Qfrelated interest:
THE COSMIC CALCULATOR (A Vedic Mathematics Course for Schools) Kenneth R. Williams & Mark Gaskell The Cosmic Calculator course offers a complete system of mental mathematics which can be taught in (\ holi~tit way. This course consists of three textbooks an Answer Book and a Teacher's Guide. The course is aimed at 11-14-year-oid pupils though some O'l it is very suitable for children from 8 years. Vedic Mathematics is being taught in many schools world-wide with great success: many top mathematics prizes have been won by st'ldents of this system. .
GEOMETRY IN ANCIENT AND MEDIEVAL INDIA
........ilavati - aracarya ( f Vedic Tradition
TA. Saraswati Amma This book is a geometrical survey of the SaJTlskrt and Prakrt scientific and quasi-scientific literature of India beginning with the Vedic literature and ending with t~e early part of the 17th centufV. It deals in detail with the Sulbastltras in the Vedic literatl~re, with the mathematical parts of Jaina Canonical works and of the Hindu Siddhantas and with the contributions to C\ geo,metry II, Sripati, Bhaskara I & II, Sangamagrama Madhava, ParameSvara, Nilakantha, his discipies and a host of others. The works of the mathe~~ticiam Mah'avira, Sridhar?. and Narayal).a Pal)c.lita and the Bakshali Manusc~ pi. have also been studied.
VEDIC MATHEMATICS Jagadguru Swami Sri Bharali Kr~1Ja TirlhajZ MahclTiiJa The book brings to light how great and true knowledge is born of intuition, quite different from modern Western method. The ancient Indian method and its secret techniques are examined and shown to be capable of solving various problems of mathematics. The universe we live in has a basic mathematical structure obeying the rules of mathematical measures and relations. All the subjects in mathematics-Multiplication, Division, Factorization, Equations, Calculus, Analytical Conics, etc. - are dealt with in forty chapters, vividly working out all problems, in the easiest ever method discovered so far.
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hankara Patwardhan Naimpally . Shyam Lal Singh
Ary;,hhatil (h. 476 All) is credited for his piom'crinl( wurk of extra Cling tantrajiis/ra, (i .e. ;Istronomy and mathematics) from Yl'dfl.1 (gIt¥of'1 1!. .ih.,iUjiQ) In the Hemanta (winter) season the earth looks more enchantin( i because of the plenty of stored grains. The drops of water, in the fonq; of dew, appear like pearls and add to the beauty. The well-fed con! I tented cows and bulls move about in flocks and the farmers enjoy lhil beautiful sight. Teaching Ability In the LWivatf Bhaskaracarya has displayed his various skills. Whil~ teaching Mathematics he also wanted to convey informations on reli! gion, the Vedas, Puriinas, Epics etc. In the stanza 76 the informations! ' . are given regarding a chariot and the fact that Kan:Ja. though a brother of Arjuna, was his enemy. The entire battle-scene is graphically d~ picted. This stanza can be compared with Kalidasa's Sakunta/am. l Grivabharigabhiramam. The graphic descriptions of a drove of swans ; a flock of elephants; a colony of bees; the attack of a snake by a d~, mesticated peacock; sinking of a lotus in water owing to a strong wind and so on: were to train students in the appreciation of nature and to make mathematics interesting rather than tiresome. Bhaskaracarya haS seldom given proofs or derivations following the contemparory tradi~ tion. Still, he gave a large number of examples. He was a leading Astronomer of his time and had earned the well deserved title of "Jewel among the mathematicians". Bhaskaracarya handled determination of sines of angles much more skilfully than predecessors. Although he took earth to be stationary and the centre of , the universe, it did not adversely affect his calculations. Bhaskaracarya was familiar with many results that were later discoyered by Copernicus and Tayko Brahe. He also had some idea of limi~ which was (500 years) later discovered by Newton. He moulded a difficult language like Sanskrit to express a scientific subject like Mathematics. By using tenns from fine literature, ~e
hi.
l
pl'aelical too as he was of the opinion that if eclipses, • do not occur as given in ancient almanacs, then the I.lO .·h.mged. Orthodox people used to stick to false , '-'10n!! Rahu and Ketu swallow the Moon and the Sun Whal actually happens is that at the time of Lunar Cnlers the shadow of the Earth and so it becomes . carya knew this fact but found it difficult to con0:....1'--'.,- - 1\1 l;Irge. He used to say:
I. Hot a demon. It is the shadow of the earth that MOOn invisible. If you still believe in Rahu, at least ICNd the shadow of the Earth and then swallowed
1II1
Til ~
found a way to reconcile the old and the. new disi'I practical scholar, he was not to compromise with 1'0 was a great Indian Astronomer Lalla (A.D. 768) Xlbook on Mathematics which was used until
- - '• •• UIIIC. That text contained some incorrect propositions his hook on Astronomy Bhaskaracarya corrected I olllilac like the correction of the fonnula for the I ~ surface area of a sphere. He has given this in u h daimed himself to be a "student of Brahmaillllcd to criticize the latter's long or wrong or teOlYlclimcs he also criticized Aryabha~ .ll il conjectured that Bhaskaracarya might have vislOiuC lUlonymous Indian astrologers had visited TurMil1illlto4lUll as honourable guests according to some Araftom Sanskrit.
"
xxiv
LIliivali ofBhaskariiciI'yrA l
If/.f
l.ife and Work
xxv
1-'1
bian works. If this view is acceptable, then there should have beeJt.: j some reference to his · works to the Arabian mathematicians. Hence 1 the above view that Bhaskariicarya had visited the middle-east is nol' ) very convincing. Social Conditions •1
It is rather difficult to draw a picture of the social conditions of the)' Bhaskariicarya's period by way of a book on Arithmetic. IJ11' Bhaskariicarya's times, scholarship was assessed by the study of ten 'l works. All trade was in the hands of Lamanas. The wealthy wore! garments made of silk imported from China. Young women were sold , as slaves. Scholars were patronized by the rulers. Travel at night was I risky. Ni~ka, a silver coin, was the principal currency. Peacocks were ~ common pets. There was abundance of lotus flowers, bees, elephants . , 'I and birds. Siva, Vi~~u, and Ga~eSa were universally worshipped. Tho) RamayaJ.li1 and Mahabharata were studied regularly. On the whole, ' there was prosperity. Gold was expensive but there were plenty of .' food items. Astronomers were more involved with Astrology than its mathematical applications. Although the scholars were generally pa- ,I tronised by rulers, Bhaskaracarya seems to be an exception. He wasI hale and hearty, and died in 1193 at the age of 79. After Bhaskaracarya we don't find well-known learned (traditional); mathematicians in Maharashtra. Though, most students of Indian math- \ ematics merely studied Bhaskaracarya's books still no research was done until seventeenth century. Though a great mathematician, he did not know decimal fractions; which were discovered in Italy in the sixteenth century. It is surprising): that Bhaskaracarya did not discover Newton's Binomial Theorem (in' its general form). Had he discovered logarithms, determination of chord\ , would have been considerably simplified. He knew that the earth attracted; other bodies but he missed Force = Mass x Acceleration. For the' computation of daily motion of planets he introduced the concept 0[. ' instantaneous velocity by dividing the day into large number of smal ~ intervals. The formula &(sin 8) = cos 8 88 is implicitly available in Siromani. He was aware that the differential coefficient vanishes at al'l' . " extreme value of the function. He appears to be the first mathematicianl who could perceive the ideas of differential calculus 500 year~ I
11 ' I
I' I,
IIlId Leibniz. He had little knowledge of ellipse, parho III. Had he proved theorems in Algebra and GeomeII hll\' achieved a greater status as a mathematician. III < 'hievements put him on the top of the list of
'unullatC to have had such a great mathematician in India . " I\( 'ligent that we forgot even to celebrate the SOOth composition of his SiddhantasiromaI.Ji. Let us Bhaskariicarya's eight hundredth
ur
Lilavati (Pali Mathematics)
~~ ~ ift1.ff......{ I ·9'(dl~i~ ....."~~rcft;iiIPJ~ II I II _ _"""'"
~
Sillutations to the elephant-faced Lord who creates Ii I ,by remembering whom all obstacles are der-~ I are revered by the community of gods. Here I l illIe mathematics - LJUivatl - which is loved by hccause of its clarity, brevity as well as its liter-
CHAPTER 1
'cfinitions and Tables 'fIl ~ ~ qUlllq('lu: I
.-.m.
~ ~ ~:II 1111
Jrm make one kakiIJ.I Four
kaki~lis
make one paIJ.a. mll'ke one dramma. And sixteen drammas make one
:'IIIIIIWllclIrya gives the following table concerning the . Ieventh and twelfth centuries: (C wri ) c:: 1 kakil)J (davaifi> . . .iD. I fHI(lO (pa/slf)
......."'.,~ h century. the above coins were more or less used in . Oriiish regime coins pai. ruka and dhabu paisa were ' \joll coins were also in use during the rule of the Petr w possible to get 40 Cowries for one Paisa and U.cd In procure sundries such as hot peppers and corird (h71/1f1 for price has evolved from 'dramma'.
3
4
5
LilavatT ofBhaskaracJ
Measure for gold
• I .'."/11. 24 arigulas = I hasta (forearm). .. I ".~,1t" 2000 da¢as I kroSa or kosa. II ,'I IIIhi " 'aIIlS of barley-com are kept be11ywise close to Ih.' lenl'" is one arigula, i.e., the phalanx of the fingers. . . . . III' , ,"nl.lnnl measures in India. There were many kingi flli'il~II, each having its own weights and measures . ....11 til 11'1'111'" was not the same all over India. However, as~'I¥hl or a man is 31h hastas (foreanns), the above t,t \..' IIl1irunllly IIsed throughout India. Ordinarily I hasta = \ ,'" \• • 2.:'1400 cms. So the height of a man equals 5 ft. 10 A ,'",III,,1'd lilli' of length came into use all over India only .... IItlll'," 1~lIll~. Old measures remained in the literature as III IIItllll )1'11111' transactions. Now miles have given way to • '''"' 1r"...,,1 I II IWO miles but now both kosa and mile will go.
=
~ ~ ~ ~ q~fa~s:ffl ~ 1f a~ I 'Itllolq)fi1C;:Q O. He also takes only positive square roots, whereas we
Quadratic Equation
~ ~I"'N'tnl('j~1
t1MI 0, p - ~ > O. Also in2 stead of solving for x, Bhaskariicarya solves it for x . 2 (LXXII): Here the equation is x2 ± x ± bx = c. When simplified n this becomes x2 ±~ = _c_ and then (LXXI) can be used.
take
73
81 =2 + -49 =16 16
litml
~ 1fi: II LXXV II
A flock of swans contained x2 members. As the clouds gathered, lOx went Miinasa Lake, and
~
2 x flew away to a garden of Hibiscus Mu-
ta/is. The remaining three amourous couples played about in the water. 0 young woman, how many swans were there in that lake full of beautiful little lotuses? Comment: Here the equation is x2 - lOx x2 = 6. 8
.!.
7
2
:. - x - lOx = 6
8
•
2
80x
48
.. x ---=7 7
Lilavati of Bhaskaracarya
74
Quadratic Equation 1936 48 (40)2 (40)2 2 80x -=-. x - - + - - = -7+ -49 49 49 7 ., 2 40 44 + - 7 = 12 and x =144 . ... X = - 7
(2X2) = 2 i e , Comment: Here the equatio n is 2X2. - X-! ' .. 9 2x2 --x=2 9
Example 4
9
2 :. x - - x =9
m
qpj: ~ -qJ.lo,"cri ~ ~ ~ ~ (f1Ui(.,oj 'l~'Ifig11l'''''1'
:. x2
~ ~ 1U'Ji tJf\ii ~ ~uhil~ ~:
~
tR1II
a ~: m
I, LXXVI, ,
2
X
Conr",ent: Here the equation is x - i.e., x
:. x
2
-
:. x =
2
-
2
- 4x = 6 + 3 + I
•.
= 10
r (~r
f
16
16
225 16
=
"4
Examp le 6
m 't'~ji(":'C::"~:
~:
~'f1~,:q,
urn; tlt1C,c::t,4S ;qg""'jHl ~ ~1.:'.~qsq~'RIC:l ~ I I LXXXVI I I
If the price of a sixteen-year-old female slave is 32 ni!jkas, find the price ci a twenty-year-old one. An ox which has been yoked for two years cost's 4 ni!jkas: what will an ox which has been yoked for six
CHAPTER 23
years cost?
The Rule of Five
Comment: (i) Price decreases as the age increases. Sod=
~= 128=25~N. 20
5 5 . 4x2 4
(ii) Here the price IS - 6 -
1
='3 = 1 '3 N .
Elample 2 ~ ~ ~ fmlvl ... q~IQJ~ I cro « ~~~d't I I LXXXVII I I
~ ~Iilqoi~
One 'gady8r)aka' (= 48 'guifjas') of 10 carat gold can be bought for 1 N. How much 15 carat gold can be bought for the same price?
Comment: Finer the gold, less can be bought for the same price. So quantity of 15 carat gold =
10
48
15 x 1=32 gUifjas.
.j.,tfIC'I ... q'lJftlIlS'~¥~~ 'M'~C:l' I ~
..!(,~~ ri
~ ~II LXXXIX I I
In the case of examples on the rules of five, seven, nine, etc. keep the antecedents of all proportions in the numerator. All the other terms, except the desired result. should be kept in the denominator. The product of the numerators divided by the product of the denominators is the required result.
Example 1 ~ ~ ~ q3iqctii"lR'l~ ~ ~ ~ ~ ~ ..).nli.", I
1f11 Example 3 ~~mfI~~1 ~ ~ Uffii cro q~ICi¥" ~ I I LXXXVIII II
m
{tID
lP.Rl
'1i'f lIS i"Il "'d'lf'Ql *(
witA~~~IIXCII
There are three problems in this.
With a measure of 7A (acJhakas), a certain quantity of grain measures 100 units. How many units will there be if the measure is 5A?
Comment: This is also an example of inverse proportion. The answer . 100 x 7 -140 uru'ts.
IS - - -
S
83
LTlavati ofBhaskaracarya
84
The Rule ofFive
(1) If 100 ni!jkas (= N) fetch 5 N interest per month, find the interest on 16 N for one year.
Interest
85
, , 26.
, . "'5 . x
16 x 12 x 5 __ 48 __ 9 _3 N =---100 x 1 5 5
direct} direct
In a tabular form, the rule of five is: 100 N Principal: 16 N princiPal} I month: 12 months : : 5 N interest: x
Example 3
(2) The above problem is altered: at the same rate as in (1), find the
period to fetch 9 ~ interest on 16 N.
In the above, the first proportion is inverse since if the principal is less, the period is longer to gain the same interest. 100 xl 48 So x = - - x - = 4 x3=12 months. 16x5 5 (3) Suppose we are given the period and interest and we have to
1 M : 12 M x=
For 100 N (ni!jkas) 8 pieces of multi-coloured embroidered (clothing) material, each measuring 3 cubits x 8 cubits, are available. 0 businessman, if you are proficient in business, tell me quickly the price of
. 3 -1 cu b'Its x -1 cubit, a pIece 2 2' Comment: This is an example of the rule of seven.
3 cubits: .!. cubit 2
find the principal (x). Here 100 N : x
~ ~~f?q ~
tul~~~ ~~ ~ ~ CJfOjiq,fO,"'Q* ~ ~'I XCII Ii
100 N : 16 N ) ' mverse } 48 : : 1M : x 5 N:"'5 N direct
N) : :
(fi(jliCtilfi4n'
~~,\O lUmbers is multiplied by their difference, the result is the-differen:e of their squares. [Thus (1+» (a - b).J
o'fc'Jqg'Cfi
i
+ b2 :: (a - b)2 + 2ab and
Comment:
~ =~. -Vb b.JX [l69 13.fi.
i - b2 = In the above Example 2,
= - 4- =
13(1·414) _ 13(141·4) 4 400
.= {4 476 + _l_} approx . 800 200
til ~ i'r.t Cfi1 ~: I
2
By Bhaskarlicarya's method, choose x = 10000 = 100 and
~~:~~~~~IICXLIVII
J169 = 8
File the hypotenuse if the base is 3 and altitude 4. I f the hypotenuse !~ffii1I§t '!J""6lIa ~ I ~ lf5I ~ ~er-i
faf.lflfict II
CLXX I I
If an idiot says that there is a quadrilateral of sides 2, 6, 3, 12 or a triangle with sides 3, 6, 9, explain to him that they don't exist.
t
I. = FD . r
t
y Y BF + FD sr . Adding - + - = I :. y = - - . Another proof can be given s+r s r t by co-ordinate geometry. Clearly t is not needed to compute y.
Example ~.~,q. .qmj('~: I
$('~i('l
Comment: Clearly the quadrilateral is a trapezium since one side and the altitude are equal. Also we can break the trapezium into a rectan-
~: ClioiQ)4qi 31~gt
