = √2-1/√2

So, AD = √2 –1

In triangle ODB, BC2 = OD2 + OB2 = (1 – 1/√2)2 + (1/√2)2 = [(√2-1)/√2]2 + (1/√2)2.

So, BD2 = ½ (2 – 2√2 +1 ) + ½ = 2 – √2

So, BD = √(2-√2)

Drop DG perpendicular to AB

DG2 = AD2 – AG2 = BD2 – BG2, BG = AB – AG

So, AD2 – AG2 = BC2 – (AB – AG)2 = BD2 – AB2 + 2(AB)(AG) – AG2

So, AG = (AD2 – BD2 + AB2)/2AB = [(√2 - )2 – (2-√2) + 12]/(2)(1) = (2 – √2)/2

DG2 = AD2 – AG2 = (√2 – 1) – [(2 – sqrt2)/2]2 = 1 ½ - √2

So, DG = √(3/2 – √2) = 1 – 1/√2

Taking D as centre and DG as radius, draw an arc, which cuts AC at H.

So DH = DG

Drop HJ perpendicular to AG

AH = AC – CD – DH = √2 – 1 – (1 – 1/√2) = (3 – 2√2)/√2

Triangle AHJ and ADG are similar.

So, AH/AD = HJ/DG or HJ = (AH/AD)DG

So, HJ = (5√2 – 7)/2(√2-1) and HJ = HK

Drop KL perpendicular to AJ

Triangles AKL and AHJ are similar

AK = AC – CD – DH – HK = (17√2 – 24)/2√(√2 – 1)

So, KL = (338 – 239√2)/(68 – 48√2)

AC = CD + DH + HK =KL

So, √2 = 1 + 1 + 1/√2 + (5√2 – 7)/2(√2-1) + (338 – 239√2)/(68 – 48√2)

So, √2 = 1 + 1/3 + 1/12 – 1/(12)(34)



This last method yields the same result as the last method 1.41421568627 whereas today’s value of √2 is 1.41421356237, a difference of .0000021239.
III


Sulbasutra Constructions

Following are a selection of constructions given in the Sulbasutras (Amma 23-46).



i) Drawing two perpendicular diameters in a circle in Baudhayana’s recipe for drawing a square.

Drawing a line one fixes a pin at its middle. Slipping the end ties on to this pin, one draws a circle with the mark (the middle of the cord) and fixes pins at the ends of the diameter. With the end-tie on the eastern pin one draws a circle with the whole cord. Similarly at the western pin. The second diameter should be stretched through the points where these (circles) intersect.

This method is well known in later Indian mathematics as the ‘fish’ method due to its shape.
'Fish' Method



ii) The construction of a square with a given side results in a beautiful geometrical pattern.

Wishing to construct a square one should make ties at both ends of a string as long as the desired side and make a mark at its middle. One should draw a line and fix a pin at its middle. Fixing the ties on this pin one should draw a circle by the middle mark (of the cord) and at the ends of the diameter (formed by the praci) one should fix pins. Fixing one tie on the eastern pin one should draw a circle with the other tie. Similarly round the western pin. Through the points where they meet the second diameter should be drawn and pins should be fixed at this end. With the ties on the eastern pin a circle is to be drawn with the middle mark. Similarly round the southern, western and northern (pins). Their outer points of intersection form the square.

This method is found in the Baudhayana Sulbasutra.
Construction of a Square


Transformation of Figures

Different shapes were prescribed for the fire-altars depending on the benefit sought: the falcon or syenacit for attaining heaven, the isosceles triangle or praugacit for destroying enemies etc. However, the shapes had to have the same area of 7 ½ square purusas and therefore the Sulbasutras described different methods of changing the shapes of figures while retaining the same areas.


To convert a square into a circle

The Sulbasutras give an approximate construction, which results in a value of π of 3.088. Whether or not it was known at the time that this method indeed results in an approximation is not known.

If a is the side of the square and r the radius of the circle then r = a(2 + Ö2)/6.

If a = 1, then the area of the square is 1, r = (2 + Ö2)/6 = 0.5690355 and the area of the circle (using today’s value of π) is 1.0178256.
Convert a Square to a Circle


To convert a square into a rectangle

One method is given by:

Wishing to transform a square into a rectangle one should cut diagonally in the middle, divide one part again and place the two halves to the north and east of the other part. If the figure is a quadrilateral one should place together as it fits. This is the distribution.
Convert a Square to a Rectangle


Rectilinear Figures and the Circle

Knowledge of determining the areas and measuring of rectilinear figures it seems is derived from a functional need in terms of land or ownership thereof. In Rg Veda there are many instances where the division of land by Asuras and Gods are mentioned. Likewise, the three fires or agni, Garhapatya, Ahavaniya and Daksina were developed from pre-Vedic times and their shapes, circular, square and semicircular, are of equal area (Kulkarni 68).


The area of a regular pentagon, hexagon etc. inscribed in a circle

Area of pentagon = ABCDE = 5 (area of triangle AOB).

OF2 = OB2 – BF2 = OB2 – ¼ AB2

The construction of a pentagon inscribed by a circle was known.

Area of pentagon ABCDE = 5 ( ½ OF AB)



By following this procedure the area of any sided figures inscribed in a circle can be found (Kulkarni 70).
Area of a Regular Pentagon


Area of a citi

The area of a normal citi is 7 ½ square purusa which is equal to 108 000 square angula (7 ½ times 1202). The area of a citi is divided into 15 equal parts or 7200 square angula.

The cross line of the Smasana cita towards east is 3 purusa in length and the one on the western side is 2 purusa and praci (east-west) is 6 purusa long. The area of this trapezium is ½ (3 + 2) 6 = 15 square purusa or 108 000 square angula as all citi.

Likewise, the volume of all citi is expected to be equal. As other citi are the same height, this problem does not arise but for the Smasana cita its height on the eastern side is greater than the western side.

The volume of a standard citi is 108 000 times 32 = 3 456 000 cubic angula. The Smasana cita height is increased by one fifth = 32 + 32/5 = 38.4 angula and then divided by three = 38.4/3 = 12.8 angula. Twice this height is divided by 4 = 6.4 angula. Four layers of these bricks are laid. The height of theses 4 layers of bricks = 6.4 times 4 = 25.6 angula. The fifth layer of bricks is placed with bricks of thickness 1/3 times 38.4 = 12.8 angula. The total height of the citi will be 25.6 + 12.8 = 38.4 angula.

The thickness of bricks of the top layer (fifth layer) is decreased by drawing a diagonal plane through it so that the height of citi on one side is 38.4 angula and 25.6 angula on the other side. Thus the volume of the Smasana citi = (108 000)(25.6) + (108 000)(12.8)( ½ ) = (108 000)(32) = 3 456 000 cubic angula which is equal to the standard volume of citi (Kulkarni 71-71).
Smasana citi



Note on Measurements

The length of measure angula occurs frequently in the three Vedas and seems to have been in use from ancient times. The Sulbasutras give the relations of different units to each other and in particular the angula for example the purusa is said to be 120 angula. However there is a dissenting measurement of the purusa, which is 125 angula or the height of a man with upraised hands when standing on toes (Kulkarni 30-31).




Works Cited

Amma, T. A. Sarasvati. Geometry in Ancient and Medieval India. Motilal Banarsidass. Delhi: 1979.

Kulkarni, R. P. Geometry According to Sulba Sutra. Vaidika Samsodhana Mandala. Pune: 1983.

Tirthaji, Swami Sri Barati Krsna. Vedic Mathmatics: Sixteen simple mathematical formulae from the Vedas. Motilal Banarsidass Publishers. Delhi: 1989.





I hope those help!! :-)