**book excerptise: a
book unexamined is wasting trees **

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Rao, S. Balachandra;**

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Indian Mathematics and Astronomy: Some Landmarks**

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Jnana Deep Publications, Bangalore, 1994 (rev 98)**

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ISBN 81-9100962-0-6
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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 books. 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.

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 stated. 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. ]

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] ELLIPSE 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: form ashrautapart ofkalpa vedAnga- nine texts - mathematically most imp -baudhAyana,Apastamba, andkAtyAyanashulvasUtra. [12] Rules for constructingyajNa 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 rectangle.

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]

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. ]

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.

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

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

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]. 5 2.gaNita pAdais 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

inkAlakriyA 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]

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 not. 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

चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम। अयुतद्वयविष्कम्भस्यासन्नो वृत्त- परिणाहः।।caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇāmayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ || 10[gaNita pAda, 10] word by word:chatuH+adhikaM shatam: 4 in excess of 100;aSTa guNam: eight, multiply (by)dvASaSTiH sahasraNAmद्वाषष्टि 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." i.e. 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)

p.52 AryabhATiya : gaNitapAda 6tribhujasya falasharIraM samadalakoTI bhujArdhasaMvargaH_ त्रिभुजस्य फलशरीरं समदलकोटी भुजार्ध संवर्गः। the body of a triangle (its area) is half the side multiplied by the perpendicular (समदलकोटी)

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 225,224,222,219.215,210,205, 199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7 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 tables.

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 7. 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-- Thegolachapter deals with astronomy.bhUgrahabhAnAM golArdhAni svacchAyayA vivarNAni|ardhAni yathAsAraM sUryabhimukhAni dIpyante|| 5भूग्रहभानां गोलार्धानि स्वच्छायया विवर्णानि । अर्धानि यथासारं सूर्याभिमुखानि दीप्यन्ते ॥ ५ ॥

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.

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.

[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 (ryze.com), 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?

From http://nrich.maths.org/public/viewer.php?obj_id=330&part=index&refpage=monthindex.php 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 diamond? 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.]

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...

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 it. 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. [...]

[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 rectangle. 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 byOld-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 bookGeometry and Algebra in Ancient Civilizationsby Van der Waerden puts forward similar views which were inspired by Seidenberg. ]

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