book excerptise:   a book unexamined is wasting trees

Indian Mathematics and Astronomy: Some Landmarks

S. Balachandra Rao

Rao, S. Balachandra;

Indian Mathematics and Astronomy: Some Landmarks

Jnana Deep Publications, Bangalore, 1994 (rev 98)

ISBN 81-9100962-0-6

topics: |  india | mathematics | astronomy | history

S. Balachandra Rao wrote this volume while he was the Principal at the
National College, Bangalore, where he had been professor of mathematics for
several decades.  For most of this period, he has been working on Indian
mathematics.  He has been associated with the Indian J. of History of
Science, and is presently an associate of the National Institute of
Advanced Studies (NIAS) at Bangalore.

This book came out in 1994, and this is a revised edition from 1998. 
Two years later, Balachandra Rao brought out a separate volume 
focusing on Indian astronomy.

The last chapters of this book, which deals with astronomy, are more
elaborated there.  On the whole, I felt that the some of the arguments are
presented with more evidence in that later volume.

In addition to these two volumes, he has written as many as 20 other

In this work, sometimes the points made to support Indian primacy are not
sufficiently justified, leaving the reader unsure.  One must tread
carefully especially in domains as contested as that of ancient india.
But on the whole, he makes a number of important points, which are very
useful for people like me who have little background in the texts and the
tradition being talked about. 

Excerpts: Mathematics in India

the credit for giving for the first time the value of pi correct upto
four places of decimal as pi = 3.1416 goes to AryabhaTA I (476 AD). - p.2

 In the west, Archimedes inscribed polygons in and around a
 circle - and showed that 223/71 < pi < 22/7. 

 There are some claims that Ptolemy gave the same value of pi in 150 AD,
 but this seems to be based on shaky evidence.   Victor Katz
 agrees with Rao in his History of Mathematics (1989) that "the earliest
 known occurrence of the approximation 3.1416, was in India, in the work of
 AryabhaTa." (p.268) However, the method used by AryabhaTa is not clearly

 But it seems that around the same time as Aryabhata (end of 5th c. AD), a
 Chinese father-son duo - Tsu Ch'ung-chih and his son Tsu Keng-chih,
 extended Archimedes' idea - drawing polygons upto nearly 25,000 sides --
 to determine the value of pi as 3.1415926 < pi < 3.1415927.  


In Jain mathematical texts, dating from 500 BC to 200 BC (e.g.
jambu-dvIpa prajNapati and sUryaprajNApati), 
PI is approximated to the square root of 10 and calculated correct
up to 13 places of decimal! - p.3

 [AM: I think what is meant here is that the sqrt of 3 was computed
	correctly to 13 places.  Initially I read it as PI being computed
	correctly, which would have been self-contradictory since only on
	page 2, he writes abt Aryabhata in 476 AD giving PI to four places
	for the first time.  It took a second reading after several years to
	see that there may be a second meaning.  

other contributions

zero symbol (a dot) was used in metrics (chhandas) by Pingala (before
200 BC) in his chhandah-sUtra. 

	the first inscription with zero in it, c. 850 AD, gwalior
	can you see the "270" appearing on line 2, just right of center?
	[taken from the excellent 

The formula nCr is attributed to Herrigone (1634 AD) by D.E. Smith in
his History of Mathematics, 1925 (vol.2, p. 527).  Ironically,
Mahavira's gaNita sAra saMgraha (GSS) edited 1912 by M. Rangacharya
carries a foreword by Prof Smith himself.  In the same History of
mathematics, DE Smith remarks that BhAskara (1150 AD) gave
formulae for both nCr and nPr to find combinations and
permutations. - p.4

equations of the type Nx² +1 = y²  :%

In 1657 Fermat proposed to his friend Frenicle to solve in integers
the indeterminate equation 61x² + 1 = y², but the solution was not
found until 1732 by Euler.  But coincidentally, the same equation was
v* completely solved by Bhaskara who obtains the lowest values x =
226153980 and y = 1766319049. [p.7]

Mahavira (9th c AD) gives a formula for area which turns out to be
incorrect, but his circumference - sqrt (6z² + 4b²) is a very good
approximation. -p.7


  On hearing the distinct sound caused by the drum made up of clouds in
  the rainy season, 1/16th and 1/8th of a collection of peacocks,
  together with 1/3d of the remainder and 1/6th of th remainder
  thereafter, gladdened with joy, kept on dancing in the big stage of
  the mountain top; and 5 times the square-root (of that collection)
  stayed in an excellent forest of vakula trees; and the remaining 5
  were seen on a punnAga tree.  O mathematician, tell me, how many
  peacocks were there in the collection? [Mahavira, GSS, 9th c.] - p.9

shulva sutras

shulva-sUtras: form a shrauta part of kalpa vedAnga - nine texts -
mathematically most imp - baudhAyana, Apastamba, and kAtyAyana
shulvasUtra. [12]

Rules for constructing yajNa bhUmikAs -- older contributions referred
by statements such as "iti abhyupadishanti" "iti vijNAyate" etc. [13]

[shulva = cord.  many instructions use cords for taking measurements in
altar construction.]

The most ancient of the Shulva is the baudhAyana sUtra (3 chapters),
with theorems such as: diagonals of rectangle bisect each other,
diagonals of rhombus bisect at right angles, area of a square formed
by joining the middle points of a square is half of original, the
midpoints of a rectangle joined forms a rhombus whose area is half the

dates for the shulva sUtras

Rao does not suggest any date. 

The shulva-sUtras are thought to date between 1500 to 500 BC, and the
baudhAyana sUtra is the earliest extant, an opinion formed based on.  
the nature of the old sanskrit, which which is late vedic.

Based on this, it has been suggested that Baudhayana lived in the 8th or
9th c. BC.  However, it can clearly be a century here or there.

The MacTutor biography assigns a date of 800 BC for Baudhayana.  
An article by Seidenberg [Seidenberg, A. 1978, "The Origin of Mathematics"]
suggests 600 BC but is open to having it earlier; he makes it clear, in a
fascinating argument of great scholarship, saying at one point
that the constructions described in the earlier work, the Satapatha BrahmaNa
(1000BC-500BC) - indicate a knowledge of pythagoras: 

	I therefore regard it as certain that the Satapatha BrahmaNa
	knows the Theorem.

The main point of his work is that indian thoughts on pythagoras
pre-date the greek, and are animated by 
theoretical (or abstract) considerations as much as by the need for laying
out altars.   Some aspects of both Indian and Greek are predated by old
babylonia (c. 1700BC) - but that is too practical and does not satisfy the
requirements for abstraction. 

Datta and Singh (v.1 p. 247) gives a date of 800 BC, but again without
discussion.  However, I personally have faith in Datta and Singh's
judgment, though the text is now nearly 80 years old.

Here Rao doesn't bother discussing the issue of the date, other than to say
that these sUtras are "far more ancient than Pythagoras" (6th c. BC) [p.1]

Early statement of Pythagoras

But the most notable sUtra in Baudhayana is:

	dIrghasyAkShaNayA rajjuH pArshvamAnI tiryaDaM mAnI.
	cha yatpr^thagbhUte kurutastadubhayAM karoti.

The diagonal of a rectangle produces both areas which its length and
breadth produce separately.

Interestingly, A. Burk even argues that the much travelled Pythagoras
borrowed the result from India.
		 [p.15; but Burk text is not listed in references. ]

"It is interesting that this theorem is stated by the Vedic authors
far earlier than Pythagoras (6th c. BC). There is, therefore, a strong
case to rename this famous theorem after the shulvasUtras"
  [AM; But no dates are given for the Shulva-sUtras in general or the
  baudhAyana in particular.  Are they indeed part of the vedic period,
  dating to before 1400 BC, say?  Furthermore, there may be many
  versions of the text with parts that may have been added at later
  times; the dating of this particular shloka will require far more
  scholarly treatment in order to substantiate such a claim. ]

Squaring the circle (p.18)

baudhAyana i.58 gives this formula:

	Draw half its diagonal about the centre towards the East-West line;
	then describe a circle together with a third part of that which lies
	outside the square.

i.e. draw the half-diagonal of the square, which
is larger by
x = (a/2.sqrt(2)-a/2). Then draw a circle with radius a/2 + x/3, or
a/2 + a/6.(sqrt(2) -1) = a/6(2 + sqrt(2)). Now (2+sqrt(2))² ~= 10, so,
this turns out to be a² * pi/4 * 10/9 which is abt a².

Square Root (2) [p.20]

baudhAyana i.61-2 and Ap. i.6 give this formula for sqrt:

samasya dvikaraNI. pramANaM tritIyena vardhayet
tachchaturthAnAtma chatusastriMshenena savisheShaH.

sqrt(2) = 1 + 1/3 + 1/(3.4) - 1(3.4.34) -- correct to 5 decimals
	= 1.41421569

(a+b)³ and (a+b)⁴ are given in shulva-shAstra

Yajurveda - Calendar [p.25]

The solar year was 365 days and a fraction more. krshNa yaJurveda,
tattirIya saMhitA 7.2.6 says that 11 days more than the 354 days in
the 12 lunar months are the ekAdasharAtra or elevn-day sacrifice.

yuga = 5 years - samvatsara, pari, kRA, anu, id-vatsara.

vasanta	       madhu and mAdhava
grIShma	       sukra / shuci
varShA	       nabha / nabhasya
sarad	       isha /  Urja
hemanta	       saha / sahasya
shishira       tapa / tapasya

Chapter 3: Aryabhata [32]

AryabhATiyam - four parts:

1. gItikA pAda - 13 stanzas, ten in gItikA metre - astronomical
   figures, calendar. circular units of arc, units of length [yojana,
   hasta, anguli].
2. gaNita pAda is the second part of the text, with 33 stanzas dealing
   with arithmetic. -- geometry, shadow computation for gnomon, simple
   and compund interest, simple, simultaneous, quadratic, and linear
   indeterminate equations (e.g. ax+y = b).

3. kAlakriyA pAda - 25 shlokas - units of time and planetary positions,
   speeds, etc.

4. golAdhyaya: celestial sphere / astronomy - 50 stanzas - celestial
   equator, the node, shape of earth, cause of day and night


in kAlakriyA pAda, AryabhaTa gives his his age on a particular date: 

	षष्ट्यब्दानां षष्टिः यदा व्यतीताः त्रयः च युगपादाः ।
	त्र्यधिका विंशतिः अब्दाः तदा इह मम जन्मनः अतीताः ॥ १० ॥

	When six times sixty years (3600) and three quarter yugas have
	elapsed (in the running kaliyuga), twenty-three years have passed
	since my birth.  This period is dated to 499 AD.

KALIYUGA: started at midnight after 17 Feb 3102 BC .  -- so 500 AD is
3624 + 3/4 kaliyuga; 2000 AD is 5124 + 3/4 kali-yUga.

If the "23 years" is exact, then AryabhaTa's birth date would be March 21, 476.   
     [K.S. Shukla, trasn. and editor of AryabhaTiyam, publ K
     Sambasivasastri, Trivandrum]

Most commentators take this date (499 AD) to be about the time when he was
composing the text.

However, there have been several interpretations for the 3600 years date. 
According to one school, this should be the date from which the the bIja
correction for the mean longitudes of the moon and planets should be
computed.  Thus the moon's motion correction is -25/250 per annum, while
Mercury is +420/250.   

Other groups suggest slightly later dates for the zero of the
computation; these base date differences have resulted in different schools
of astronomical calculation in India. 

[see K.S. Shukla's Critical edition of the AryabhaTiya, publ. INSA 1976]

Letters for Numerals code p.36

varga letters = numbers from 1 to 25:
		ka-N~ 1-5, 
		ca-n~a: 6-10, 
		Ta-Na 11-15, 
		ta-na 16-20, 
		pa-ma 21-25
the avargiya vyanjanas are:
y = 30, r = 40, l=50, v=60, Sh=70, sh=80, s =90 and h=100

In a word representation, the odd positions are varga (square)
positions (since 1, 100, 10000 etc are squares).  The evens are
avarga.  The vargiya numbers appear in the vargiya (odd) positions,
and the avargiyas in the even positions.  The positions are given by
the nine svaras - au au ai ai O o E e L. l. R. r. U u I i A a,
So gr. is 3x10⁶; whether it is a hrasa or dIrgha vowel does not
matter; the position depends on whether the consonant is vargiya or

ravi's revolutions in a yuga - 4,320,000 years -- is khyughr. or
2x10⁴ + 30 x 10⁴(khyu) + 4 x 10⁶ (ghr.)
 its own axis (number of days in year) in a yuga

Value of Pi p.39

	चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम। 
	अयुतद्वयविष्कम्भस्यासन्नो वृत्त- परिणाहः।। 

	caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
	ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ || 10
			     [gaNita pAda, 10]


word by word: 

	chatuH+adhikaM shatam : 4 in excess of 100; 
	aSTa guNam : eight, multiply (by)
	dvASaSTiH sahasraNAm द्वाषष्टि 62 thousand; tathA  : to that (add)
	Ayuta dvaya : a pair of 10,000s   viSkambha विष्कम्भ : diameter -sya : gen.,
		[of a 10,000 diameter (vr^tta वृत्त,circle)]
	Asanna : approaches pariNAha,परिणाह : circumference

"Add 4 to 100, multiply by 8 and add to 62,000.  This approaches
the circumference of a circle with diamenter 20,000."

		pi*20000 = (104*8 + 62000)
			 = 62832 

	  --> 	pi = 3.1416

correct to four places.  

Even more important however is the word "Asanna" - approximate, indicating
an awareness that this is an approximation.  Commentator nIlakaNTha of
Kerala, (1500 AD) makes a case for AryabhaTa's conjecture tha PI is
incommensurable (or irrational).  In the west, it was shown to be
irrational in 1761 {Lambert) and transcendental (not a soln to any
algebraic eqn with rational coeffs) in 1882 (Lindemann)

Area of a triangle


AryabhATiya : gaNitapAda 6

	tribhujasya falasharIraM samadalakoTI bhujArdhasaMvargaH_

 		त्रिभुजस्य फलशरीरं समदलकोटी  भुजार्ध संवर्गः। 
	the body of a triangle (its area) is half the side 
	multiplied by the perpendicular (समदलकोटी)

Sine Tables p.42

These sine tables as documented by AryabhaTa (possibly known in earlier
times) were translated into arabic in the 8th c.  The name "jya", half
chord, was mis-translated 

jyA = sine, koTijyA = cosine

jyA tables :
Circle circumference = minutes of arc = 360x60 = 21600.
Gives radius R = radius of 3438; (exactly 21601.591)
   [ with pi = 3.1416, gives 21601.64]

note: 3438 = 360 x 60 = 21600 / 2*pi

The R sine-differences (at intervals of 225 minutes of arc = 3:45deg),
are given in an alphabetic code as
which gives sines for 15 deg as sum of first four = 890 -->
sin(15) = 890/3438 = 0.258871 vs. the correct value at 0.258819.
sin(30) = 1719/3438 = 0.5

Expressed as the stanza, using the varga/avarga code:
ka-M 1-5, ca-n~a: 6-10, Ta-Na 11-15, ta-na 16-20, pa-ma 21-25
the avargiya vyanjanas are:
y = 30, r = 40, l=50, v=60, sh=70, Sh=80, s =90 and h=100

vowels: a i u -> mult by 1, 100, 10000, etc.

makhi (ma=25 + khi=2x100) bhakhi (24+200) fakhi (22+200) dhakhi (219)
Nakhi 215, N~akhi 210, M~akhi 205, hasjha (h=100 + s=90+ jha=9)
skaki (90+ ki=1x00 + ka=1)  kiShga (1x100+80+3), shghaki, 70+4+100
kighva (100+4+60) ghlaki (4+50+100) kigra (100+3+40) hakya (100+1+30)
dhaki (19+100) kicha (106) sga (93) shjha (79) Mva (5+60) kla (51)
pta (21+16, could also have been chhya) fa (22) chha (7).

makhi bhakhi dhakhi Nakhi N~akhi M~akhi hasjha
  225,   224    222   219    215     210    205
skaki kiShga shghaki kighva ghlaki kigra hakya
  199    191     183    174    164   154   143
dhaki kicha sga  shjha Mva kla pta fa chha kalA ardha jyaH
  119   106  93    79   65  51  37 22    7

given radius R = radius of 3438, these values give the differences for
Rxsin(theta) at steps of 3 deg 45 min; within one integer value;
e.g. sine (15deg) = 225+224+222+219 = 890, modern value = 889.820.

Both the choice of the radius based on the angle, and the 225 minutes
of arc interpolation
interval, are ideal for the table, better suited than the modern

Indeterminate Equations p.44

e.g. ax + c = by - determine integer solutions for x and y.
[Diophantine equation]
OR, from Bhaskara I commentary (621AD) on AryabhaTiyam:

Find the number which gives 5 as the remainder when divided by 8, 4 as
the remainder when divided by 9 and 1 as the remainder when divided by

i.e. N = 8x+5 = 9y+4 = 7z+1  --> smallest value of N is 85

ALSO from Bhaskara I (621AD), and also dealt with by Ibn-al-Haitam (c.
1000 AD), Leonardo Fibonacci (1202) and also others:

Find the number N which leaves a remainder 1 when divided by 2,3,4,5,6
and is exactly divisible by 7.

[ERROR: Gives solution as (6!+1 = 721) - but 301 is smaller by 420!!]

method: kuTTaka (breaking or pulverizing)

AryabhaTa I gives a systematic method. [This part very unclear]

[ e.g. 3x+5y = 1

5 = 3.1 + 2
3 = 2.1 + 1

then, work backwards with the factors above:
0.1 + 1 = 1     +
1.1 + 0 = 1	-
1.1 + 1 = 2	+

and the answer is 2,-1

==AryabhaTa's astronomy : Moon, Earth, planets - halves lit by sun p.46--

The gola chapter deals  with astronomy. 

	bhUgrahabhAnAM golArdhAni svacchAyayA vivarNAni |
	ardhAni yathAsAraM sUryabhimukhAni dIpyante ||  5

भूग्रहभानां गोलार्धानि स्वच्छायया विवर्णानि । अर्धानि यथासारं सूर्याभिमुखानि दीप्यन्ते ॥ ५ ॥

Earth is spherical; star motions are relative p.46

AryabhaTa I (476 AD) - bhugolaH sarvato vr.ttaH  - golapAda.6
	  (earth is circular in all directions)

	vr.ttabhapan~jaramadhye kakSyAparibeSTitaH khamadhyagataH |
	mr.jjalashikhivAyumayaH bhUgolah sarvataH vr.ttah ||

	 वृत्तभपञ्जरमध्ये कक्ष्यापरिवेष्टितः खमध्यगतः । 
	 मृज्जलशिखिवायुमयः भूगोलः सर्वतः वृत्तः ॥ ५ 

	vr.ttabha+ pan~jara + madhye : inside a spherical cage (of nakSatras, bhagola)
	khamadhyagataH 		     : in space (free, unsupported)
	kakSyAparibeSTitaH  	     : surrounded by orbits (of planets)
	mr.t+jala+shikhi+vAyu-mayaH  : earth-water-flame-air filled
	bhUgolaH sarvato vrttaH	     : the earth, circular from all sides

	the globle (of earth) stands (freely) in space inside a circular frame
	(of nakSatras),  surrounded by the planets.  it is filled with earth,
	water, fire and air and is round all over. 

rotation of the earth as cause for planetary motion

AryabhaTa is the first among the Indian astronomers in stating that the
rising and setting of the sun, the moon and other heavenly bodies is due to
the [earth rotating] about its own axis. 

      anulaumagatirnaursthaH achalAni bhAni samapashchimagAni

	अनुलोमगतिर्नौस्थः पश्यत्यचलं विलोमगं यद्वत्।
	अचलानि भानि तद्वत् समपश्चिमगानि लंकायाम्।। 9
		golapAda 9

	just as a mand in a a boat moving forward sees the stationary objects
	(on either side of the river) as moving backward, just so are the
	stationary stars seen by the people at Lanka (i.e. on the equator) as
	moving exactly towards the east. p.47

[achalAni bhAni samapashchimagAni - golapAda.9]

Sidereal day - AryabhaTa = 23h 56m 4.1 s; modern value 23:56:4.091

Later astronomers, Varahamihira (d. 587AD), Brahmagupta (628AD)
severely criticized him, because of the contrariness of his views to
the overwhelming tradition.

The moon eclipses the sun, and the great shadow of the earth eclipses the
moon.   - AryabhaTIya IV.37

==bhAskara : lIlAvatI problems--

bhAskara's remarkable mastery over language and poetic imagination coupled
with mathematical ingenuity can be discerned from the very interesting
problems posed in his text, lIlAvatI.  146

An example: 

	A beautiful pearl necklace of a young lady was torn in a love quarrel
	(mithuna kalahe) and the pearls were all scattered on the floor.
	One sixth was found by the pretty lady, one-tenth was collected by
	the lover, and six pearls were seen hanging in the thread.  Tell me
	the total number of pearls in the necklace. 

Lilavati problems in verse [original poetry from caferati

[this is inspired by the lilavati text by bhAskara (1150 AD], in which he
poses several mathematical problems in verse.  Here are some modern
renderings from members of the literary group, caferati.  the website where
these were posted (, is now lost. ]

Shankar Hemmady :

  "Whilst making love a necklace broke.
  	A row of pearls mislaid.
  One sixth fell to the floor.
  	One fifth upon the bed.
  The young woman saved one third of them.
  	One tenth were caught by her lover.
  If six pearls remained upon the string
      How many pearls were there altogether?"

Response by Raamesh Gowri Raghavan

	Thirty pearls did the woman fritter,
	Six upon the bed did glitter,
	Ten she on her bosom bore,
	Three snatch'd by her paramour,
	Five descended upon the floor.

answer in verse by SOEB FATEHI

	if it just six that remained
 	then three the lover retained
	the woman skilled in gathering wealth
	to get back ten did strain her health
	on the bed did fall not nix
	i say the actual count was six
	and five that fell down on the floor
	thank heavens didn't roll through the door
	does that not a tidy sum up make
	and the total up to thirty take?

The four jewelers problem


Four jewellers R,S,P,D.  R owns eight rubies, S, ten saphires, P, a hundred
pearls and D, five diamonds.

Now, they presented each of the others with one of their jewels.

After this, they found they each own jewels of precisely equal value.

How much is a saphire worth in terms of pearls? And a ruby, or a

100 p  --> 96p + (p+s+r+d)
10 s  --> 6 s  + (p+s+r+d)
8 r   --> 4 r  + (p+s+r+d)
5 d   --> 1 d  + (p+s+r+d)

therefore diamond = 96p, ruby = 24p, saphire = 16p

[This problem comes from Lilavati, a standard work on Hindu
mathematics written by Bhasakaracharyya, who lived in the twelfth
century of the Christian era. The book is written as instruction for a
young and beautiful woman called Lilavati and it is thought that she
was Bhaskaracharyya's daughter.]

Common origin for Indian, Babylonian and Greek mathematics?

This is an unusual point of view, which Seidenberg presents with immense
multi-cultural scholarship. 

One of the problems of Indian research - not only in Indian science or
history, but in any field including technical - is that our vision is
relatively limited in terms of exposure to alternate cultures. 

His points on the two aspects of Indian and Greek mathematics - the more
abstract of which he does not find in Babylon - is also of interest...

from Abraham Seidenberg, A. 1978, "The Origin of Mathematics"

2. View on the origin of geometry in 1900

Not so long ago, say about 75 years, the thesis that mathematics had a single
origin would have been taken as a foregone conclusion, since with some minor
exceptions, or what were taken to be minor exceptions, there were no
competitors to Classical Greece. For example, W.W.R.  BALL could easily bring
himself to write: 2

	The history of mathematics cannot with certainty be traced back to
	any school or period before that of the Ionian Greeks,

and this statement corresponds to a large extent with what was known about
ancient mathematics in 1900.   Not entirely, however, for there were the
shulvasUtras, ancient Indian sacred works on altar constructions. 

BALL does not mention the SulvasUtras and it is hard to say whether he had
ever heard of them, but M.  CANTOR, a leading historian of mathematics of the
day, had. In 1875 G. THIBAUT had translated a large part of the SulvasUtras,
and these showed that the Indian priests possessed no little mathematical
knowledge.  In 1877 CANTOR, realizing the importance of THIBAUT'S work, began
a comparative study of Greek and Indian mathematics. 5 He concluded that the
Indian geometry was a derivative of Alexandrian knowledge, an opinion he held
for some twenty-five years before finally renouncing it.

THIBAUT was a Sanskrit scholar and in translating the Sulvas0tras his
principal object was to make available to the learned world the mathematical
knowledge of the Vedic Indians; but that wasn't his only object. After
commenting that a good deal of Indian knowledge could be traced back to
requirements of ritual, THIBAUT adds:
	[...]  While therefore unable positively to assert that the treasure
	of mathematical knowledge contained in the LilAvati, the
	VijagaNita, and similar treatises, has been accumulated by the
	Indians without the aid of foreign nations, we must search whether
	there are not traces left pointing to a purely Indian origin of these
	sciences. And such traces we find in a class of writings, commonly
	called S'ulvasfitras, that means 'sutras of the cord,' which prove
	that the earliest geometrical and mathematical investigations among
	the Indians arose from certain requirements of their sacrifices ...

THIBAUT himself never belabored, or elaborated, these views; nor did he
formulate the obvious conclusion, namely, that the Greeks were not the
inventors of plane geometry, rather it was the Indians. At least this was the
message that the Greek scholars saw in THIBAUT's paper. And they didn't like

R.J. BRAIDWOOD, the well-known archeologist, has remarked that a hazard of
his profession is for the archeologist to think that the place he himself has
dug up, especially if it's older than anything else that's been dug up,
represents the beginning of things.

THIBAUT in 1875 had assigned no absolute date to the SulvasUtras, thereby
showing proper scholary restraint. Therefore CANTOR felt free to press his
own chronology.  9

CANTOR had been struck by the analogy of the Indian altar problems to the
Greek duplication of the altar and grave problems, problems he assigns to the
fourth and fifth centuries B.C. 1° Now according to CANTOR HERON'S geometry
intruded about 100 B.C. into India, where it was given a theological
form. This theologic-geometry then left traces in Greece in poetry ascribed
(by CANTOR himself) to EURIPEDES (485-406)--a clear contradiction. Anyway, as
already remarked, CANTOR eventually renounced his view and conceded a much
earlier date to Indian geometry. Even so, he did not believe that PYTHAGORAS
got his geometry from India: he preferred to believe it was Egypt

What then was the view on the origin of geometry in 1900, or even in 19047
The Greeks themselves had supposed, or conjectured, that they had received
their intellectual capital, especially in geometry, from the more Ancient
East, but modern historians have been hard put to corroberate their
views. CANTOR with great acuteness conjectured (op. cir. 1904) that in very
ancient times ("roughly speaking three or four thousand years ago") there
already existed a not altogether insignificant mathematical knowledge
common to the whole cultured area of that time; but this was based on most
scanty materials, indeed. So the Ancient East may have made some minor
contributions, often referred to in the literature as "empirical," but the
prevailing view was (and indeed remains) that we owe geometry as a Science
to the genius of the Greeks.


5. Comparison of Greek and Vedic mathematics

[Though he admires van der Waerden]
 The main fault in VAN DER WAERDEN'S analysis, as I see it, is that at all
vital points he takes into account only Old-Babylonia and Greece: if one
includes the Vedic mathematics, one will get quite a different perspective on
ancient mathematics.

The main issue is the origin of geometric algebra. The SulvasUtras have
geometric algebra, and I will first show that Greece and India have a common
heritage that cannot have derived from Old-Babylonia, i.e., the Old-Babylonia
of about 1700 B.C. as portrayed in Science Awakening.  

The Indian priests in their altar rituals had to convert a rectangle into a
square.  "If you wish to turn an oblong into a square (see Fig. 3), take the
tiryaNmAni, i.e., the shorter side of the oblong, for the side of a square,
divide the remainder (that part of the oblong which remains after the square
has been cut off) into two parts and inverting [one of them] join these two
parts to the sides of the square. (We get then a large square out of which a
small square is cut out as it were.) Fill the empty space (in the corner) by
adding a small piece (a small square). It has been taught how to deduct it
(the added piece)." Cf BAUDHAYANA SulvasUtra 154, APASTAMBA SulvasUtra II 7,
or KATYAYANA SulvasUtra III 2. 38

This is entirely in the spirit of The Elements, Book II, indeed, I would say
it's more in the spirit of Book II than Book II itself. The problem and its
solution are precisely that of II 14, except that the diagram of II 6
intervenes instead of that of II5. In any case the Theorem of PYTHAGORAS and
the identity xy= 1/2 [(x+y)^2- (x-y)^2] are the key facts in the solutions.

The Old-Babylonians could have had no use for such a procedure: they would
simply multiply the two sides and take the square root.

Let us consider now the Theorem of PYTHAGORAS, and under two aspects: 

*  Aspect I : the theorem is used to construct the side of a square equal to the
   sum or difference of two squares; 

*  Aspect II, the theorem is used (say) to compute the diagonal of a

Aspect II comes in, for example, when one uses the (3, 4, 5) triangle to
construct a right angle. 39 The SulvasUtras know both aspects. The Elements
has only Aspect I, but Classical Greek geometry presumably also realized
Aspect II since it had Pythagorean number triples. Now the Old-Babylonians
had Aspect II, but they would have had no use for Aspect I: they would simply
square the lengths of the sides of the given squares, add, and take the
square root.

I could give further common elements of the Greek and Indian mathematics not
shared by Old-Babylonia, for example, the gnomon; or the problem of squaring
the circle. In Ap. SS. III 9 and in The Elements II 4 the gnomon is analysed
into two rectangles and a square (see Fig. 4); and the propositions amount to
our rule: (a+b)^2 = a^2 + b^2 + 2ab. 
The Old-Babylonians know this rule, but they do not have
the gnomon (though it lies at hand to conjecture they once did).

The squaring of the circle is a true geometrical problem in Greece and in
India; in Babylonia, either it does not exist or is to be considered
trivially solvable: the circle has area 3 r 2 and the side of the required
square is ]f3 r.  Conclusion: Either the geometric algebra of Greece came
from India or that of India came from Greece or both came from a third source
different from Old- Babylonia of 1700 B.C.

The altars were, for the most part, composed of five layers of 200 bricks
each, which reached together to the height of the knee; for some cases ten or
fifteen layers and a corresponding increased height of the altar were
prescribed. Most, though not all, of the altars had a level surface, and
these were referred to in accordance with the shape and area of the top (or
bottom) face. The basic falcon-shaped altar had an area of 71 square purusas:
the word "puruSa" means man and is, on the one hand, a linear measure, namely
the height of a man (the sacrificer) with his arms stretched upwards (about
7½ feet, say), and, on the other, an areal measure (about 56 1/4 square feet).

O. BECKER (Geschichte der Mathematik, with J.E. HOFMANN, pp. 39~41) accepts a
date before 600 B.C. for the Theorem of PYTHAGORAS in India.

The conclusion is that Old-Babylonia got the Theorem of PYTHAGORAS from India
or that both Old-Babylonia and India got it from a third source. Now the
Sanskrit scholars do not give me a date so far back as 1700 B.C. 43 Therefore
I postulate a pre-Old-Babylonian (i.e., pre-1700 B.C.) source for the kind of
geometric rituals we see preserved in the Sulvasutras, or at least for the
mathematics involved in these rituals. This sort of hypothesis is made in the
physical sciences. Why not in history, too?

[In other work, e.g.  The diffusion of counting practices (1960) Seidenberg
argues that counting 
	was diffused from one centre and was not discovered again and again
	as is commonly believed.
In general, Seidenberg has argued for a common source prior to Greek,
Babylonian, Chinese, and Vedic mathematics.

The book Geometry and Algebra in Ancient Civilizations by Van der Waerden
puts forward similar views which were inspired by Seidenberg.

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This review by Amit Mukerjee was last updated on : 2015 Aug 03