science in ancient india - 105

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Landmarks of Science in Early IndiaIndras PearlsIn 1971 Roger Billard (a French mathematician and Sanskritist) did astatistical study of the deviations of longitudes of ryaU.S. astrophysicist McKim Malville, with Indian scholars, studiedIndias sacred geography: at Varanasi, Chitrakut, Vijayanagaryabhatasorbit of the skyFind:

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1. Indus-Sarasvati CivilizationFirststepsof technology and science inthe protohistoric era*The east-westalignmentofthemain streets of Mohenjo-daroscitadel (oracropolis,left) wasbased on the Pleiades star

cluster(Krittika), which rose dueeast atthe time;itno longer

does because ofthe precessionof the equinoxes.(GermanarchaeologistHolger Wanzke,Axis systems and orientation at

Mohenjo-daro, 1987)The mystery of Mohenjo-darosring stones(above right):thesmall drilled


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holes (see redarrows),showed thestoneswere usedto track the pathofthe sun through the year, as seen fromMohenjo-daro. Such evidences demonstrate the firststeps inobservational astronomy. There are ot

her hints, such as possibleastronomical symbolism on a fewseals.(Finnish scholar ErkkaMaulan, The Calendar Stonesfrom Mohenjo-daro, 1984)* For techn

ology, pleaseseeseparatepdf file onthis

civilization.A rudimentary decimal systemThe standardized Harappan system ofweights followed a dual binary-decimal progression:


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1 (= 0.86 g), 2, 4,8, 16, 32, 64;then, instead ofcontinuing withthe geometric progression:160, 200, 320, 640, 1,600, 3,200, 6,400, 8,000, 12,800;therefore tens, hundreds andthousands of previous units.Note: This does not

mean thattheHarappans noted their numbers in adecimal manner (thatis virtually impossible, as this developmentcomesmuch later). Several ot

her ancient civilizations also used multiples of10without a decimal systemof numeral notations.2. Historical Era:Pre-Siddhantic PeriodGeometry of theShulbastras

(6thto 10thcenturyBCE, possibly earlier):t


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hese fourancient texts deal with complex fire altars ofvarious shapes constructed withbricks ofspecific shapes and area:the total area ofthealtarmust always be carefully respected.This leads to

precise butpurely geometricalcalculations (algebra does not existyet).TheShulbastrasg

ive a precisegeometric expression of the so-called Pythagorean theorem.Right angles were made by

ropes marked to give the triads3, 4, 5 and 5, 12, 13 (32+ 42= 52


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,52+ 122= 132).We should renamethis theoremthe Shulba theorem!Examples of other geometric operations in theShulbastras:Squaring the circle(and vice-versa):geometrically constructing a square

having the same area as a given area.Adding or subtracting the areas oftwo squares (toproduce a single square).Doubling the area of a square.

In the last construction,2 works out to577/408 or1.414215, correctto the 5thdecimal. (Same precision with3.)Pingalas

treatiseChhandashstraor thescience ofverse meters(a few


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centuriesBCE,Pingalabeing perhaps the brother of thefamous grammarian Panini).Notationofverse meters for verses withvarious numbers ofsyllables (6inthis case).Syllables are light (laghu

) orheavy (guru). Thesystem of notation, spelling outeverypossible combination of light and heavysyllables, is identical to the modern binarynotation of numbers used in computers.Earlie

st inscriptions(first centuriesBCE and AD): numerals without

decimal place value. See for instancehow in the first column, 40 is formedby repeating the symbol for


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20 twice.There is also no symbol for 0 and nomathematical concept of zero.It is well established that so-called Arabic numeralsoriginated in India. Their evolutionhas been traced tothe BrhmscriptofMauryan times.The Webster dictionary gives thesynonym

of Hindu-Arabicnumerals.All Indian numerals are alsoultimately derived from Brhmnumerals (except for Tamil, whichhad a differentsystem usingletters).Ghati yantra,a type of

water clock: thebowl, with a small hole atits bottom, sinksafter 24 mn(a unit of time calledghati,equal to 1/60th

of a day).Ancient texts refertovarious other devices(gnomons, sun dials etc.)which have


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disappeared, but point toa long tradition ofobservation.(Below:)A sun dial (Jantar Mantar, Jaipur,18thcentury).Such massive structures arefound only in recent times;ancientobservatories musthave consisted ofsimple implements made of wood.Judeo-Christian

time scaleIndian time scalecompareSatya:1,728,000 yearsTret:1,296,000 yearsDvpara:864,000 yearsKali:

432,000 years_______________Chaturyuga:4,320,000yearsDuration of a day of Brahm=

onekalpaor 1,000chaturyuga=4.32 billion years.Anno mundi(year of theworld


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s creation):3761 BC (Judaism)4004 BC (Christianity)The Hindu religion is the only one oftheworld s great faiths dedicatedtothe idea that the Cosmos itselfundergoes an immense, indeed aninfinite, number of deaths and rebirths. It is the only religionin which thetimescales correspond, no doubt

by accident, tothose ofmodernscientific cosmology. Its cycles run fromour ordinary day and night to adayand

night of Brahma, 8.64 billionyears long. Longer than the age oftheEarth or the Sun and about halfthe time since the Big Bang.Andthere are much longer time scales still.U.S. astronomerCarl Sagan,Cosmos

Ancient Indians conceivedthe infinityof time and space:Cyclic time.Limitless space B


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hskaraI:The sky is beyondlimit;itisimpossible tostate its measure.The concept ofinfinity underlies muchoflaterIndian science: Brah

magupta first spelt out themathematical definitionof infinity.S. Ramanujan: The manwhoknewinfinity isthetitleof one

of hisbiographies.Concept of evolution:The notion ofDashavatar(10incarnations of the divineconsciousness)containsthe

seed of the concept ofevolution: the firstbody is afish, the second an amphibian,the third a mammal, t


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he fourthhalf-man half-animal, the fifth ashort man, etc. (laterstagesreflecting a spiritual evolution).Aconceptualization orintuition ofthe truth expressedby Darwinian evolution.3. Highlightso

f theSiddhantic Period(from the 5thcenturyCE):the golden age of Indian mathematicsand astronomyEarly Indian scientistsThis map (adapted from the w

ebsite ofSt.Andrews University, Scotland) lists themainfigures ofearly Indian science. (Theexact p

lace orepoch of manyofthemremainsuncertain).Note the shif


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ttothe South, especiallyKarnataka and Kerala, afterthe 12thcentury.The first known inscription with adecimal place-value notation(Sankheda, Gujarat,dated 346Chhedi era, orAD 594):forthe fi

rsttime, 3 stands for hundreds, 4for tens and 6 for units.Georges Ifrah:The UniversalHistoryof Numbers,in 3 volumes.Volume 2 is mostly about Indiascontributions to mathematics.Testimonies from two French mathematicians:

The ingenious method ofexpressing every possible number using aset often symbols (each symbolhaving a place value and anabsolute value)emerged in India. The idea seems so

simplenowadays that its significance andprofound importance is no longerappreciated. Its simplicity lies in the way it facilitated calculation andplaced arithmetic foremostamongst useful inventions. The


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importance of this invention is more readily appreciated when oneconsiders thatitwas beyond the twogreatestmenofAntiquity,Archimedes and Apollonius.Laplace(early19thcentury)The Indian mind has always had for calculations and the handlingo

fnumbers an extraordinary inclination, ease and power, such as noother civilization in history ever possessed to the same degree.Somuch so thatIndian culture r

egarded the science ofnumbers as thenoblest of its arts.... A thousand years ahead of Europeans, Indiansavants knewthat the zero and infinitywere mutually inverse

notions....Georges Ifrah(1994)ryabhata was a brilliant scientist who lived at Kusuma


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pura(probably todays Patna). In 499 CE, he wrote theryabhatya,a briefbut extremely important treatise ofmathematics andastronomy, atthe age of 23!A fewhighlights:ryabhataabout the earth:

The earth is a rotating sphere: the stars do notmove, itisthe earth thatrotates.Its diameteris 1,050yojanas.Its circumferencei

stherefore1050 x 13.6x= 44,860km, about 12%off.(1yojana=8,000 human heights)ryabhata

on eclipses:The moon eclipses the sun,and the greatshadow oftheearth eclipses the moon.(ryabhat


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s ofthe typeax+ c = bythrough thekuttakaor pulverizing method.A thousand years ahead of Europeans,Indian savants knew that the zero andinfinity were mutually inverse notions....Georges IfrahKhachhedameans divided bykha;Kha(space) stands for zero;Divided by zero = infinity.

Brahmagupta,BrahmasphutaSiddhnta(628CE)Foundationsof modern algebra

Solutions in integers forNx2+ 1 = y2were proposed by Brahmagupta(thebhvanmethod).Mahvira(9

thcentury):approximate formulasfor the area andcircumference of anellipse;wo


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rkon permutations and combinations.BhskaraII (12thcentury)developed the improved cyclic method(chakravla);e.g., smallestsolutions to61x2+ 1 = y2are 226153980 &

1766319049. Lagrange reached the same solutions in the 18thcentury,but througha muchlonger method.BhskaraII also workedon derivatives (of

asinefunctionin relationto the velocityof pl

anets).The Kerala SchoolParameswara(1360-1455):detailed observatio


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ns ofeclipses over 55years and consequent correction techniques; minute corrections for theposition of planets after long periods of time.Infinite series, especially oftrigonometric functions.Mdhava(14thcentury): power series expansions for sine and cosine

(correctto 1/3600thofa degree).Infinite seriesof(resulting in values wi

th 10 correctdecimals).Nlakantha (15thcentury):formula forthe sumof

aconvergent infinitegeometric series. Concept ofheliocentrism (building on Paramesw


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ara).The calculus controversy:Indian mathematics had on Arabic mathematics, and ultimately, through Latintranslations, on European mathematics,an influence that is considerablyneglected.... Ifindeed itis true thattransmission of ideas and results between

Europe and Kerala occurred[about calculus], then the role of later Indianmathematics is even more important than previously thought.... The work ofIndian mathematicians has been severely neglectedby Western historians.Brit

ish mathematician IanPearce(www-history.mcs.st-andrews.ac.uk/history/Projects/Pearce/index.html)Specificities of Indian scientific method

Littleinterestin axiomatics, more in pragmaticmethods.Nevertheless, a notion of proof


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(upapatti)does exist,especially in the commentaries(e.g. JyesthadevasYuktibhsof1530 CE). Observed results mustbevalidated.Great skill for developing efficient algorithms (the termsetymology has an Indian connection, through the PersianmathematicianAl-Khwarizmi(800-847 CE). Especiallyvisible in astro

nomical calculations.India was a pioneer in manytechnologies.Metallurgy(bronze, iron, wootz,zinc...)Pottery (ceramic, faience...)Pigments (painting, dyeing...)

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