Walter Eugene Clark - The Aryabhatiya of Aryabhata_ An Ancient Indian Work on Mathematics and Astronomy-University of Chigago Press (1930).pptx

THE ARYABHATI
of
ARYABHATA
An Anc en nd an Work on
Mathematics and As ronomy
T R A N S L A T E D W IT H N O TES B Y
WALTER EUGENE CLARK
Professor of Sanskrit in Harvard University
THE UNIVERSITY OF^CHICAGO PRESS
CHICAGO ILLINOIS
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PREFACE
In 1874 Kern published at Leiden a text called the
Aryabhatlija which claims to be the work o Arya
bhata and which gives III 10 the date o the birth
o the author as 476 a d I these claims can be sub
stantiated and the whole work s genu ne, the
text s the earliest preserved Indian mathematical and
astronomica text bearing the name o an individua
author the ear iest Ind an text to dea specifically
with mathematics, and the ear est preserved astro
nomical text rom the third or scientific period o
Ind an astronomy The on y other text which might
dispute this last claim s the Suryadddhanta (trans
ated with e aborate notes by Burgess and WT tney
in the sixth vo ume o the Journal of the American
Oriental Society). The o d Suryasiddhanta undoubt
ed y preceded Aryabhata but the abstracts rom t
g ven early n the sixth century by Varahamihira n
his Paficasiddhantika show that the preserved text
has undergone considerable revision and may be later
than Aryabhata O the o d PaulUa and Romaka
Siddhantasy and o the transitional Vdsidha Si-
ddhanta, noth ng has been preserved except the short
abstracts given by Varahamihira. The names o sev
era astronomers who preceded Aryabhata or who
were h s contemporaries, are known but nothing has
been preserved rom their writings except a few brief
fragments.
The Aryabhatlya, therefore, s o the greatest im
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VI PREFACE
portaiice n the history o Indian mathematics and
astronomy The second section which deals with
mathematics (the Ganitapuda), has been translated
by Rodet n the Journal asialique (1S79), I, 393
434 and by Kaye n the Journal of the Asiatic Society
of Bengalf 190S pages 111-41 O the rest o the work
no translation has appeared, and on y a ew o the
stanzas have been discussed. The aim o this work s
to give a comp ete translation o the Aryabhatlya with
references to some o the most important paralle
passages which may be o assistance or further study
The edition o Kern mnkes no pretense o giving a
really critical text o the Aryabhatlya. It gives merely
the text which the sixteenth century commentator
Paramesvara had before him There are several un
certainties about th s text Espec a y noteworthy s
the considerable gap after IV 44 which s discussed
by Kern (pp v v The names o other commenta
tors have been noticed by Bibhutibhusan Datta n
the Bulletin of the Calcutta Mathematical Society,
XVIII (1927), 12 Al available manuscripts o the
text should be consulted, a the other commentators
should be studied, and a careful comparison o the
Aryabhatlya with the abstracts rom the old si-
ddhantas given by Varahamihira, with the Suryasi-
ddhanta, with the Sisyadhlvrddhida o Lalla and with
the Brahmasphutasiddhanta and the Khandakhadyaka
o Brahmagupta should be made. All the later quota
tions rom Aryabhata especially those made by the
commentators on Brahmagupta and Bhaskara, should
be collected and verified. Some o those noted by
Co ebrooke do not seem to fit the published Arya-
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PREFACE V
bhatlya. I so, were they based on a lost work o
Aryabhata on the work o another Aryabhata or
were they based on later texts composed by followers
o Aryabhata rather than on a work by Aryabhata V PREFACE
himself? Especially valuable would be a careful study “ warrior,” and bhalla means “ learned man ” “ scho
o Prthudakasvamin or Caturvedacarya the eleventh ar.” Aryabhatta s the spelling which would natural
century commentator on Bralimagupta, who, to judge y be expected t owever a the metrical evidence
rom Sudhakara’s use o him in his edition o the seems to avor the spelling with one t. It s claimed
Brdhmasphutasidclhdnta, frequent y disagrees with by some that the metrical evidence s inclusive that
Brahmagupta and upholds Aryabhata against Brah bhata has been substituted or bhalta or pure y
magupta's criticisms. metrical reasons and does not prove that Arya
The present translation wdth its br e notes bhata s the correct spelling It s po nted out that
makes no pretense at completeness It is a prelimi Kern gives the name o the commentator w'hom he
nary study based on inadequate material. O severa ed ted as Paramadisvara. The name occurs in this
passages no translation has been given or on y a ten orm in a stanza at the beginning o the text and tat ve
tianslation has been suggested. A year s work in another at the end, but n the prose co ophons at in India with
unpublished manuscript material and the ends o the rst three sections the name s given
the help o
competent pundits would be required or as Paramesvara, and this doubtless s the correct orm
the product on o an adequate translation. I have However until more definite historica or metrica
thought it better to publish the material as it s rather ev dence avor ng the spelling Aryabhatta is produced
than to postpone pub cat on or an indefinite period. I prefer to keep the orm Aryabhata
The present translation will have served its purpose The Aryahhatlya s d v ded into our sections
t succeeds in attracting the attention o Indian which conta n n a on y 123 stanzas. It s not a com
scholars to the problem arousing criticism, and en p ete and detailed working manua o mathematics
couraging them to make available more adequate and astronomy It seems rather to be a br e descrip
manuscript material. t ve work ntended to supplement matters and proc
There has been much discussion as to whether the esses which were generally known and agreed upon
name o the author should be spelled Aryabhata or to g ve on y the most distinctive features o Arya
Aryabhatta.* Bhata means “hireling,” “ mercenary ” bhata s own system. Many commonplaces and many
*See especially Journal of the Royal Asiatic Society, 1S65 pp. simple processes are taken or granted For instance
392-93 Journal asiatique ISSO II 473 S Sudhakara DyivedI, there are no rules to indicate the method o calculat
Gai^alarangini, p 2 • ing the ahargana and o finding the mean places o the
planets. But ru es are given or calculating the true
places rom the mean places by applying certain cor
rections, although even here there s no statement o
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PREFACE IX PREFACE
the method by which the corrections themselves arc Aryabhata o Kusumapura cannot be the later
to be calculated It s a descriptive summary rather Aryabhata who was the author o the Mahasiddhanta. than a
full working manual like the later i'ara/ia- The many quotat ons given by Alberun prove con
granthas or the Suryasiddhania in its present orm c us ve y that his second Aryabhata was dent ca
It s questionable whether Aryabhata himself com w th the author o our Aryabhatiya o Kusumapura
posed another treatise, a karanagrantha which m ght as stated at II 1). Either there was a stil earlier
sere directl} as a basis or practical ca cu at on or Aryabhata or A berun mi&taken y treats the author
whether h s methods were confined to oral trad tion o our Aryabhatiya as two persons. I this author
handed down in a school. really composed two works which represented two
Brahmagupta* implies knowledge o two works by s ght y different po nts o v ew t s easy to explain
Arj'abhata one giving three hundred sdvana da^^s n A berun s mistake.*
a yuga more than the other, one beginning the yuga The published text begins w th 13 stanzas, 10 o
at sunr se the other at midnight He does not seem wh ch g ve in a peculiar alphabetica notat on and n
to treat these as works o two different Aryabhatas a very condensed orm the most mportant numerical
This s corroborated by Pancasiddhantika, XV 20 elements o Aryabhata s system o astronomy In
“ Aryabhata maintains that the beginning o the day ord nary anguage or in numerical vords the materia
s to be reckoned rom midnight at Lanka and the wou d have occup ed at least our times as many
same teacher sa eva] again says that the day begins stanzas. Th s section s named DasagiHkasutra in the
rom sunr se at Lanka ” Brahmagupta however conc ud ng stanza o the sect on Th s final stanza,
names only the Dasagltika and the Arydsiasata as the wh ch is a sort o co ophon the first stanza, wh ch s
works o Aryabhata and these const tute our Arya- an nvocat on and which states the name o the
hhatlya. But the word audayikatantra o Brahma- author and a paribhdsd stanza, which explains the
sphutasiddhanta, XI 21 and the w ords audayika and pecu ar a phabetica notat on which is to be em
drdharatrika o X I 13 14 seem to mp y that Brahma p oyed in the o owing 10 stanzas are not counted gupta
s distinguishing between two works o one I see noth ng suspicious in the d screpancy as Kaye
Aryabhata The published Aryabhatiya I 2 begins does. There s no more reason or quest on ng the
the yuga at sunrise The other work may not have authent c ty o the paribhdsd stanza than or ques
been named or criticized by Brahmagupta because o t on ng that o the nvocat on and co ophon Kaye
the act that it ollowed orthodox tradition *For a d scussion o the whole problem o the two or three Vr *a
Alberun refers to two Aryabhatas. H s later bhatas sec Kaye, Bibliotheca mathcmatica, X 2S9 and Bibhutibhusan
Datta Bullctin.of the Calcutta Mathematical Society, XVII 192G 59.
^Brahmasphu^asiddhanta, XI o and 13-14.
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p r fac XI XU PREFACE
would like to e m ru te t .- ncc it seems to furnish Brahmagupta refers to Aryabhata some s xty times
evidence or Aryabhata s knowledge o place-value Most o these passages contain very general criticism Noth
ng is gained by do ng so since Lalla gives n o Aryabhata as departing rom smrti or being igno numerical
words the most mportant numerica ele rant o astronomy but or some 30 stanzas t can be
ments o Aryabhata w thout change, and even w th shown that the identical stanzas or stanzas o iden out this
paribhdsd stanza the rationale o the alpha- tical content were known to Brahmagupta and
b
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0tical notat on in general cou d be worked out and ascribed to Aryabhata In X I 8 Brahmagupta names
ust as satis actory ev dence o place-value furnished; the Arydstasata as the work o Aryabhata and X I
Further, Brahmagupta {BrdhmaspJmtasiddhdnta, XI 43, jdndiy ekam api yato ndryabhato ganitakdlago-
8) names the DasagUika as the work o Aryabhata Idndm, seems to refer to the three sections o our
gives direct quotat ons X I 5 I 12 and X I 4 X I Arydstasata. These three sections contain exact y
17 o stanzas 1 3, and 4 o our DasagTtikaj and X I 108 stanzas N o stanza rom the section on mathe
15 (although corrupt almost certa n y contains a mat cs has been quoted or criticized by Brahma quotat
on o stanza 5 o our Dasagitika. Other stanzas gupta but it is hazardous to deduce rom that, as
are clearly referred to but w thout d rect quotat ons Kaye does ^ that this section on mathematics s
Most o the Dasagitika as we have it can be proved spurious and s a much later addition.^ To satis y the
to be earlier than Brahmagupta 62S a d.) cond t ons demanded by Brahmagupta s name Aryd
The second section in 33 stanzas deals with stasata there must have been in the work o Arya
mathematics. The third section n 25 stanzas s bhata known to him exactly 33 other stanzas orming
called Kdlakriyd, or “ The Reckon ng o T m e ” The a more primitive and ess deve oped mathematics, or
ourth section in 50 stanzas s called Go a or “ The these 33 other stanzas must have been astronomical
Sphere.” Together they conta n 108 stanzas. in character, either orming a separate chapter or
The Brahmasputasiddhanta o Brahmagupta was scattered through the present third and ourth sec
composed in 628 a d just 129 years after the Anja- tions. This seems to be most unlikely I doubt the
hhatiya, if we accept 499 a d the date given n III va d ty o Kaye s contention that the Ganitapada was
10 as being actua y the date o compos t on o that later than Brahmagupta. His suggestion that it s by
work The e eventh chapter o the Brdhrnasphuta- the later Aryabhata who was the author o the
siddhdnta, which is called “ Tantrapar ksa ” and s Mahasiddhanta (published n the “ Benares Sanskrit
devoted to severe criticism o previous works on *Op. dl., X 291-92.
astronomy is chiefly devoted to criticism o Arya * For cr tic sm o Kaye see B bhutibhusan Datta op cit.,
bhata In this chapter, and in other parts o h s work. X V III 1927), 5
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PREFACE x i u x v PREFACE
Series” and to be ascribed to the tenth century or no ogy and expression between the uller text o even
later s impossible, as a comparison o the two Brahmagupta X V III 3 5 and the more enigmatical
texts would have shown, text o Aryobhataja, II 32 33 n their statements o
I feel justified n assum ng that the Aryahharuja the famous Indian method (kuUaka) o solving inde
on the whole s genuine It s o course, possible that terminate equations o the first degree. It seems prob
at a later period some few stanzas may have been able to me that Brahmagupta had be ore him these
changed in word ng or even supplanted by other two stanzas in their present orm It must be e t to
stanzas. Noteworthy is I 4 o which the true reading the mathematicians to decide which o the two ru es
hhuh, as preserved in a quotat on o Brahmagupta s ear er
has been changed by Paramesvara or by some pre The on y serious internal discrepancy which I have
ceding commentator to bham in order to eliminate been able to d scover in the Aryabhatiya is the o ow
Aryabhata s theory o the rotat on o the Eatth ing Ind an astronomy in general maintains that the
Brahmagupta criticizes some astronom ca mat Earth is stationary and that the heavenly bodies
ters in which Aryabhata s wrong or in regard to which revo ve about it, but there s evidence in the Arya-
Aryabhata s method differs rom his own but h s bhatlya tself and in the accounts o Aryabhata given
bitterest and most frequent criticisms are directed by later w r ters to prove that Aryabhata- maintained
against po nts in which Aryabhata was an innovator that the Earth which s situated n the center o
and differed rom smrti or tradition Such criticism space, revo ves on its ax s and that the asterisms are
would not ar se in regard to mathematical matters stationary. Later writers attack him b tter y on this
which had nothing to do with theologica tradition. po nt Even most o h s owm o ow^ers notab y Lalla,
The silence o Brahmagupta here may merely indicate refused to o ow him in this matter and reverted to
that he ound nothing to criticize or thought criticism the common Ind an tradition Stanza IV 9 in spite
unnecessary. Noteworthy s the act that Brahma o Paramesvara, must be interpreted as maintaining gupta
does not g ve rules for the vo ume o a pju’amid that the asterisms are stationary and that the Earth
and or the volume o a sphere which are both given revolves. And yet the very next stanza IV 10 seems
ncorrectly by Aryabhata II 6 7 Th s s as likely to describe a stationary Earth around which the
to prove ignorance o the true values on Brahma asterisms revo ve Quotations by Bhattotpa a the Va
gupta s part as lateness o the rules o Aryabhata sanavarttika, and the M a c indicate that this stanza
WT a other ru es o the Ganitapdda coxild be open to was known in ts present orm rom the eleventh cen
adverse criticism? On the pos t ve side may be tury on. Is it capable o some different interpreta
pointed out the very close correspondence n ter iii- t on? Is t intended merely as a statement o the
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XVI PREFACE
PREFACE XV
dakhadyaka seem to differ much rom those o th
popular view? Has its wording been changed as has Aryabhallya.^ Is this to be taken as an indication
been done with I 4? I see at present no satis actory that Brahmagupta here s following an older and a d
solution o the problem ferent Aryabhata? I so the Brdhmasphulasiddhdnta
Colebrooke^ gives calurviyhsalij amsais cakram gives no clear ndication o the act Or is he o
vhhayalo gacchet as a quotat on by Munlsvara rom low ng another work by the same Aryabhata? Ac
the Aryastasata o Aryabhata Th s would ind cate a cord ng to Drk§it,- the Khandakhadyaka agrees n a
knowledge o a ibration o the equinoxes. No such essentials with the old orm o the Silryasiddhanta
statement s ound in our Arydstasala. The quotat on rather than with the Brahmasphutasiddhanta. Just as
should be verified in the unpublished text n order Brahmagupta composed two different w orks so
to determine whether Co ebrooke was mistaken or Aryabhata may have composed twO works which
whether we are aced by a rea discrepancy. The represented two different points o view The second
words are not ound in the part o the MarTc which work may have been cast n a traditional mold ma
has already been published in the Pandit. have been based on the old Suryasiddhdnta, or have
The ollow ng problem also needs elucidation. A ormed a commentary upon it.
though Brahmagupta X I 43 44 The Mahasiddhanta o another Aryabhata w^ho
anaty ekam ap yato naryabhato ganitaka ago anam ved in the tenth century or later declares X III 14
na maya proktarii tatab prthak p thag du§anany e§am |
I
aryabhatadu§apa,nam sariikhya vakturh na sakyate yasmat
vrddharyabhataproktat siddhantad yan mahakalat
patha r gatam ucchedam visc§itarii tan maya svoktya |
|
tasmad ayam uddeso buddh mataiiyan yo yan |
sums up h s criticism o Aryabhata in the severest But this Mahasiddhanta differs in so many particulars
possible way yet at the beginning o h s Khanda- rom the Aryabhatiya that it s difficult to believe that
khadyaka, a karanagrantha which has recently been the author o the Aryabhatiya can be the one referred
edited by Babua M sra Jyot shacharyya University to as Vrddharyabhata un ess he had composed an
o Calcutta, 1925), we find the statement vaksydmi other work which differed n many particulars rom
khan^dakhadyakam dearydryabhatatulyaphalayn. It s the Aryabhatiya. The matter needs careful nvestiga
curious that Brahmagupta n h s Khandakhadyaka t on ®
should use such respectful language and should fo ow Cf Pancasiddhdnlika, p xx and Bulletin of the Calcutta Mathe
matical Society, X V II (1920j, 69
the authority o an author who was damned so un *As reported by Thibaut, Astroriomie, Astrologie und Mathematik,
mercifully by him in the Tantrapariksd o h s Brahma- pp 55, 50.
sphutasiddhanta. Moreover the elements of the Khan- *See Bulletin of the Calcutta Mathematical Society, X V II (1926)
*
■
Miscellaneous Essays, II 378
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XX TABLE OF CONTENTS
II G a NTTAPADV ok M VTI E I TIC5 21
TABLE OF CONTENTS 1* I n v o c a t o n .................................................... 21
List of Abbr eviat io ns............................... xxvii 2 Name s and Values o Classes o Numbers Increas
I D s c Itika or the T en GIti Stanzas 1 ng by Powers o T e n ..............................
21
A I n v o c a t o n .............................. 1 3 De initions o Square (varga) and Cube (ghana) 21
B System o E.vpressing Numbers b y Letters o 4 Square R oot 22
A p h a b e t ................................................................... 2 5 Cube R o o t ...................................................... 24
1 Revo ut ons o Sun, Moon Earth and Planets 6. Area o Tr ang e Volume o Pyram d 26 in
a yvga . . . . . . . . . . . 9 7 Area o C rc e Vo ume o S p h e re .................27
2 Revo ut ons o Apsis o M oon Con unct ons o 8 Area o Trapez um Length o Perpendiculars rom
Planets and Node o M oon in a yuga; T me Intersect on o D agona s to Parallel Sides 27
and P ace rom TVT ch Revo ut ons Are To 9 Areao Any Plane Figure Chord o One-sixth Cir
Be Calculated 9 cum erence Equa to R a d u s ........................ ...... 27
3 Number o Manus in a kalpa; Number o yvgas 10. Re at on o C rcumference o Circle to D ameter 28
in Per od o a M anu Part o kalpa Elapsed up 11. Method o Construct ng Sines by Form ng Tr
to Bharata Batt e 12 angles and Quadrilaterals in Quadrant o C rc e 28
4 D v s ons o C rc e C rcumference o Sky and 12. Ca cu at on o Tab e o Sine-Dif erences rom
Orb ts o Planets in yojanas; Earth Moves One First One ........................................................... 29
kald in a prQna; Orb t o Sun One-sixtieth 13 Construct on o Circles, Triangles and Quadri
That o Asterisms 13 laterals Determinat on o Hor zonta and Per
5. Length o yojana; D ameters o Earth Sun pend cular .................................................................... 30
Moon Meru and Planets Number o Years 14 Rad us o khairtta (or svavrtla)] Hypotenuse o
in a y u g a .................................... 15 Right-.Aaigle Tr ang e Formed by Gnomon and
6. Greatest Dec nat on o E c pt c Greatest Shadow ................................................ 31
Dev at on o M oon and Planets rom Ec pt c 15 16 Shadow P r o b e m s .............................. 31 2
Measure o a nr .................................... 16 17 Hypotenuse o R ght A g e Tr ang e Re at on o
7 Pos t ons o Ascend ng Nodes o Planets, and Ha Chord to Segments o D ameter Wh ch
■o Aps des o Sunand P a n e t s ................................. 16 Bisects C h o r d ................................................. 34
8 9 D mensions o Ep cyc es o Aps des and Con 18 Ca cu at on o sampatamras W en Two Circles
unct ons o Planets Circumference o Earth Intersect ........................................................................ 34
W nd 18 19 20 Arithmet ca Progression 35 6
10 Tab e o S n e D e r e n c e s .................................. 19 21. Sum o Series Formed by Tak ng Sums o Terms
C C o o p h o n ..........................................................................20 o an Ar thmet ca Progression 37
x x 22. Sums o Series Formed b y Tak ng Squares and
Cubes o Terms o an .Arithmetica Progression 37
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xxu TABLE OF CONTENTS
TABLE OF CONTEXTS XXI
11. Yw /a Year Month and Day Began at First o
23. Product o Tw o Factors Ha the DilTercncc be Cailra; End ess T me Measured by ^Movements o
tween Square o The r Sum and Sum o The r Planets and A s t e r s m s ........................ ...... 55
Squares 38 12 Planets M ove v,ith Equa Speed T me in W h ch
24 To F nd Tw o Factors When Product and D er They Traverse D stances Equa to Orb t o Aster
ence AreKnown .................................. 38 isms and C rcum erence o S k y ......................."55
2 5 Interest ......................................38 13 Periods o Revo ut on D er because Orb ts D er
26. Ru e o Three P ro p o rt o n ............................... 39 in S z e ................................................. 56
27. Fract ons ......................... 40 14 For Same Reason Signs, Degrees and Alinutes
28. Inverse Method 40 D er in L e n g t h .................................................. 56
29 T o F nd Sum o Severa Numbers When Resu ts 15 Order in Wh ch Orb ts o Planets Are Arranged
Obta ned b y Subtract ng Each Number rom The r (beneath the Asterisms aroundEarth as Center 56
Sum Are I v n o w n .......................................... ...... 40 16. Planets as “ Lords o Days” o Week 56
30. T o F nd Value o Unknown V'hen Two Equa 17. P anets M ov e w th The r Alean M ot on on Orb ts
Quantities Cons st o u ow ns and Similar Un and Eccentr c Circles Eastward rom Aps s and
knowns .............................. 41 Westward rom Con unct on 57
31. To Ca cu ate The r Past and Future Con unct ons 18 19 Eccentr c C rc e Equa in Size to Orb t Its
rom D stance between Tw o Planets 41 Center D stant rom Center o Earth by Rad us
32 33 Indeterm nate Equat ons o First Degree o E p c y c e .......................................................58
(hu(iaka) .................................... 43 20. M ovem ent o Planet on E p cyc e "W en ahead o
III Kalkriya or the R eckoning of T ime 51 and When beh nd Its Mean Pos t on 58
1 2 D v s ons o T m e D v s ons o Circle Corre 21 M ovem ent o Ep cyc es Mean P anet on Its
spond ................................................................................ 51 Orb t at Center o E p c y c e ...................................... 59
3 Con unct ons and v^yafipdtas o Two Planets in a 22 24 Ca cu at on o True P aces o Planets rom
y u g a .................................... 51 M ean Places .................................................................... 60
4 Number o Revo ut ons o Ep cyc es o Planets 25 Ca cu at on o True D stance between P anet and
Years o Jup ter 51 E a r t h ................................ .......... 61
5 De n t on o So ar Year Lunar Month C v Day IV Gol or the Sp h e r e .........................................................63 and
Siderea D a y ............................................ 52 1 Zod aca Signs in Northern and Southern Ha ves
6. Interca ary M onths and Omitted Lunar Days 52 3 o E c pt c Even Dev at on o E c pt c rom
7 8 Year o Men Fathers and Gods yuga o A Equator .......................................................................... 63
the P anets D ay o B r a h m a n ....................53 2 Sun Nodes o M oo n and Planets and Earth s
9 Ulsarpinl, avasarpinl, su§amd, and du^-iama as Shadow M ov e a ong E c p t c .......................................63
D v s ons o yuga ...................................................... 53 3 M oon Jup ter Mars and Saturn Cross Ec pt c
10 Date o Wr t ng o Aryabha(lya; Age o Author at at The r Nodes Venus and M ercury at The r
T m e .................................... .............................. 54 Con unct ons .................................................................... 63
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XXIV TABLE OF CONTEXTS
TABLE OF CONTEXTS XXIn 20. Prime Vert ca Merid an and Perpend cu ar rom
Zenith to Nad r Intersect at Place Where Observer
4 D stance rom Sun at W'iiich M oon and Planets
Is 70
Become Vis b e .............................. ...... 63 4
21. Vert ca Circ e Pa ss ng through Planet and P ace
5. Sun Illumines One Ha o Earth Planets and
Aster isms Other Ha Dark 64 VTiere Observer Is (drhinandala Vert ca C rc e
6 7 Spher ca Earth Surrounded b y Orb ts o Pa ss ng through Nonagesima Po nt (drkk^epa-
Planets and b y Asterisms, S tuated in Center o mandala) .................................................................. 70
Space Cons sts o Earth Water Fire and A r 64 22 Construct on o Wooden G obe Caused T o R e
8. Rad us o Earth Increases and Decreases by a vo ve So as T o K eep Pace w th Revo ut ons o
yojana during D ay and N ght o Brahman 64 Heaven y Bod es ...................................................... 70
9 At Equator Stat onary Asterisms Seem T o M ove 23 Heaven y Bod es Dep cted on Th s Equ noct a
Straight Westward Sim le o M ov ng Boat and Sine (Sine o Lat tude Is Base Sine o C o
Ob ects on S h o r e ........................................................64 atitude {sahku at M dday o Equ noct a D ay
10. Asterisms and Planets Dr ven b y Proviector Is co Perpendicular to Base 70
W nd M ov e Straight Westward at Equator— 24. Rad us o Day C rc e 71
Hence R s ng and Sett ng ........................ 66 25. R ght Ascens on o Signs o Zod ac 71
11 12 M ount Meru and Vadavamukha North and 26. Earth-Sine Wh ch Pleasures Increase and De
South Po es Gods and Demons Th nk the Others crease o D a y and N ght 71
beneath Them 68 27. Ob que Ascens on o Signs o Zod ac
72
13 Four Cit es on Equator a Quadrant Apart Sun 28 Sahku o Sun (Sine o A t tude on Vert ca Circle
rise at First Is M dday Sunset— M dn ght at Passing through Sun) at Any G ven T me 72
Others 68 29. Base o sahku D stance rom R s ng and Sett ng
14 Lanka (on Equator 90® rom Po es U a n 22 ° L n e .............................. 73
North o Lanka ..............................................................68 30 Amp tude o Sun (agm) 73
15 From Leve P ace Ha o Stellar Sphere minus 31 Sine o A t tude o Sun When Crossing Prime
Rad us o Earth Is V s b e Other Ha plus Rad us Vert ca ............................................................ 74
o Earth Is Cut O b y Earth ................................. 68 9 32. M dday sahku and Shadow 74
16. At Meru and Vadavamukha Northern and South 33 Sine o E c pt c Zen th D stance (drkk^epajyd) 74
ern Ha ves o Stellar Sphere Visib e M o v n g rom 34. Sine o Ec pt c A t tude (drggatijyd); Parallax 75
Le t to R ght or V ce Versa 69 35 36 Drkkarman {dk§a and dyana) . 76 7
17. At Poles the Sun a ter It Rises Visib e or Ha 37 M oon Cau ses Eclipse o Sun Shadow o Earth
Year on M oon the Sun Visib e or Ha a Lunar Causes Ec pse o M o o n ..................................................78
38. T me at Wh ch Eclipses O c c u r ......................................78
M o n t h .......................................... 69
39 Length o Shadow o E a r t h ......................................78
IS De n t on o Prime Vert ca Mer d an and 40 D ameter o Earth s Shadow n Orb t o M oon 79
Hor zon ..........................................................................69
41. Sthityardha Ha o T me rom First to Last
19 East and West Hour C rc e Passing through Poles C o n t a c t .................................... 79
(unmar),dala)............................................................ 69
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81

X X V l l l L ST OF ABBREVIATIONS
C o e b r o o k c H T Co cbrookc Algebra, with
Arithmetic and Mensuration,
from the Sanscrit of Brahmegupta
LIST OF ABBR EVIATION S and Bhdscara. London 1817
Co cbrooke Essays.................. MisceUaficoiis Essays 2d ed b y
A berun .................................... A berun 's India. Trans ated by H T Co ebrookc Madras
E C Sachau London 1910 1872.
Barth Zutres ........................ QLuvres de Auguste Barthc. 3 vo s Hemacandra Abhidhand- Ed ted b y Boht gk and R eu
.Paris, 1917 cintdmaiii. .............................. St Petersburg 1847
B C M S . . . . , .............................Bulletin o f the Calcutta Mathe l A ................................... .IndianAntiquary.
matical Society. I H Q .............................. IndianHistoricalQuarterly.
Bhaskara, Gan,itddhydya Ed ted b y B apu Deva Sastr re J A .......................................... .Journalasiatique.
v sed b y Ramachandra Gupta JA S B .......................... Journal and Proceedings of the
Benares no doite). Asiatic Society o f Bengal.
Bhaskavdi., Goladhydya E d te d b y Bapu Deva Sastri re JBBRAS...................................Journal of the Bombay Branch of
v sed b y Ramachandra Gupta the Royal Asiatic Society.
Benares no date JBORS.............. ........................Journal of the Bihar and Orissa
Ed ted b y G r a Prasad Dv ved Research Society.
Lucknow Newu K shore Press J IM S ........................... .Journal of the Indian Mathematical
1911. Society.
Bhattotpa a.............................. Th e Brhat Samhita b y Varuha JRA S ........................................ Journal of the Royal Asiatic
m h ra w th the commentary Society.
o Bhattotpa a “ V zianagram Kaye Indian Mathematics.. G R Kaye Indian Mathematics.
Sanskrit Series,” V o X B e Ca cutta 1915
nares 1895 97 K&ye, Hindu Astronomy “ Memo rs o the Archaeo og ca
Bihl. math.................. ................Bibliotheca mathematica. Survey o Ind a No 18 Ca
Brahmagupta...................... ... Re ers to Brdhmasphutasiddhdnta. cutta 1924
Brdhmasphu(asiddhdnta Ed ted by Sudhakara Dv ved n in Khandakhddyaka......................By Brahmagupta Ed ted by
the PandtY N S Vo s X X II I Babua Misra Jyot shachar>ya
X X IV Benares 1901 2 Un vers ty o Ca cutta 1925
Brennmd, Hindu Astronomy. .W . Brennand Hindu Astronomy. La a.................................... T h e Si^yadhurddhida o Lalla
London 1896. Ed ted by Sudhakara Dv ved n
Brhat Samhita. .................... The Brhat Sarhhitd by Varaham Benares no date
h ra Vr th the commentary o Mahdsiddhdnta. ...................... By Aryabhata Ed ted by Sudha
Bhattotpa a “ Vizianagram San kara Dv ved n in the “ Benares
skrit Series,” Vo X Benares, Sanskrit Series.” 1910
1895 97
xxvii
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.

2 ARYAB IAT VA
present the ev dence s too scanty to allow us to
spec y the sources rom which Aryabhata drew.
The stanza has been translated by F eet As
C HAPTE R I po nted out first by Bhau DajI, a passage o Brahma
d a Ba g I ti k a o r t he t en gupta X II 43 jdndty ekam api yalo ridryabhato
G T I STANZAS ganitakdlagoldndm, seems to refer to the Ganitapdda,
A Hav ng paid reverence to Brahman, who s one (in causal the Kdlakriyapada, and the Golapdda o our Arya-
ty as the creator o the universe, but many (in his manifesta bhatlya (see also Bibhutibhusan Datta).^ Since
t ons the true de ty the Supreme Spirit Aryabhata sets forth Brahmagupta XI 8 names the Dasagllika and the
three things mathematics ganita], the reckoning o time Arydstasata (108 stanzas) as works o Aryabhata and
[fca ofcnya and the sphere gofa]. since the three words o X I 43 refer in order to the
Baidyanath suggests that salya devatd may denote last three sections o the Aryabhatiya which contain
Sarasvatl, the goddess o learning For th s I can exact y 108 stanzas), their occurrence there in this find
no support, and therefore o ow the commen order seems to be due to more than mere coincidence tator
Paramesvara in translating “ the true de ty ” As F eet remarks ^ Aryabhata here claims specifically G od in the
highest sense o the w ord as referring to as his work on y three chapters. But Brahmagupta Pra apati,
Pitamaha, Svayambhu, the ower ndi (628 A D actua ly quotes at least three passages o vidualized
Brahman, who is so called as be ng the our Dasagltika and ascribes t to Aryabhata There creator o the
universe and above al the other gods. is no good reason or refusing to accept t as part o Then this lower
Brahman is dentified with the higher Aryabhata s treatise.
Brahman as be ng on y an individualized manifestar B Beg nn ng w th ka the varga letters (are to be used) in the
t on o the latter. As Paramesvara remarks the use varga places, and the avarga letters (are to be used in the avarga o
the word kam seems to indicate that Aryabhata places. Fa s equa to the sum o no and ma. The nine vowe s
based his work on the o d Pitamahasiddhanta. Sup (are to be used in two nines o places varga and avarga. Navdntya-
varge vd.
port or this view is ound in the concluding stanza
o our text IV 50) dryabhatiyarh ndmnd pi'irvam Aryabhata s system o expressing numbers by
svdyambhuvani sadd sad yat. However, as shown by means o letters has been discussed by Wh sh ® by
Thibaut^ and Kharegat,^ there s a close connect on ^JRAS, 1911 pp 114-15. *BCMS, XVIII (1927), 16
between Aryabhata and the o d Suryasiddhdnta. At » Ibid., 1865 p 403 *JRAS, 1911 pp 115 125
^Pahcasiddhanlika, pp xv xxv * Transactions of the Literary Society of Madras, I (1827) 54
trans ated with additiona notes by Jacquet, JA (1835) II 118
* XI X 129-31.
1
I I
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if f
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: f f ,
f ], [
f ,
f (
f .
ll
' i l
f .
i , l ,
f ' , l
j l i l f
. .)
i i .
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. iii, ii.
l l , ,
,

T O TEN GiT STANZAS ARYAB AT YA
I rockhaus * by Kern,- by Barth by Rodet by which are arranged n five groups o five letters each Iv ye ^
by F eet ® by Sarada Kanta Ganguly,^ and by -The avarga letters arc those rom y to h, which are S k imar
Ranjan Das ® I have not had access to the not so arranged n groups. The phrase “ beginning Vrlhiinr Itihdsa
o Durgadas Lah r ® w th Aa” s necessary because the vowels also are
The words varga and avarga seem to re er to the d v ded nto vargas or “ groups ”
I u an method o extracting the square root which There ore the vowe a used n varga and avarga s
described n detail by R odet “ and by Avadhesh places with varga and avarga etters refers the varga
Narayan Singh.“ I cannot agree with K aye s state letters A to w to the rst varga place, the unit place, ment^
that the ru es given by Aryabhata or the multiplies them by 1 The vowe a used with the extraction o
square and cube roots II 4 5 “ are avarga letters y to h re ers them to the first avarga perfectly general
(i.e., a gebra ca ” and app y to a place, the p ace o ten s multiplies them by 10 In arithmetical notations
nor with h s criticism o the like manner the vowe i refers the letters k to m to foregoing stanza “ Usually
the texts give a verse the second varga place, the place o hundred s multi explaining this notat on but this
exp anatory vCrse plies them by 100 It re ers the avarga letters y to h
s not Aryabhata s ” ^ Sufficient evidence has not been to the second ayar^o place, the place o thousand s
adduced by him to prove either assertion multiplies them by 1 000 And so on with the other
The varga or “ square” places are the rst third, seven vowe s up to the ninth varga and avarga places. th
etc., count ng rom the right. The avarga or From Aryabhata s usage it s clear that the vowels to
“ non-square” places are the second, ourth sixth be emp oyed are a i, u, r, I, e, ai, o and au. No
etc., counting rom the right. The words varga and d st nct on s made between long and short vow'els
avarga seem to be used n this sense in II 4 There From Aryabhata s usage t s clear that the letters s no
good reason for refusing to take them n the same k to m have the values o 1 25 The letters y to h
sense here As applied to the Sanskrit alphabet the wou d have the values o 3 10 but since a short a s
varga letters referred to here are those rom k to m regarded as inherent n a consonant when no other
*Zeilschrifl fur die Kunde des Morgenlandes, IV 81 vowe sign s attached and when the virdma s not
*JRAS, 1SG3 p. 3S0 ' IHQ, III 110 used, and since short o re ers the avarga letters to the
*CEuitw III 182 »I II 332 p ace o ten s the signs ya, etc., really have the values
*JA (1880) II 440 “ Op. oil. (1879) I 406-8 o 3 K100 ^ The vowels themselves have no numerical
1907 p. 478 “ BCMS. X Y l ll 1927), 128 values. They merely serve to refer the consonants
*Op cU., 1911 p 109 “ Op cil., 1908 p 120
(wh ch do have numerica values) to certain places.
*BC.MS, XVII (1926), 195 “ Ibid., p. 118
*See Sarada Kanta Ganguly, op. cit., XVII (1926), 202
III I II l
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ii i
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6 ARYAB AT VA
THE TKX G T STANZAS
terpretation s acceptable. However I know no other
The ast clause, which has been e t untranslated passage which would wan ant such a translation o
Ucrs great d T qu ty The commentator Paraincsvara antyavarge.
takes it as affording a method o expressing stil Sarada Kanta Ganguly trans ates “ [Those nine
» hcr numbers by attaching anusvdra or visarga to [vowels] [should be used n higher places n a similar
the vowels and using them n n ne further varga (and manner.” It s possible or vd to have the sense o
cKirga) places. It s doubt u whether the word “ be eb g ” “ aku tat v ” and or nava to be sepa
cuirga can be so supplied in the compound Fleet rated rom antyavarge, but the regular meaning o
wou d translate “ in the varga place after the n ne” antya s “ the last ” It has the sense o “ the o ow ng”
as giving directions or referring a consonant to the on y at the end o a compound and the dictionar}'
n neteenth place In view o the act that the plura g v es on y one example o that usage I navdntyavarge
subject must carry over into this clause F eet s n is to be taken as a compound the translation “ n the
terpretation seems to be mposs b e Fleet suggests as group o ow ng the nine” s a right. But Gangu y s
an a ternate interpretation the emendation o vd to translation o antyavarge can be maintained on y he
bm. But, as explained above au refers h to the produces evidence to prove that antya at the begin
e ghteenth place It would run to nineteen places on y ning o a compound can mean “ the ollowing ”
when expressed in digits. There s no reason w hy such I nava s to be separated rom antyavarge it s
a statement should be made n the rule Rodet possible to take t with what precedes and to trans
trans ates (without rendering the word nava), “ (sep- late, “ The vowe s (are to be used) in two nine s o
nre ncnt ou k un groupe term n par un varga ” That places, nine in varga places and nine n avarga p aces ”
s to say the clause has nothing to do with the ex but antyavarge vd remains enigmatical.
press on o numbers beyond the eighteenth place, The translation must remain uncertain unti but
merely states that the vowels may be attached further ev dence bearing on the meaning o antya
to the consonants s ng * as gara or to a group o con can be produced TM atever the meaning may be, the
sonants as gra, in which latter case t is to be under- passage s o no consequence or the numbers actually
*>tood as applying to each consonant in the group. So dealt with by Aryabhata n th s treatise. The largest
giri or gri and guru or gru. Such indeed, s Arya number used by Aryabhata himsel 1 1 runs to only
bhata’s usage and such a statement s really nec ten places.
essary in order to avo d ambiguity but the words do Rodet Barth, and some others would translate “ in
not seem to warrant the translation g ven by Rodet the two nine's o zero s ” instead o “ in the two nine s
I the words can mean “ at the end o a group and o p aces ” That s to say each vowe would serve to
nava can be taken with what precedes, Rodet s n
Il l
l I
i ,
f
l l f ,
l if i l .
l , ]
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f
f
i i
l li i
f
i
, l i f
i f
f
.
. f f ll i
l i
l
i '
f ,
f
l f . f
i . f f l
,
i ll
i
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f ll i
f
l ’
l if
i i l .,
l f
,
f f .
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f i f i
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l
i l ,
i )
,
i i
i
l
i f
i f
. i
i lj f
i f f
i i
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, i
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i
f
.
f l . i , l
f
if
,’’
’ i

THE TEN GiTI STANZAS 8 ARYAB AT YA
a t tuo zero’s to the numerica value o the con unwieldy numbers n verse n a very brie orm
N»n nt This o course, w ll work rom the vowe i The alphabetical notation s emp oyed only n the
on but the vowe a does not add two zero s It adds Dasagitika. In other parts o the treatise, where only
no zero s or one zero depending on whether it s used a ew numbers o small size occur the ordinary words
w th larga or avarga letters. The act that khachi- which denote the numbers are employed
mnake s amplified by varge ^varge'is an added difficulty As an illustration o Aryabhata s alphabetica
to the translation “ zero ” It seems to me therefore, notat on take the number o the revolutions o the
pre erable to take the word kha n the sense o “ space” M oon n a yuga I 1), wd ch is expressed by the word
or better “ p ace ” ^ Later the word kha s one o the cayagiyihumchlr. Taken syllable by syllable th s
commonest words or “ zero,” but it s st disputed gives the numbers 6 and 30 and 300 and 3,000 and
whether a symbo or zero was actually in use n 60,000 and 700,000 and 7,000,000 and 50,000,000.
Aryabhata s time. It s possible that computation That s to say 57,753,330 It happens here that the
may have been made on a board ruled nto columns. d g ts are given in order rom right to left, but they
On y n ne symbols may have been in use and a blank may be given in reverse order or n any order w^h ch
column may have served to represent zero will make the syllables fit into the meter. It s hard
There s no evidence to indicate the way n which to be eve that such a descriptive alphabetica nota
the actua calculations were made, but it seems cer t on was not based on a place-value notation
ta n to me that Aryabhata cou d write a number n Th s stanza as being a technica paribhdsd stanza
s gns which had no absolutely xed values in them which indicates the system o notation emp oyed n
se ves but which had value depending on the places the Dasagitika, s not counted The nvocat on and
occupied by them (mount ng by powers o 10). Com the co ophon are not counted There s no good reason
pare II 2 where in giving the names o classes o why the thirteen stanzas should not have been named
numbers he uses the e xpress on sthdndt sthdnam Dasagitika (as they are named by Aryabhata himself
da^dgumm &ydt, “ rom place to p ace each s ten times n stanza C rom the ten central stanzas n G t
the preceding.” meter which give the astronomical elements o the
There s nothing to prove that the actual calcula system. The discrepancy offers no firm support to the
tion was made by means o these letters. It s prob content on o Kaye that th s stanza s a ater addition
ab e that Aryabhata was not inventing a numerical The manuscript referred to by Kaye^ as containing
notation to be used in calculation but was devising fifteen instead o thirteen stanzas s doubtless com
a system by means o which he might express arge »See 7A (1880), II 454 and BCMS, X V II 1926), 201
*Cf F eet op. cit., 1911 p 116 » Op a 1908 p 111
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10 ARYABIIATlYA
THE TEN G T STANZAS 9
Here and elsewhere in the DdsagUika words are
o raMc to the one referred to by Bh iu Da * as having used n their stem orm without declensional endings.
o introductory stanzas “ ev dent y an after-add Lalla {Madhyamadhikdra, 3 6 8) gives the same
t on and not in the Aryii metre ” numbers or the revolutions o the planets, and differs
1 In a yitga the revo ut ons o the Sun are 4,320,000, o the on y in g v ng “ revolutions o the asterisms” instead
»7 733 330 o the Earth easUvard 1,582,237,500, o Saturn
150 01 o Jupiter 364,224 o ?v ars 2,296,824 o Mercur and o “ revo ut ons o the Earth ”
Wnii the same as those o the Sun The Suryasiddhanta I 29 34 shows slight varia
2 o the apsis o the M oon 488,219 o (the con unct on o tions (see Pancasiddhdniikd, pp xv x x and
MtTcur>* 17,937,020, o (the con unct on o Venus 7,022,358, o Kharegat* or the closer relationship o Aryabhata
conjunctions o the others the same as those o the Sun, o to the*old Suryasiddhanta).
tl.c node o the M oon westward 232,226 starting at the beginning
Bibhutibhusaii Datta ® in criticism o the number
o Mcâ at sunrise on Wednesday at Lanka
o revo ut ons o the planets reported by Alberun II
The so-called revo ut ons o the Earth seem to 16 19 remarks that the numbers given or the
re er to the rotat on o the Earth on its ax s The revo ut ons o Venus and Mercury really refer to the
minbcr given corresponds to the number o s derea revo ut ons o their apsides. It would be more accu
days usually reckoned in a yuga. Paramesvara, who rate to say “ con unct ons ”
ollows the normal trad tion o Ind an astronomy Alberun I 370, 377 quotes rom a book o
and believes that the Earth s stat onary', tries to Brahmagupta s which he calls Critical Research on the
prove that here and in IV 9 (which he quotes Basis of the Canons a number or the civil days accord
.ryabhata does not really mean to say that the Earth ng to Aryabhata Th s corresponds to the number o
rotates His effort to br ng Aryabhata nto agreement siderea days g ven above c the number o siderea
w th the views o most other Ind an astronomers days g ven by Brahmagupta [I, 22])
ôems to be misguided ingenuity. There s no warrant Compare the figures or the number o revolutions
for treating the revolutions o the Earth given here o the planets g ven by Brahmagupta 1 15 21 which
• ls based on false knowledge {mithijdjhana), which differ in detail and include figures or the revolutions
causes the Earth to seem to move eastward because o the apsides and nodes. Brahmagupta I 61
of the actua westward movement o the planets (see
akrtaryabhatab sighragam nduccaih patam alpagara svagate |
note to I 4 tithyantagrahananari ghunak§arani tasya satiivadah
In stanza 1 the syllable $u in the phrase which
g ves the revolutions o the Earth s a misprint or criticizes the numbers given by Aryabhata or the
lu as given correct y in the commentary ® revo ut ons o the aps s and node o the Moon.^
yibid., IS60 p. 397 * See ibid,, 1911 p. 122 n
» JBBRAS, X V III 129-31. * BCMS, X V l l 192G 71
*See further Bragma/îpta (V 25 and A berun (I, 376)
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12 ARYABHAT YA
THE TEN G T st a 3 11 3 There arc 14 Manus in a day o Brahman a kalpa], and
73 yugas const tute the period o a Manu Since the beginning
Brahmagupta II 4G-47 remark iiat accord ng o this kalpa up to the Thursday o the Bharata battle GManus
to Aryabhata a the planets were not at the rst po nt 27 yugas, and 3 yugapddas have clap.scd.
o Mesa at the beginning o the yuga, I do not know The word yiigapdda seems to indicate that Arya on
what evidence this criticism s basedd bhata d v ded the yuga nto four equal quarters.
Brahmagupta XI 8) remarks that accord ng to There s no d rect statement to th s effect, but a so the
Aryasiasata the nodes move while accord ng to there s no reference to the traditional method of the
DasagUika the nodes except ng that o the Moon d v d ng the yuga into four parts n the proport on of are fixed:
4, 3 2 and 1 Brahmagupta and later tradition firyaijtasate pata bhramanti dasagitike sthiruh patalj |
ascribes to Arj^abhata the division o the yuga into muktvendupatam apamandale bhramanti sthira natal?. H four
equa parts. For the traditiona division see This re ers to I, 2 and IV 2 Aryabhata ^I 7 gives
Suryasiddhanta I 18 20 22 23 and Brahmagupta
the location at the time-his work was composed o I 7 8 For discussion o this and the supposed
the apsides and nodes o a the planets, and I 7 and divisions o Aryabhata see Fleet.* Compare III 10
IV 2) implies a knowledge o their mot on But he wh ch gives data or the calculation o the date o
gives igures on y or the aps s and node o the Ioon the compos t on o Aryabhata s treatise. It s clear
This may be due to the act that the numbers are so that the fixed po nt was the beginning o Aryabhata s
sma that he thought them negligible or h s purpose ourth yugapada (the later Ka yuga at the time o
Brahmagupta X I 5) quotes stanza 1 o our text the great Bharata battle in 3102 b c
Compare Brahmagupta I 9
'ug 'ira'ibhaganah khyughr t yat proktam tat tayor yuga h
spa§tara J yugapadan aryabhatas catvar sam5n Iq-tayugadlni
tr iat ravyudayanam tadantaram hetuna kena *
* yad abh h tavaa na te§am smrtyuktasamanam ekam ap
*See SHryasiddhdnta, pp 27-28 and JRAS, 1911 p. 49 1
and X I 4
^ * Cf. JRAS, 1865 p. 401 This imp es as Sudhakara says that ryabhato yugapada iis t n yatan aha.kaliyugSdau yat
Brahmagupta knew two works by Aryabhata each giving the revolu tasya krtantar yasraat sva ugadyantau na tat tasmat 1
t ons o the Sun as 4,320,000 but one reckoning 300 sdvana days more
op cit., X V II 11920 00 74 The Paticasiddhantika also XV 20),
than the other. Cf. Kharegat (op. cit., X I X 130 Is the re erence to
“ Aryabhata mainta ns that the beginning o the day s to be reckoned
another book by the author o our treatise or was there another
from midnight at Lanka and the same teacher aga n says that the
ear
er Ary abhata? Brahmagupta X I 13-14 further imp es that day begins from sunr se at Lanka,” ascr bes the two theor es to one
he knew two works by an author named Aryabhata n one o which Aryabhata.
the yuga began at sunr se n the other at midnight see JRAS,
1S63 p. 3 U ; JBBRAS, X IX 130-31; 5 1911 p. 494; IHQ, IV *Op. c ^ 1911 pp. I 4S6
506 At any rate Brahmagupta docs not imply knowledge o a
second .ryabhata For the whole problem o the two or three Arya
bhatas see Kaye (Bibl. math., X 289 and Bibhutibhusan Dutta
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THE TEX G T STANZAS 13
14 ar y a b a tTy a
with the commentary o Sudhfikani. Brahmagupta ows ekapariirtlau grahasya javo gatimdnam yojand-
I 12 quotes stanza I, 3 tmakaju bhaiati.
mauusandhiih yugarn cchaty aryabhatas tanmanur yata The word yojandrii must be taken as given a figure
skhajoiga | in yojanas or the circumference o the sky (dkdsa-
kalpas caturyuganarti sahasram a?tadhikaii tasya |I kaksyd). It works out as 12,474,720,576,000 which s
Brahmagupta I 28 re ers to the same matter, the exact figure given by Lalla {Madhyamddhikdra
adh kab smrtyuktamanor aiy^abhatoktai caturja gena manu | 13 who was a ollower o Aryabhata Compare
adhikam vimsarhsayutais tribhir yugais tasya kalpagatam H Suryasiddhanta, X II 80 82 Brahmagupta X X I
Brahmagupta X I 11 criticizes Aryabhata or be 11 12 Bhaskara Goladhyaya, Bhiivanakosa, 67 69
ginning the Ka yuga w th Thursday (see the com and Ganitddhydya, Kaksddhydya, 1 5
mentary o Sudhakara). The statemciR o Alberun I 225) wdth regard to
Bhau Daji* irst po nted out the pa a e s in the ollowers o Aryabhata
Brahmagupta I 9 and X I 4 and X I 11 * I t s sufHciont or us to know the space wh ch is reached b
4 The revolut ons o the IMoon (in a yuga) multiplied b y 12 the solar rays. We do not want the space wh ch is not reached
are signs rds * The signs multiplied by 30 are degrees. The b y the solar rays, though t be in itse o an enormous e.xtent.
degrees multiplied by 60 are minutes. The minutes multiplied
That wliich s not reached by the rays is not reached by the per
by 10 are yojanas o the circum erence o the sky The Earth
cept on o the senses and that wh ch is not reached b y per
cept on s not knowablc
moves one m nute in a yrdna} The c rcum erence o the sky (in
yojanas) d v ded b y the revolut ons o a planet in a yuga gives may be based ultimately upon this passage
the yojanas o the p anet s orb t The orb t o the Sun s a s .xt eth The reading hham oi our text must be ncorrect
part o the circ e o the asterisms. It s a reading adopted by Paramesvara who was de In
translating the words saHrdsayas tha cakram termined to proye that Aryabhata did not teach the
I have ollowed Paramesvara’s interpretation sasinas rotat on o the Earth This passage could not be ex
cakram bhagand dvddasagunitd rdsayah. The Sanskrit plained away by recourse to false knowledge {mith'
construction is a harsh one but there s no other way ydjndna) as cou d I, 1 and IV 9 and therefore was o
niaking sense Sasi (without declensional ending) changed. The true reading s bhuh, as s proved con
s to be separated. c usively by the quotation o Brahmagupta XI 17
Paramesvara explains the word grahajavo S
ls o pranenaiti kalam bhuryad tarh kuto vra et kam adhvanam |
» Cf. I 8 * Op cil., 1865 pp 400-401. fivarttanam urvya cen na patant samucchrayab kasmat. |
|
* Cf, A beruni, I, 370 373-74. Compare Brahmagupta XXI 59 and Alberuni I
*A Td§i s a s gn o the zodiac or one-twelfth o a c rcle 276 77 280).
*For prana see III 2
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THE TEN GiTI STANZAS 15 G ARYABHAT YA
5. A yojrma consists o S,000 times u nr [the liei" it o a man] passage I 244—
16 quotes Bahibhadra on Arya The
diameter o the Earth s 1 050 yojana.-u T c diameter o the bhata's concept on o Meru Its height s said to be
Sun s 4,410 yojanas. T e diameter o the Moon s 315 ynjnnnfi. a yojana. The context o the foregoing stanza seems
Meru s one yojana. The diameters o Venus Jupiter Mercury
to mp y that ts diameter s a yojana, as Paramesvara
Saturn and Mars are one-f fth one-tenth one-f fteenth one
twent eth and one twenty th o the diameter o the Moon takes it. It s probab e that its height s to be taken
The years o a yuga are equa to the number o revolutions o the as the same
Sun in a yuga. I Paramesvara s correct in interpreting samdrka-
As po nted out by Bliau Da ^ Brahmagupta XI samdh as yiigasanid yugdrkabhaganasamd, the nom
15 16 seems to quote rom th s stanza in h s criticism native plura sa?ndh has been contracted after sandhi.
o the diameter o the Earth given by Aryabhata 6. The greatest declination o the ec pt c s 24 degrees. The
godasagav yojana parldhim pratibhuvyasarh pulavadata |
greatest den at on o the M oon rom the ec pt c s 41 degrees o
fitmajnanara khyap tam aniscayas tanik takanyat J
Saturn 2 degrees, o Jupiter 1 degree o Mars degrees o
bhuvyasasya nanad vyartharh desantaram tadajhanat |
Mercury and Venus 2 degrees. N nety s x ahgulas or i hastas
make 1 nr.
sphutatithyanta uanarii tithinasad grahanayor nasab. 1
1
The text o Brahmagupta s corrupt and must be Paramesvara explains the words hhdpakrarno
emended See the commentary o Sudhakara who grahdmsdh as o ows grahdndm bha amsds catur-
suggests or the first stanza vimsatibhdgd apakramah. paramdpakrama ity arthah.
The construct on s as strange as that o stanza 4
nr§iyojanabhuparidhirh prati bhuyasarh punar fli a vadata |
atmajnanarh khyap tam aniscayas tatk tavyasab above *
7. The ascending nodes o Mercury Venus Mars Jupiter
Lalla {Madhyamadhikara, 56 and Candragrahand- and Saturn hav ng moved (are situated at 20 60, 40 80 and
dhikdra, 6 gives the same diameters or the Earth 100 degrees rom the beginning o Me§a The apsides o the Sun
and the Sun but gives 320 as the diameter o the and o the above ment oned planets (in the same order (are
M o o n and {Grahayutyadhikdra, 2 gives or the situated at 78, 210 90, 118 180 and 236 degrees rom the
planets the same fractions o the diameter o the beginning o Mc§a
Moon.2 I have o owed Paramesvara s explanation o
Alberun I, 168 quotes rom Brahmagupta gatvdrhsakari as uktdn elan evdrhsakdn mesadiio gatvd
Aryabhata s diameter o the Earth, and a confused vyavaslhitdh.
*JRAS, 1865 p 402 In view o IV 2 “ the Sun and the nodes o the
*C Suryasiddhanta, I, 59; IV 1 VII 13-14; Brahmagupta planets and o the M oon move constantly along the
X X I 32 Kharegat {py. cit., X IX 132 34 Suryasiddhiirila, *Cf. Suryasiddhanta, I, 68-70 and II 28 Brahmagupta IX 1
IX 15-16). and X X I
52
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S A YABHAT YA
THE TEN G T STANZAS 17
indicate a knowledge o the motion o the nodes and ec pt
c ” and o I, 2 which gives the number o revo apsides o the other jdancts too I Aryabhata had
utions o the node, o the M oon n a ywja, the word ntended to say merely that the nodes and apsides are
galva “ having gone” seems to imp y as Paramc situated at such-and-such places the word gated s
svara sa ^s a knowledge o the revolution o the nodes superfluous. In a text o such studied brev ty every
o the planets and to indicate that Aryabhata n word s used with a very definite purpose. It s true
tended merely to give their positions at the time h s that Aryabhata regarded the movement o the nodes
treatise was composed The orce o gatvd continues and apsides o the other planets as negligible or pur
nto the second line and indicates a knowledge o the poses o calculation, but Brahmagupta s criticism
revolutions o the apsides. seems to be capt ous and unjustified (see also Bra
Aryabhata gives figures or the revolutions o the hmagupta X I 6 7 and the commentary o Sudha-
apsis and node o the Moon Other siddhantas g ve kara to X I 8 Barth s cr t c sm^ s too severe.
figures or the revolutions o the nodes and apsides o Lalla {Spa^stddhikdra, 9 and 28 gives the same
a the planets. These seem to be based on theory pos t ons or the apsides o the Sun and ive planets
rather than on observat on since their mot on (except (see also Pancasiddhdnlikd, X V II 2
in the case o the Moon s so slow that it wou d take For the revo ut ons o the nodes and apsides see
several thousand years or them to move so far that Brahmagupta 1 19 21 and Suryasiddhdnta, 1 41 44
their mot on could easily be detected by ordinary and note to I 44
methods o observation.^ Aryabhata may have re 8 D v ded b y 41 the ep cyc es o the apsides o the M oon
frained rom giving igures or the revolutions o nodes the Sun, Mercury Venus Mars Jup ter and Saturn (in the first
and third quadrants are 7, 3, 7, 4, 14 7, 9 the ep cyc es o the
and apsides (except n the case o the Moon because con unct ons o Saturn Jupiter Mars Venus and Mercury (in
he distrusted the igures given n ear er books as the first and third quadrants are 9 16 53, 59, 31
based on theory rather than upon accurate observa 9 the ep cyc es o the apsides o the planets M ercury Venus tion
Brahmagupta X I 8 quoted above to stanza 2 Mars Jupiter and Saturn in the second and ourth quadrants remarks in
criticism o Aryabhata that in the Dasa- are 5 2 IS 8 13 the ep cyc es o the con unct ons o the planets
gltika the nodes are stationary while in the Aryd- Saturn Jupiter Mars Venus, and Mercury in the second and
ourth quadrants are S 15 51, 57, 29. The circum erence w thin
stasata they move Th s re ers to I 2 and IV 2 In wh ch the Earth w nd b ows s 3,375 yojanas.
the Dasagitikd only the revolutions o the nodes o
the Moon are g ven in the Arydstasata the nodes and The criticism o these stanzas made by Brahma
apsides are said explic t y to move along the ec ptic gupta II 33 and X I 18 21 s as po nted out by
In the present stanza the word gatvd seems clearly to *Op c 7 I 154
*Cf. Suryasiddhanta, pp 27-28
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THE T ON o r n STANZAS 19 20 ARYABHAT YA
Sudluikara not justifiable. For the dimensions of in a ootnote to the stanza and Ayyangar p 125 n.
Brahmagupta s epicycles sec II 34 39 po nt out that the text-reading or the sixteenth and
Lalla {SpaslCidhikara, 28 agrees c ose y with seventeenth s nes violates the meter. Th s however
stanza S and (Grahabhramana, 2 gives the same figure may be remedied easily without changing the va ues ^
or the Earth wind Compare also SurijaaiddhCinla, C Whoever knows this DasagUika Sutra wh ch dc.scribes II 34
37 and note, and PahcasiddhCinlikCi, XVII 1 3 the movements o the Earth and the planets in the sphere o the
10 The twenty our s nes reckoned n minutes o arc are asterisms passes through the paths o the planets and asterisms
225 224 222 219 215 210, 20 5 199 191 1S3 174 1G4 154 and goes to the higher Bralm an
143 131 119 IQG 93 79, 65, 51, 37, 22, 7 *C JRAS, 1910 pp 752 754 and lA , XX 228
In Indian mathematics the “ ha chord” takes
the place o our “ sine.” The s nes are g ven n minutes
o which the radius contains 3,438 at intervals o 225
minutes The numbers given here are in rea ty not
the values o the s nes themselves but the differences
between the s nes
Compare Suryasiddhanta II 15 27 and Lalla
(Spastadhikdra, 1 8 and Brahmagupta II 2 9
Bhaskara (Ganitddhydya, Spastadhikdra, Vdsandhhd-
^ya to 3 9 re ers to the Suryasiddhdnta and to
Aryabhata as furnishing a precedent or the use o
twenty our s nes ^
Krishnaswami Ayyangar furnishes a plausible
explanation o the discrepancy between certain o the
values given in the foregoing stanza and the values
as calculated by II 12^ Some o the discrepancies
may be due to bad readings o the manuscripts. Kern
^For d scussion o the stanza sec Barth, ibid., I 150 n and
JRAS, 1911 pp. 123-24.
XV (1923-24), 121 20
*Sec also Naraharajya “Note on the Hindu Table o S nes,”
ibid., pp 105-13 o “ Notes and Questions.”
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22 ARYAB l.VrIVA
4 One should always d v de tlie avarga by twice the (squat
root o the preceding varga. A ter subtract ng the square o t
quot ent rom the varga the quot ent will be the square root t
CHAPTER II the next place
GANITAPADA OR MATHEMATICS Count ng rom right to left, the odd places arc
1 Hav ng paid reverence to Brahman the Earth tlie M oon called varga and the even places are called avarga
Mercury Venus the Sun, Mars Jupiter Saturn and the aster- Accord ng to Paramesvara, the nearest square root sms
Aryabhata sets orth here [in this work the science wh ch to the number in the last odd place on the left is set
Is lionored at Kusumapurad down in a place apart, and after this are set down the
The translation here at Kusuraapura the revered successive quotients o the d v s on performed The
science” is possible. A t any rate, Aryabhata states number subtracted s the square o that figure in the the
schoo to which he belongs. Kusumapura may or root represented b " the quotient o the preceding may not
have been the place o his birth division. The divisor s the square o that part o the
2. The numbers eka [one], daso [ten], kata [hundred], sahasra root wluch has already been found I the ast sub
[thousand], ayula [ten thousand] niyuta [hundred thousand traction leaves no remainder the square root s exact.
prayuta [million], ko{i [ten million], arbuda [hundred millionj, and ^A ways” indicates that if the divisor s larger than
rr/u/a [thousand million are rom place to p ace each ten times
the number to be d v ded a zero s to be placed in the
the preceding.*
line or a blank space le t there). Sthdndntare Tn an
The names or classes o numbers are given on y other p ace” s equivalent to the pahkti ^T ne” of
to ten places, although I B describes a notat on the later books
which reaches at least to the e ghteenth place The Th s process seems to be substantially correct but
highest number actually used by Aryabhata himsel there are severa difficulties Sthdndntare may mean
runs to ten places simp y “ to another p ace ” that s to say, each
3 A square the area o a square and the product o two division performed gives another figure o the root
equa quantit es are called varga. The product o three equa
quantities and a solid wh ch has twe ve edges are called ghana.^ Nityam “ always” may merely indicate that such s
the regular way o performing the operation
*Translated by F eet JRAS, 1911 p 110 See Kern s Pre ace to
h s edition o the Brhal Sariikita, p 57 and BCMS, X V III (1927) 7 All the translators except Saradakanta Ganguly
* See JRAS, 1911 p. 116 IHQ, III 112 BCMS, X V II 1926), translate vargdd varge snddhe with what precedes I
193 For the quotation n Alberun 1 176) which d ers in the ast think he s correct in taking it with what ollows In
two names see the criticism in BCMS, X V II (1926), 71
that case the parallelism with the o ow ng rule s
*Read diddasasraji with Paramesvara For akra in the sense o
“ edge” see Colebrooke, Algebra, pp. 2 n and 2S0 n The trans at ons exact Otherwise the first ru e would g ve the opera
g ven by Rode and Kaye are naccurate
21
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GANITAPADA OR MATHEMATICS 23
24 ARYABHAT YA
t "U for the mr ya place and then that for the avarga
no example o the w'ork ng o the ru e accord ng t**
p ace wh e the second ru e would give rst the opera
his nterpretation. To what do the wôrds “ square”
t ons or the aghana places and then that or the
and “ non-square” o h s translation refer? The Words
ghina place However or purposes o description, t
o Aryabhata exactly it the method emp oyed n ate
na kcs no difference whether the operations are given
Ind an mathematics. A though Brahmagupta docs
n one or the other o these orders.
not g ve a rule or square root his method or cube
Paralle ism wdth ghariasya mulavargcna o the
following ru e seems to indicate that vargumulena s
root s that described below although the word ng of
not to be translated “ square root” but “ root o the
his rule s different rom that o Aryabhata s I a to
(preceding) varga ” see any similarity to the rule and method o Theon
I the root is to conta n more than twô figures the o Alexandria.
varga o vargamulena s to be interpreted as applying In the o owing example the sign ° indicates the
to a the preceding figures up to and including the varga places, and the sign indicates the avarga
varga place which s being worked wdth That s to places.
say the word mula would refer to the w^ho e o that 15129 root = 1
part o the root which had already been ound Square o the root 1
For discussion see Kaye,* Avadhesh Narayan Tw ce the root 2 0 5 2 = quot ent or next dig t o rôt)
S ngh * Saradakanta Ganguly.^ I cannot agree with 2 X 1 4
Ganguly's discussion o the words hhdgarh hared
evargdt. I see no reason to question the use oihhdgam 11
Square o the quot ent
harali with the ab at ve in the sense o “ d v de ”
Brahmagupta X II 7) in his descr pt on o the Tw ce the root 24 72 3 = quotient (or next d g t o root
process o extracting the cube root has chedo *ghandd 2 X 1 2 72
dvlCiyat, which means “ the d v sor o the second 09
aghana.” Square o the quot ent
Kaye® ns sts that this ru e and the next are per
fectly genera (i.e. algebraical) and app y to a
Square root is 1 2 3
arithmetica notations He offers no proo and gives
•See Colebrooke, op. cil., p. 2S0 n 5 One should d v de the second aghana by three times t
1907 pp 493 94 * JBORS, X ll , 7S. square o the cube root o the (preceding ghana. The square
*BCMS, X V III (1927), 124 Op. cit., 1908 p. 120 o the quot ent multiplied by three times the jmrva (that part
o the cube root already ound is to be subtracted rom the irst
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GANITAPADA Oil MATIIi'.MATICS 25
26 ARYABII.VTlVA
and tlic cul)c o the quotient o the above div s on s to In the ollowing c xamp e the s gn ° indicates the
btracted rom the ghtina. ghana places and the sign indicates the aghana The
translation given by Avadhesh Narayan places.
o — 0 —d
S u ^ d as a “ correct litera render ng" is inaccurate 1SG0SG7 root = 1
Tlicre s nothing n the Sanskrit which corresponds Cube o root 1
to “ after having subtracted the cube o the quo Three t mc s square o root 3 dS 2 = quot ent or ne t dig t o
tient) rom the ghana p ace" or to “ the quot ent 3 X P root
placed at the next place gives the root " The latter 26
thought, o course, does carry over into this ru e rom Square o quot ent multiplied 12 the
preceding ru e In the same article (p 132 the b y three times the purva
Sanskr t o the rule s inaccurately pr nted with 22X3X1 140
Cube o quot ent 8
trighanasya or trigunena ghanasya.^
Kaye remarks that this rule s g ven by Brahma Three times square o root 432 1328(3 = quot ent or ne t digit
gupta “ word or word " As a matter o act the 3X122 1296 o root
Sanskr t o the two rules s very different, although 326
t e content s e.xact y the same Square o quot ent multiplied 324
Counting rom right to left, the first ourth etc., b y three times the yurva
places are named ghana cub c the second, fifth, (32X3X12 27
Cube o quot ent 27
etc places are called the first aghana non cub c
places; and the third, sixth etc., places are called the 0.
second aghana non cub c places. The nearest cube Cube root s 1 2 3
root to the number n (or up to and including the ast 6 The area o a triangle s the product o the perpendicular
ghana place on the left s the rst figure o the cube and half the base. Ha the product o this area and the height s
root A ter t are p aced the quot ents o the succes the vo ume o a solid %vh ch has six edges pyram d
sive d 'is ons I the last subtract on leaves no I sarnadalakotl can denote, as Pa rame svara says
remainder the cube root s exact. a perpendicular which s common to two tr ang es the
» B CilS, X V III (1927), 134 rule re ers to a triang es. I sarnadalakotl re ers to a
* The ru e has been discussed in JBORS, X II 80 Cf Brah- perpendicular which bisects the base t re ers only to
n a^upta XII 7) and the trans ation and note o Colebrooke {op. isosceles tr ang es ^
cit., p 2S0 *For (lira or nsri n the sense o “edge sec note to stanza II 3
* Op d 190S p 119 See JBORS, XII S4 S5 for discussion o the naccurate va ue g ven
in the second part o the ru e
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2S A VA H AT YA
GAXITAPADA OR MATHEMAT C.:; 27
The very genera ru e given in the rst hal o t ^
7 H:ilf o the circum erence multi])!it.d b half tlic J a nc cr
stanza seems to mean as Paramc.svara exp a n s n
t .o area o a circle. Th s area riiultiplicd by its own square root
e exact vo ume o a sphere.* some detail, that the mathematician s to use h s in
S The tw o s c c s (.separately multiplied by the pcrpcnd cu genuity n determining two sides which wil rcprc ?ent
I and d v ded b y their sum wil g ve the perpendiculars rom the average length and the average breadth o the
nt where the two diagonals intersect to the paralle sides. figure The r product w ll be the area Methods t o
The area Is to be known b y mu t p ing half tlic sum o the be employed with various kinds o figures were doubt
t« s des by the perpendicular.
ess handed dow n by oral tradition
Rodet thinks that the ru e directs that the figure he
a Xc broken up into a number o trapeziums It s doubt u
a+6
bXc whether the words can bear that interpretation.
a+6 * 10 Add 4 to 100 mu t p y b y 8, and add 62,000. The
c o+6 result s appro.ximately the circum erence o a circ e o wh ch the
Area = diameter is 20,000.
The circumference s 62,832 The diameter s
The ru e applies to any four-sided plane figure o 20 000
which two sides are parallel, i.e. trapezium. The
By this ru e the relation o circumference to
word translated “ sides” re ers to the two parallel
diameter s 3 1416 ^
s des The perpendicular s the perpendicular be
Bha skara Goladhydya, Bhuvanakosa (stanza 52),
tween the two paralle s des
Vasandbhdsya, re ers to this ru e o Aryabhata
11 One should d v de a quarter o the circum erence o
In the example given above a and 6 are the circ e nto as many equa parts as are desired From the tri
para e s des c s the perpendicular between them angles and quadrilaterals wh ch are ormed one will have on the
and d and e are the perpendiculars rom the po nt radius as many sines o equa arcs as are desired.*
o intersection o the tw
^o diagonals to the sides a and The exact method o working out the table s not
b, respectively. know n It s uncertain what s intended by the
9. The area o any plane figure s found by determ ning two triangle and the quadrilateral constructed rom each
s des and then multiply ng them together. po nt marked on the quadrant ®
The chord o the sixth part o the circum erence s equa to
■
»See JBORS, XII 82; JRAS, 1910 pp 752 754
rhe n d Ls
* See the table given n I, 10 o the d Tcrenccs between the s nes
•See ibid, and Bibl. math., IX , 19G for d scussion o the nac Twenty-four s nes taken at intervals o 225 minutes o arc are regu
n n e va ue given n the second part o the ru e For a poss ble arly given in the Indian tab es
r cn ncc to this pasSc ge by Bhaskara Goladhydya, Bhucanakosa, * Note the methods suggested by Kaye and Rodet and cf JIMS,
• C (Vdsandbhd^tja) (not stanza 52 as stated), see BCMS, X V (1923-24) 122 and 108-9 o “ Notes and Questions,"
W 1927), 10
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30 ARYAB AT YA
GAXITAPADA OR MATHEMATICS 2 0 How Kaye gets “ I the first and second be bisected
12 Ity whiit umbc t!ic second sine s ess than t o
n succession the s ne o the half chord s obtained*’
and by the quot ent obta ned by div d ng the sum o the s a puzzle to me It s impossible as a translation
*
< I ding -incs by the irst sine by the sum o these two quant o the Sanskr t.
ty th e o ow ng sines are c ss than the r t sine 13 The circle s made by turning and the tr ang e and the
The hint phrase may be translated “ the sinc quadrilatera by means o a kan^a; the horizonta s dcteriu ned
d Terc ces are c ss than the first sine.’ * by water, and the perpend cular b y the p u nb nc
This rule describes how the table o s nc d Tcr Tribhuja denotes triangle in general and calur-
ences given in I 10 may be calculated rom the rst bhiija denotes quadrilateral in general The word
t>nc (225 The irst sine means always th s rst s ne kania regularly denotes the hypotenuse o a right-
2*25 The second sine means any particular sine with angle triangle and the diagonal o a square or rec
wh ch one is working in order to calculate the o ow tangle I am not sure whether the restricted sense o ng s
ne harm limits tribhuja and calurhhuja to the right-angle
Subtract 225 rom 225 and the remainder s 0 D triangle and to the square and rectangle or whether v de
225 by 225 and the quot ent s 1 The sum o 0 and the general sense o tribhuja and caturbhuja general
1 s subtracted rom 225 to obta n the second s ne 224 izes the meaning o karna to that o one chosen side o
Subtract 224 rom 225 and the remainder s 1 a triangle and to that o the diagonal o any quadri
Divide 225 plus 224 by 225 and the nearest quotient
s 2 Add 2 and 1 and subtract rom 225 The third
lateral. At any rate, the context shows that the
s ne w be 222. Proceed in like manner or the o rule deals with the actual construct on o plane
owing s nes gures
I this method s o owed str ct y there resu ts Paramesvara interprets it as referring to the con
severa slight divergences rom the values given in I, struction o a triangle o which the three sides are
10 It s possible to reconcile most o these by assum known and o a quadrilateral o which the our s des
ng as ICr s maswam Ayyangar does, that rom time and one diagona are known. One side o the triangle
to time the neglected ract ons were distributed s taken as the karna. Two sticks o the length o the
among the sines But o this there is no indication n other two s des one touching one end and the other the ru
e as given the other end o the karna, are brought to such a posi
*For discussion o the Indian s nes see the notes o Rodet and tion that their tips o n The quadrilatera s made
Kaye; Pancasiddhanlikd, chap v Suryasiddhdnta, II, 15-27 La a by constructing two triangles, one on each side o the
p 12 Brahmagupta, II 2 10 JRAS, 1910 pp. 752 754 lA, X X
-V'' Brennand Hindu Astronomy, pp. 210 13 JIMS, X V (1923-24), diagonal.
121 2* with attempted explanation o the variation o severa o t
va ues given in the table from the va ues calculated by means o tr
ru e and ibid., pp 105 13 o “Notes and Questions ”
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32 ARYAB ATiVA
GAN TAPADA OR MAT KMAT CS 31
Because o the usc o the word kotl n the ollowing
The circle s made by the turning o the karkuld
rule Rodet s inclined to think that the gnomon and
coinpa.'.'d
the bhujd were not perpendicular but pro ected hori
11 A'Kl the square o the height o the gnomon to the square
zonta y rom a wall. Bhujd denotes any side o a
* >hadov T e square root o this sum s the radius o the
i.h'.ir'Jn. triangle but kotl usually refers to an upright. It s
The text reads khavrtta “ sky c rc e” Para possible however or kotl to denote any perpendicu r
csvara reads svavrlla “ its c rc e” I do not know ar to the bhujd whether horizontal or upright.
wh ch s correct.
Kaye remarks that in order “ to mark out the
hour angles on an ord nary sun-dial, it s necessary to
h*scr be two circles, one o which has its radius equal
to the vertical gnomon and the other w th radius
c< ua to the hypotenuse o the triangle ormed by the
equinoctia shadow^ and the gnomon ” It may be that
th s second circle s the one referred to here Para-
ncsvara has chdy&gramadhyam sankusirahprdpi yan BA s the bhujd which holds the ight,
mandalam urdhvddkahsthitam tat svavrttam ity iccyate, DE s the gnomon,
“ the circle which has its centre at the extrem ty o DC =
D E X B D
t c shadow and which touches the top o the gnomon AF
Is called the svavrtta” As Rodet remarks, it s d 16. The d stance between the ends o the two shadows mu t
cu t to see or what purpose such a circle cou d serve p ed by the length o the shadow and d v ded by the difference
15 Mu t p y the ength o the gnomon by the distance be
in ength o the two shadows gives the koHi. The kotl mu t p ed
tv.een the gnomon and the bhujd and d v de by the d Terence
b y the length o the gnomon and d v ded by the length o the
« tveen the length o the gnomon and the length o the bhujd. shadow gives the ength o the bhujd.
Tiie quotient will be the ength o the shadow measured from The literal translation o chdydgunitam chdyd- the
base o the gnomon.® gravivaram unena bhdjiid kotl seems to be “ The dis For
para e s to the stanza see La a {YanlrddhyCuja, 2 and tance between the ends o the two shadows multiplied
H ah uagupta XXII 7 See BCMS, XVIII (1927), 6S G9 which s by the length o the shadow s equal to the kotl
tvo emphatic n ts assertion that kari,ia must mean “diagonal” and
d v ded by the difference in length o the two shad r
ut "h^ potenuse ”
* See Brahmagupta X II 53; Colebrooke op. cit., p 317 ows ” Th s is equivalent to the translation g ven
Brvnnand op. cit., p. 166
above
II
I II I
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GAN TAPADA OR MATHEMAT CS 33 34 ARYA R AT VA
Or the rst pos tion o the gnomon may be C'D'
and the second CD. To find AE' and AB.^
17 The square o the bhujd plus the square o tlic kofi s the
square o the karna.
In a circle the product o two saras s the square o the ha
chord o the two arcs
The bhujd and kotl are the s des o a right-angle
triangle The I arna s the hypotenuse.
AB s the bhujd,
AE s the kotl, The saras or “ arrows” are the segments o a
CD s the gnomon n ts rst position, diameter which b sects any chord
C'D' s the gnomon n ts second position
CE and C'E' are the rst and second shadows
CEXEE'
C 'E '-C E '
AB = AE X C D a X b =
c?
CE * where c s the half-chord.
The length o the hhuja which holds the light and
the distance between the end o the shadow and the
base o the bhujd are unknown. In order to find them
the gnomon s p aced n another pos t on so as to give 18 The diameters o two circles (separate y minus the
a second shadow. grdsa, multiplied by the grdsa, and div ded separate y by the sum o
the diameters o the two circ es after the gra-^a has been sub
The length o the shadow s its length when the tracted rom each will g ve respect vely the sarnpdtasaras o the
gnomon is in its first position The kotl s the d s two circles.
tance between the end o the shadow when the gno T en two circles intersect the word grdsa “ the
mon is in its irst pos t on and the base o the bhujd. b te” denotes that part o the common diameter o
The word kotl means perpendicular or upright) the two circles which s cut off by the intersecting
and the rule might be interpreted, as Rodet takes t chords o the two circ es
as meaning that the bhujd and the gnomon extend
S e e Brahmagupta X II 54; Colcbrooke, op. oil., p. 318; Bren
hor zonta y rom a perpendicular wall But the nand op. dt p. 166
words bhujd and kotl also refer to the sides o a right- » C Brahmagupta, X II 41 See BCMS, X V III (1927) 11 71,
angle triangle w thout much regard as to which s mth d scussion o the quotation given by Colebrooke op oil., p. 309
horizonta and which s upright. from PrtUudakasvam s commentary to Brahmagupta
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GAN TAPADxV OR MATIIRMATICS 35
36 ARYABHAT YA
The second part o the ru e applies on y to the sum
AB 13 the grfUa, o the whole progrcs.s on beginning with the first
AE and BE arc the term
$ampdlaiaras.
AS=nj^a-h^~-i-hp^cfj
AE A B {d -A D ) EB =
A B (D -A B ) {a+l)n
^ D + d - 2 A B ’ — D + d - 2 A B '
where d and D are the diameters o the two c rc es
As Paramesvara says, samiikhamadhyam must be
The sampatasaras are the two distances (within taken as equivalent to samukham madhyam.
the grdsa), on the common diameter, rom the cir "WTether Paramesvara s correct in h s statement
cumferences o the two circles to the po nt o inter bahusiitrdrthapradarsakam etat sutram. ato bahudhd
section o this common diameter with the chord con yojand kdryd and subsequent exposition seems very
necting the two po nts where the circumferences doubt u
ntersect ^ Brahmagupta X II 17 has on y the second part o
19 The desired number o terms minus one ha ved plus the the ru e ^
number o terms wh ch precedes, multiplied by the common
difference between the terms, plus the first term s the middle
20 M u t p y the sum o the progression by eight times t ;e
term Th s multiplied by the number o terms desired s the sum
common difference add the square o the difference between twice
o the desired number o terms. the first term and the common difference take the square root o
Gr the sum o the irst and last terms s multiplied b y half the
this subtract twice the first term d v de by the common differ
number o terms. ence add one d v de by two The result will be the number o
terms.
Th s rule te ls how to find the sum o any desired V U S + {d -2 aY -2 a ,1
number o terms taken anywhere within an arith ■ — 5 ■
■ •
metica progression. Let n be the number o terms
extending rom the p th to the (pH n)th terms As Rodet says, the development o this ormula in an
arithmetica progression et d be the common rom the one n the preceding ru e seems to indicate difference
between the terms let a be the rst term knowledge o the solution o quadratic equations n o the
progression, and I the last term the orm ax^-]rbx--c= 0.^
^C Brahmagupta, XII 43 Colebrooke, op. cit., p. 311 * Cf. Colebrooke op. oil., p. 290
*See Brahmagupta, XII 18 Colebrooke, op. oil., p. 291
• » Sec a so JA (1878) I, 28 77 and JBORS, XII 86-87
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Walter Eugene Clark - The Aryabhatiya of Aryabhata_ An Ancient Indian Work on Mathematics and Astronomy-University of Chigago Press (1930).pptx

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