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
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
zero symbol (a dot) was used in metrics (chhandas) by Pingala (before
200 BC) in his chhandah-sUtra. Jain mathematical texts such as
jambu-dvIpa prajNapati and sUryaprajNApati date back to 500-200BC.
PI is approximated to the square root of 10 and calculated correct
upto 13 places of decimal! - p.3
[AM: This is quite unbelievable. Only on page 2, he writes abt
Aryabhata in 476 AD giving PI to four places for the first time. Even
if it is sqrt of 10 that the jains did, a claim of 13 correct places of
decimal, without further corroboration, seems rather an exaggeration. ]
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
carrie a foreword by Prof Smith himself. In the same Hostory 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 Nx^2 +1 = y^2:
In 1657 Fermat proposed to his friend Frenicle to solve in integers
the indeterminate equation 61x^2 + 1 = y^2, but the solution was not
found until 1732 by Euler. But coincidentally, the same equation was
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 (6z^2 + 4b^2) is a very good
approximation. -p.7
O tender girl, out of the swans in a certain lake, ten times the
squareroot of their number went away to mAnasa (sarovar) on the advent
of the rainy season, one-eighth the number went away to a forest by
the name sthala padminI. Three pairs of swans remained in the lake
engaged in amorous sports. What is the total number of swans?
-- Bhaskara II, Lilavati, (12th c.) [p.9]
Half the square root of a swarm of bees went to a Malati tree,
followed by another eight ninth of the total. One bee was trapped
inside a lotus flower, while his mate came humming in response to his
call. O Lady, tell me how many bees were there in all? - Lilavati,
q. in Narlikar, Sci Edge p.11
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 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] 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. But the most notable sutra 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 Pythagora 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 teh shulvasUtras" [AM; But no pdates 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 a^2 * pi/4 * 10/9 which is abt a^2.
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 ^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
wrote: 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.
[K.S. Shukla, trasn. and editor of AryabhaTiyam, publ K
Sambasivasastri, Trivandrum]
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 dealign
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 - 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
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 3x10^6; 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 2x10^4 + 30 x 10^4(khyu) + 4 x 10^6 (ghr.) its own axis (number of days in year) in a yuga
chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAm
AyutadvayaviShkambhasyAsanno vr^ttapariNahaH.
[gaNita pAda, 10]
Add 4 to 100, multiply by 8 and add to 62,000. This is approximately
the circumference of a circle whose diamenter is 20,000.
i.e. PI = 62,832 / 20,000 = 3.1416
correct to four places. Even more important however is the word
"Asanna" - approximate, indicating an awareness that even 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
AryabhATa I -- area of triangle --
p.tribhujasya falasharIraM samadalakoTI bhujArdhasaMvargaH - gaNitapAda 6
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] 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 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 119 106 93 79 65 51 37 22 7 given radius R = radius of 3438, these values give the Rxsin(theta) 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 I (476 AD) - bhugolaH sarvato vr.ttaH - golapAda.6 (earth is circular in all directions) Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in laMkA (ie. on the equator) as moving exactly towards the West. [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.
Shankar Hemmady wrote:
"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?"
[from Bhaskaracharya's Lilavati, 1150 AD]
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.
response 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.]