Skip to main content

Full text of "The_Aryabhatiya_of_Aryabhata_Clark_1930"

See other formats

The Aryabhatiya of Aryabhata 

r ^ i I V* 1 1 1 M I U I i ■ I I 1 1 1 P^ 1 1 K \ 'i 1 1 1 

on Mathematic and Astronomy 

English Translation by W. E, Clark 




An Ancient Indian Work on 
Mathematics and Astronomy 



Professor of Sanskrit in Harvard University 







A.ZjXi RIGMT& XIX:S£1KVX3X>. FXTBI^ISBX3£> J'CTlj'Z' 1930 



In 1874 Kern published at Leiden a text called the 
Aryabhatiya which claims to be the work of Arya- 
bhata, and which gives (III, 10) the date of the birth 
of the author as 476 a.d. If these claims can be sub- 
stantiated, and if the whole work is genuine, the 
text is the earliest preserved Indian mathematical and 
astronomical text bearing the name of an individual 
author, the earliest Indian text to deal specifically 
with mathematics, and the earliest preserved astro- 
nomical text from the third or scientific period of 
Indian astronomy. The only other text which might 
dispute this last claim is the Suryasiddhdnta (trans- 
lated with elaborate notes by Burgess and ^Tiitney 
in the sixth volume of the Journal of the American 
Oriental Society) . The old Sitryasiddhdnta undoubt- 
edly preceded Aryabhata, but the abstracts from it 
given early in the sixth century by Varahamihira in 
his Pancasiddhdntikd show that the preserved text 
has undergone considerable revision and may be later 
than Aryabhata. Of the old Paidisa and Romaka 
Siddhdntas, and of the transitional Vdsistha Si- 
ddhdnta, nothing has been preserved except the short 
abstracts given by Varahamihira. ' The names of sev- 
eral astronomers who preceded Aryabhata, or who 
were his contemporaries, are known, but nothing has 
been preserved from their writings except a few brief 

The Aryabhatiya, therefore, is of the greatest im- 


portance in the history of Indian mathematics and 
astronomy. The second section, which deals with 
mathematics (the Ganitapadd), has been translated 
by Rodet in the Journal asiatique (1879), I, 393- 
434, and by Kaye in the Journal of the Asiatic Society 
of Bengal, 1908, pages 111-41. Of the rest of the work 
no translation has appeared, and only a few of the 
stan2as have been discussed. The aim of this work is 
to give a complete translation of the Aryahhatlya with 
references to some of the most important parallel 
passages which may be of assistance for fm-ther study. 
The edition of Kern m^kes no pretense of giving a 
really critical text of the Aryabhatiya. It gives merely 
the text which the sixteenth-century commentator 
Paramesvara had before him. There are several un- 
certainties about this text. Especially noteworthy is 
the considerable gap after IV, 44, which is discussed 
by Kern (pp. v-vi). The names of other commenta- 
tors have been noticed by Bibhutibhusan Datta in 
the Bulletin of the Calcutta Mathematical Society, 
XVIII (1927), 12. All available manuscripts of the 
text should be consulted, all the other commentators 
should be studied, and a careful comparison of the 
Aryahhatlya with the abstracts from the old si- 
ddhantas given by Varahamihira, with the Suryasi- 
ddhdnta, with the Sisyadhlvrddhida of Lalla, and with 
the Brahmasphutasiddhdnta and the Khandakhddyaka 
of Brahmagupta should be magle. All the later quota- 
tions from Aryabhata, especially those made by the 
commentators on Brahmagupta and Bhaskara, should 
be collected and verified. Some of those noted by 
Colebrooke do not seem to fit the pubKshed Arya- 


bhatlya. If so, were they based on a_lost work of 
Aryabhata, on the work of another Aryabhata, or 
were they based on later texts composed by followers 
of Aryabhata rather than on a work by Aryabhata 
himself? Especially valuable would be a careful study 
of Prthudakasvamin or Caturvedacarya, the eleventh- 
century commentator on Brahmagupta, who, to judge 
from Sudhakara's use of him in his edition of the 
Brdhmasphutasiddhdnta, frequently disagrees with 
Brahmagupta and upholds Aryabhata against Brah- 
magupta's criticisms. 

The present translation, with its brief notes, 
makes no pretense at completeness. It is a prelimi- 
nary study based on inadequate material. Of several 
passages no translation has been given or only a ten- 
tative translation has been suggested. A year's work 
in India with unpublished manuscript material and 
the help of competent pundits would be required for 
the production of an adequate translation. I have 
thought it better to pubhsh the material as it is rather 
than to postpone publication for an indefinite period. 
The present translation will have served its purpose 
if it succeeds in attracting the attention of Indian 
scholars to the problem, arousing criticism, and en- 
couraging them to make available more adequate 
manuscript material. 

There has been much discussion as to whether the 
name of the author should be spelled Aryabhata or 
Aryabhatta.^ Bhata means "hirehng," "mercenary," 

^ See especially Journal of the Royal Asiatic Society, 1865, pp. 
392-93; Journal asiatique (1880), II, 473-81; Sudhakara Dvivedi, 
Gav^atarangiv>'i, p. 2. 


"warrior," and hhatta means "learned man," "schol- 
ar." Aryabhatta is the spelling which would natural- 
ly be expected. However, all the metrical evidence 
seems to favor the spelling with one t It is claimed 
by some that the metrical evidence is inclusive, that 
hhata has been substituted for bhatta for purely 
metrical reasons, and does not prove that Arya- 
bhata is the correct spelling. It is pointed out that 
Kern gives the name of the commentator whom he 
edited as Paramadisvara. The name occurs in this 
form in a stanza at the beginning of the text and 
in another at the end, but in the prose colophons at 
the ends of the first three sections the name is given 
as Paramesvara, and this doubtless is the correct form. 
However, until more definite historical or n^etrical 
evidence favoring the spelling Aryabhatta is produced 
I prefer to keep the form Aryabh^ta. 

The Aryahhatvya is divided into four sections 
which contain in all only 123 stanzas. It is not a com- 
plete and detailed working manual of mathematics 
and astronomy. It seems rather to be a brief descrip- 
tive work intended to supplement matters and proc- 
esses which were generally known and agreed upon, 
to give only the most distinctive features of Arya- 
bhata's own system. Many commonplaces and many 
simple processes are taken for granted. For instance, 
there are no rules to indicate the method of calculat- 
ing the ahargana and of finding the mean places of the 
planets. But rules are given for calculating the true 
places from the mean places by applying certain cor- 
rections, although even here there is no statement of 


the method by which the corrections themselves are 
to be calculated. It is a descriptive summary rather 
than a full working manual like the later karana- 
granthas or the Suryasiddhdnta in its present form. 
It is questionable whether Aryabhata himself com- 
posed another treatise, a haraiiagrantha which might 
serve directly as a basis for practical calculation, or 
whether his methods were confined to oral tradition 
handed down in a school. 

Brahmagupta^ implies knowledge of two works by 
Aryabhata, one giving three hundred sdvana days in 
a yuga more than the other, one beginning the yuga 
at sunrise, the other at midnight. He does not seem 
to treat these as works of two different Aryabhatas. 
This is corroborated by Paficasiddhantika, XV, 20: 
"Aryabhata maintains that the beginning of the day 
is to be reckoned from midnight at Lanka; and the 
same teacher [sa eva] again says that the day begins 
from sunrise at Lanka." Brahmagupta, however, 
names only the Dasagltika and the Arydstasata as the 
works of Aryabhata, and these constitute our Arya- 
hhatlya. But the word audayikatantra of Brahma- 
sphutasiddhdnta, XI, 21 and the words audayika and 
drdhardtrika of XI, 13-14 seem to imply that Brahma- 
gupta is distinguishing between two works of one 
Aryabhata. The pubhshed Aryabhatiya (I, 2) begins 
the yuga at sunrise. The other work may not have 
been named or criticized by Brahmagupta because of 
the fact that it followed orthodox tradition. 

Alberuni refers to two Aryabhatas. His later 

1 Brahmasphutasiddhdnta, XI, 5 and 13-14. 


Aryabhata (of Kusumapura) cannot be the later 
Aryabhata who was the author of the Mahdsiddhdnta. 
The many quotations given by Alberuni prove con- 
clusively that his second Aryabhata was identical 
with the author of our Aryabhatlya (of Kusumapura 
as stated at II, 1). Either there was a still earlier 
Aryabhata or Alberuni mistakenly treats the author 
of our Aryabhatlya as two persons. If this author 
really composed two works which represented two 
slightly different points of view it is easy to explain 
Alberuni's mistake.^ 

The pubUshed text begins with 13 stanzas, 10 of ,. 
which give in a peculiar alphabetical notation and in 
a very condensed form the most important numerical 
elements of Aryabhata's system of astronomy. In 
ordinary language or in numerical words the material 
would have occupied at least four times as many 
stanzas. This section is named Dasagitikasutra in the 
concluding stanza of the section. This final stanza, 
which is a sort of colophon; the first stanza, which is 
an invocation and which states the name of the 
author; and a paribhdsd stanza, which explains the 
peculiar alphabetical notation which is to be em- 
ployed in the following 10 stanzas, are not counted. 
I see nothing suspicious in the discrepancy as Kaye 
does. There is no more reason for questioning the 
authenticity of the panhhdsa stanza than for ques- 
tioning that of the invocation and colophon. Kaye 

1 For a discussion of the whole problem of the two or three Arya- 
bhatas see Kaye, BiUiotheca mathematica, X, 289, and Bibhutibhusan 
Datta, BvMetin of the Calcutta Mathematical Society, XVII (1926), 59. 


would like to eliminate it since it seems to furnisli 
evidence for Aryabhata's knowledge of place-value. 
Nothing is gained by doing so since Lalla gives in 
numerical words the most important numerical ele- 
ments of Aryabhata without change, and even with- 
out this parihhdsd stanza the rationale of the alpha- 
betical notation in general could be worked out and 
just as satisfactory evidence of place-value furnished. 
Further, Brahmagupta {Brdhmasphutasiddhdnta, XI, 
8) names the Dasagitika as the work of Aryabhata, 
gives direct quotations (XI, 5; I, 12 and XI, 4; XI, 
17) of stanzas 1, 3, and 4 of our Dasagitika, and XI, 
15 (although corrupt) almost certainly contains a 
quotation of stanza 5 of our Dasagitika. Other stanzas 
are clearly referred to but without direct quotations. 
Most of the Dasagitika as we have it can be proved 
to be earlier than Brahmagupta (628 a.d.). 

The second section in 33 stanzas deals with 
mathematics. The third section in 25 stanzas is 
called Kdlakriyd, or "The Reckoning of Time." The 
fourth section in 50 stanzas is called Gola, or "The 
Sphere." Together they contain 108 stanzas. 

The Brdhmasputasiddhdnta of Brahmagupta was 
composed in 628 a.d., just 129 years after the Arya- 
hhatiya, if we accept 499 a.d., the date given in III, 
10, as being actually the date of composition of that 
work. The eleventh chapter of the Brdhmasphuta- 
siddhdnta, which is ^lled "Tantrapanksa," and is 
devoted to severe criticism of previous works on 
astronomy, is chiefly devoted to criticism of Arya- 
bhata. In this chapter, and in other parts of his work, 


Brahmagupta refers to Aryabhata some sixty times. 
Most of these passages contain very general criticism 
of Aryabhata as departing from smrti or being igno- 
rant of astronomy, but for some 30 stanzas it can be 
shown that the identical stanzas or stanzas of iden- 
tical content were known to Brahmagupta and 
ascribed to Aryabhata. In XI, 8 Brahmagupta names 
the Arydstasata as the work of Aryabhata, and XI, 
43, jandty ekam api yato ndryabhato ganitakdlago- 
landm, seems to refer to the three sections of our 
Arydstasata. These three sections contain exactly 
108 stanzas. No stanza from the section on mathe- 
matics has been quoted or criticized by Brahma- 
gupta, but it is hazardous to deduce from that, as 
Kaye does,^ that this section on mathematics is 
spurious and is a much later addition.^ To satisfy the 
conditions demanded by Brahmagupta's name Aryd- 
stasata there must have been in the work of Arya- 
bhata known to him exactly 33 other stanzas forming 
a more primitive and less developed mathematics, or 
these 33 other stanzas must have been astronomical 
in character, either forming a separate chapter or 
scattered through the present third and fourth sec- 
tions. This seems to be most unlikely. I doubt the 
validity of Kaye's contention that the Ganitapdda was 
later than Brahmagupta. His suggestion that it is by 
the later Aryabhata who was the author of the 
MahdsiddJidnta (published in the "Benares Sanskrit 

1 Op. dt., X, 291-92. 

*For criticism of Kaye see Bibhutibhusan Datta, op. dt., 
XVIII (1927), 5. 


Series" and to be ascribed to the tenth century or 
even later) is impossible, as a comparison of the two 
texts would have shown. 

I feel justified in assuming that the Aryahhatlya 
on the whole is genuine. It is, of course, possible that 
at a later period some few stanzas may have been 
changed in wording or even supplanted by other 
stanzas. Noteworthy is I, 4, of which the true reading 
hhul],, as preserved in a quotation of Brahmagupta, 
has been changed by Paramesvara or by some pre- 
ceding commentator to hham in order to eliminate 
Aryabhata's theory of the rotation of the Earth. 

Brahmagupta criticizes some astronomical mat- 
ters in which Aryabhata is wrong or in regard to which 
Aryabhata's method differs from his own, but his 
bitterest and most frequent criticisms are directed 
against points in which Aryabhata was an innovator 
and differed from smrti or tradition. Such criticism 
would not arise in regard to mathematical matters 
which had nothing to do with theological tradition. 
The silence of Brahmagupta here may merely indicate 
that he found nothing to criticize or thought criticism 
unnecessary. Noteworthy is the fact that Brahma- 
gupta does not give rules for the volume of a pyramid 
and for the volume of a sphere, which are both given 
incorrectly by Aryabhata (II, 6-7) . This is as likely 
to prove ignorance of the true values on Brahma- 
gupta's part as laten^s of the rules of Aryabhata. 
What other rules of the Ganitapdda could be open to 
adverse criticism? On the positive side may be 
pointed out the very close correspondence in termi- 


nology and expression between the fuller text of 
Brahmagupta, XVIII, 3-5 and the more enigmatical 
text of Aryabhatiya, II, 32-33, in their statements of 
the famous Indian method (kuttaka) of solving inde- 
terminate equations of the first degree. It seems prob- 
able to me that Brahmagupta had before him these 
two stanzas in their present form. It must be left to 
the mathematicians to decide which of the two rules 
is earlier. 

The only serious internal discrepancy which I have 
been able to discover in the Aryabhatiya is the follow- 
ing. Indian astronomy, in general, maintains that the 
Earth is stationary and that the heavenly bodies 
revolve about it, but there is evidence in the Arya- 
bhatiya itself and in the accounts of Aryabhata given 
by later writers to prove that Aryabhata maintained 
that the Earth, which is situated in the center of 
space, revolves on its axis, and that the asterisms are 
stationary. Later writers attack him bitterly on this 
point. Even most of his own followers, notably Lalla, 
refused to follow him in this matter and reverted to 
the common Indian tradition. Stanza IV, 9, in spite 
of Paramesvara, must be interpreted as maintaining 
that the asterisms are stationary and that the Earth 
revolves. And yet the very next stanza (IV, 10) seems 
to describe a stationary Earth around which the 
asterisms revolve. Quotations by Bhattotpala, the VS- 
sanavarttika, and the Marici iryiicate that this stanza 
was known in its present form from the eleventh cen- 
tury on. Is it capable of some different interpreta- 
tion? Is it intended merely as a statement of the 


popular \dew? Has its wording been changed as has 
been done with I, 4? I see at present no satisfactory 
solution of the problem. 

Colebrooke^ gives caturvimsaty amsais coikram 
uhhayato gacchet as a quotation by Munisvara from 
the Arydstasata of Aryabhata. This would indicate a 
knowledge of a Hbration of the equinoxes. No such 
statement is found in our Arydstasata. The quotation 
should be verified in the unpublished text in order 
to determine whether Colebrooke was mistaken or 
whether we are faced by a real discrepancy. The 
words are not found in the part of the Marici which 
has already been published in the Pandit. 

The following problem also needs elucidation. Al- 
though Brahmagupta (XI, 43-44) 

janaty ekam api yato narj'abhato ganitakalagolanam | 
na maya proktani tatah ppthak prthag duganany e§am 1| 
aryabhatadu?ananairi samkhya vaktmh na sakyate yasmat | 
tasmad ayam uddeso buddliimatanyani yojyani || 

sums up his criticism of Aryabhata in the severest 
possible way, yet at the beginning of his Khanda- 
khddyaka, a karanagrantha which has recently been 
edited by Babua Misra Jyotishacharyya (University 
of Calcutta, 1925), we find the statement vaksydmi 
khandakhddyakam dcdrydryahhatatulyaphalam. It is 
curious that Brahmagupta in his Khandakhddyaka 
should use such respectful language and should foUow 
the authority of an author who was damned so un- 
mercifully by him in tlfe Tantraparlksd of his Brdhma- 
sphutasiddhdnta. Moreover, the elements of the Khan- 

^ Miscellaneous Essays, II, 378. 



dakhadyaha seem to differ much from those of the 
Aryabhatlya} Is this to be taken as an indication 
that Brahmagupta here is following an older and a dif- 
fereirt Aryabhata? If so the Brdhmasphutasiddhdnta 
gives no clear indication of the fact. Or is he fol- 
lowing another work by the same Aryabhata? Ac- 
cording to Diksit,^ the Khandakhddyaka agrees in all 
essentials with the old form of the Suryasiddhdnta 
rather than with the Brdhmasphutasiddhdnta. Just as 
Brahmagupta composed two different works so 
Aryabhata may have composed two works which 
represented two different points of view. The second 
work may have been cast in a traditional mold, may 
have been based on the old Siiryasiddhdnta, or have 
formed a commentary upon it. 

The Mahdsiddhdnta of another Aryabhata who 
lived in the tenth century or later declares (XIII, 14) : 

vrddharyabliataproktat siddhantad yan mahakalat ] 
patiiair gatam ucchedam vise^itam tan maya svoktya | j 

But this Mahdsiddhdnta differs in so many particulars 
from the Aryabhatlya that it is difficult to believe that 
the author of the Aryabhatlya can be the one referred 
to as Vrddharyabhata unless he had composed an- 
other work which differed in many particulars from 
the Aryabhatlya. The matter needs careful investiga- 

1 Cf. PancasiddhSntikd, p. xx, and BvUetin of tike Calcutta Maihe- 
matical Society, XVII (1926), 69. 

* As reported by THbaut, Astronor^e, Astrologie und Mathematik, 
pp. 55, 59. 

' See Bulletin of the Calcutta Mathematical Society, XVII (1926), 
66-67, for a brief discussion. 


This monograph is based upon work done with me 
at the University of Chicago some five years ago by 
Baidyanath Sastri for the degree of A.M. So much 
additional material has been added, so many changes 
have been made, and so many of the views expressed 
would be unacceptable to him that I have not felt 
justified in placing his name, too, upon the title-page 
as joint-author and thereby making him responsible 
for many things of which he might not approve, 

Hahvaed Univeesitt 
April, 1929 

While reading the final page-proof I learned of 
the publication by Prabodh Chandra Sengupta of a 
translation of the Aryabhatiya in the Journal of the 
Department of Letters (Calcutta University), XVI 
(1927). Unfortunately it has not been possible to 
make use of it in the present publication. 

April, 1930 


List of Abbreviations xxvii 

I. Dasagitika oe the Ten Giti Stanzas .... 1 

A. Invocation 1 

B. System of Expressing Numbers by Letters of 
Alphabet 2 

L Revolutions of Sun, Moon, Earth, and Planets 
ina^t^a 9 

2. Revolutions of Apsis of Moon, Conjunctions of 
Planets, and Node of Moon in a yuga; Time 
and Place from Which Revolutions Are To 

Be Calculated 9 

3. Number of Manus in a kalpa; Number of yugas 
in Period of a Manu; Part of halpa Elapsed up 

to Bharata Battle 12 

4. Divisions of Circle; Circumference of Sky and 
Orbits of Planets in yojanas; Earth Moves One 
kald in a prat^a; Orbit of Sun One-sixtieth 
That of Asterisms 13 

5. Length of yojana; Diameters of Earth, Sun, 
Moon, Meru, and Planets; Number of Years 

in a yuga 15 

6. Greatest Declination of Ecliptic; Greatest 
Deviation of Moon and Planets from Ecliptic; 
Measure of&nr 16 

7. Positions of Ascending Nodes of Planets, and 

of Apsides of Sun and Planets 16 

8-9. Dimensions of Epicycles of Apsides and Con- 
junctions of Planets; Circumference of Earth- 
Wind 18 

10. Table of Sine-Dif erences 19 

0. Colophon 20 



II. Ganitapada or Mathematics 21 

1. Invocation 21 

2. Names and Values of Classes of Numbers Increas- 
ing by Powers of Ten 21 

3. Definitions of Square (varga) and Cube (ghana) . 21 

4. Square Root 22 

5. Cube Root 24 

6. Area of Triangle; Volume of Pyramid ... 26 

7. Area of Circle; Volume of Sphere 27 

8. Area of Trapezium; Length of Perpendiculars from 
Intersection of Diagonals to Parallel Sides . . 27 

9. Area of Any Plane Figure; Chord of One-sixth Cir- 
cumference Equal to Radius 27 

10. Relation of Circumference of Circle to Diameter 28 

11. Method of Constructing Sines by Forming Tri- 
angles and Quadrilaterals in Quadrant of Circle . 28 

12. Calculation of Table of Sine-Differences from 
First One 29 

13. Construction of Circles, Triangles, and Quadri- 
laterals; Determination of Horizontal and Per- 
pendicular 30 

14. Radius of hhavrtta (or svavrtta); Hypotenuse of 
Right-Angle Triangle Formed by Gnomon and 
Shadow 31 

15-16. Shadow Problems 31-2 

17. Hypotenuse of Right-Angle Triangle; Relation of 
Half-Chord to Segments of Diameter Which 
Bisects Chord 34 

18. Calculation of sampatasaras When Two Circles 
Intersect 34 

19-20. Arithmetical Progression 36-6 

21. Sum of Series Formed by Taking Sums of Terms 

of an Arithmetical Progression 37 

22. Sums of Series Formed by Taking Squares and 
Cubes of Terms of an Arithmetical Progression . 37 


23. Product of Two Factors Half the Difference be- 
tween Square of Their Sum and Sum of Their 
Squares 38 

24. To Find Two Factors When Product and Differ- 
ence Are Known 38 

25. Interest ; . . 38 

26. Rule of Three (Proportion) 39 

27. Fractions • 40 

28. Inverse Method 40 

29. To Find Sum of Several Numbers When Results 
Obtained by Subtracting Each Number from Their 
Sum Are Known 40 

30. To Find Value of Unknown When Two Equal 
Quantities Consist of EJiowns and Similar Un- 
knowns 41 

31. To Calculate Their Past and Future Conjunctions 
from Distance between Two Planets .... 41 

32-33. Indeterminate Equations of First Degree 

(kuttaka) 43 

III, .Kalakeita or the Reckoning of Time .... 51 
1-2. Divisions of Time; Divisions of Circle Corre- 
spond 51 

3. Conjunctions and vyatlpdtas of Two Planets in a 
yuga ' . . 51 

4. Number of Revolutions of Epicycles of Planets; 
Years of Jupiter 51 

5. Definition of Solar Year, Limar Month, Civil Day, 

and Sidereal Day . . . ,. 62 

6. Intercalary Months and Omitted Lunar Days . 52-3 
7-8. Year of Men, Fathers, and Gods; yuga of All 

the Planets; Day of Brahman 53 

9. Utsarpim, avasaTpinl, sxLsamQ, and dussama as 

Divisions of yuga ^ 53 

10. Date of Writing of Aryabhatiya; Age of Author at 
Time 54 


11. Yuga, Year, Month, and Day Began at First of 
Caitra; Endless Time Measured by Movements of 
Planets and Asterisms 55 

12. Planets Move with Equal Speed; Time in Which 
They Traverse Distances Equal to Orbit of Aster- 
isms and Circumference of Sky 55 

13. Periods of Revolution Differ because Orbits Differ 

in Size 66 

14. For Same Reason Signs, Degrees, and Minutes 
Differ in Length 56 

15. Order in Which Orbits of Planets Are Arranged 
(beneath the Asterisms) around Earth as Center 56 

16. Planets as "Lords of Days" (of Week) ... 56 

17. Planets Move with Their Mean Motion on Orbits 
and Eccentric Circles Eastward from Apsis and 
Westward from Conjunction 57 

18-19. Eccentric Circle Equal in Size to Orbit; Its 
Center Distant from Center of Earth by Radius 
of Epicycle 58 

20. Movement of Planet on Epicycle; When ahead of 

and When behind Its Mean Position .... 58 

21. Movement of Epicycles; Mean Planet (on Its 
Orbit) at Center of Epicycle 59 

22-24. Calculation of True Places of Planets from 
Mean Places 60 

25. Calculation of True Distance between Planet and 
Earth 61 


1. Zodiacal Signs in Northern and Southern Halves 
of Ecliptic; Even Deviation of Echptic from 
Equator 63 

2. Sun, Nodes of Moon and Planets, and Earth's 
Shadow Move along Ecliptiij, 63 

3. Moon, Jupiter, Mars, and Saturn Cross Ecliptic 
at Their Nodes; Venus and Mercury at Their 
Conjunctions 63 


4. Distance from Stm at Which Moon and Planets 
Become Visible 63-4 

5. Sim Illumines One Half of Earth, Planets, and 
Asterisms; Other Half Dark 64 

6-7. Spherical Earth, Surroimded by Orbits of 
Planets and by Asterisms, Situated in Center of 
Space; Consists of Earth, Water, Fire, and Air . 64 

8. Radius of Earth Increases and Decreases by a 
yojana during Day and Night of Brahman . . 64 

9. At Equator Stationary Asterisms Seem To Move 
Straight Westward; Simile of Moving Boat and 
Objects on Shore 64 

10. Asterisms and Planets, Driven by Provector 
Wind, Move Straight Westward at Equator — 
Hence Rising and Setting 66 

11-12. Mount Meru and Vadavamukha (North and 
South Poles); Gods and Demons Think the Others 
beneath Them 68 

13. Four Cities on Equator a Quadrant Apart; Sun- 
rise at First Is Midday, Sunset — Midnight at 
Others . . . i 68 

14. Lanka (on Equator) 90° from Poles; Ujjain 22|° 
North of Lanka 68 

15. From Level Place Half of Stellar Sphere minus 
Radius of Earth Is Visible ; Other Half plus Radius 

of Earth Is Cut Off by Earth 68-9 

16. At Meru and Vadavamukha Northern and South- 
em Halves of Stellar Sphere Visible Moving from 
Left to Right or Vice Versa 69 

17. At Poles the Sun, after It Rises, Visible for Half- 
Year; on Moon the Sun Visible for Half a Lunar 
Month 69 

18. Definition of Prime Vertical, Meridian, and 
Horizon 69 

19. East and West Hour-Circle Passing through Poles 
(unmandala) 69 


20. Prime Vertical, Meridian, and Perpendicular from 
Zenith to Nadir Intersect at Place Where Observer 

Is 70 

21. Vertical Circle Passing through Planet and Place 
Where Observer Is {drhmandala); Vertical Circle 
Passing through Nonagesimal Point (drkk^epa- 
mandala) 70 

22. Construction of Wooden Globe Caused To Re- 
volve So as To Keep Pace with Revolutions of 
Heavenly Bodies 70 

23. Heavenly Bodies Depicted on This; Equinoctial 
Sine (Sine of Latitude) Is Base; Sine of Co- 
latitude {sahku at Midday of Equinoctial Day) 

Is koti (Perpendicular to Base) 70 

24. Radius of Day-Circle 71 

25. Right Ascension of Signs of Zodiac .... 71 

26. Earth-Sine Which Measures Increase and De- 
crease of Day and Night 71 

27. Obhque Ascension of Signs of Zodiac .... 72 

28. Sahku of Sun (Sine of Altitude on Vertical Circle 
Passing through Sun) at Any Given Time . . 72 

29. Base of sahku (Distance from Rising and Setting 
Line) . . ■ 73 

,30. Amplitude of Sun (agra) 73 

31. Sine of Altitude of Sun When Crossing Prime 
Vertical 74 

32. Midday sahku and Shadow 74 

33. Sine of Ecliptic Zenith-Distance (drkk^epajya) . 74 

34. Sine of Ecliptic Altitude (drggatijya) ; Parallax . 75 
35-36. Drkkarman {ak§a and dyana) 76-7 

37. Moon Causes EcUpse of Sun; Shadow of Earth 
Causes Eclipse of Moon 78 

38. Time at Which Eclipses Occur 78 

39. Length of Shadow of Earth <^ 78 

40. Diameter of Earth's Shadow in Orbit of Moon . 79 

41. Sthityardha (Half of Tinie from First to Last 
Contact) 79 


42. Vimardardha (Half of Time of Total Obscuration) 79 

43. Part of Moon \\Tiich Is Not Eclipsed .... 79 

44. Amount of Obscuration at Any Given Time . . 79-80 

45. Valana 80 

46. Color of Moon at Diiferent Parts of Total Eclipse 81 

47. Eclipse of Sun Not Perceptible if Less than One- 
eighth Obscured 81 

48. Sun Calculated from Conjunction {yoga) of Earth 
and Sun, Moon from Conjunction of Sun and 
Moon, and Other Planets from Conjunction of 
Planet and Moon 81 

49-50. Colophon 81 

General Index 83 

Sanskrit Index 89 


Alberuni Alberuni's India. Translated by 

E. C. Sachau. London, 1910. 

Barth {(Euvres) CEuvres de Auguste Barthe. 3 vols. 

Paris, 1917. 

BCMS Bulletin of the Calcutta Mathe- 
matical Society. 

BMskara, Ganitadhyaya Edited by Bapu Deva Sastri; re- 
vised by Ramachandra Gupta. 
Benares (no date). 

Bhaskara, Golddhy&ya Edited by Bapu Deva Sastri; re- 
vised by Ramacliandra Gupta. 
Benares (no date) . 
Edited by Girija Prasad Dvivedi. 
Lucknow: Newul Kishore Press, 

Bhattotpala. . . ; The Brhat Samhita by Varaha- 

miliira with the commentary 
of Bhattotpala. "Vizianagram 
Sanskrit Series," Vol. X. Be- 
nares, 1895-97. 

Bibl. math Bibliotkeca mathematica. 

Brahmagupta Refers to Brcthmasphuiasiddhanta. 

Brahmasphutasiddhanta Edited by Sudhakara Dvivedin in 

the PandU (N.S.), Vols. XXIII- 
XXrV. Benares, 1901-2. 

Bvennaiad, Hindu Astronomy.. 'W. Brennand, Hindu Astronomy. 

London, 1896. 

Brhat Sa'fiihita The Brhat Samhita by VaraJiami- 

hira with the commentary of 
Bhattotpala. "Vizianagram San- 
skrit Series," Vol. X. Benares, 


ColebrookCj Algebra H. T. Colebrooke, Algebra, with 

Arithmetic and Mensuration, 
from the Sanscrit of Brahmegupta 
and Bhdscara. London, 1817. 

Colebrooke, Essays Miscellaneous Essays (2d ed.), by 

H. T. Colebrooke. Madras, 

Hemacandra, AbhidhdTia- Edited by Bohtlingk and Rieu. 
cintdmani St. Petersburg, 1847. 

I A Indian Antiquary. 

IHQ Indian Historical Quarterly. 

J A Journal asiatique. 

JASB Journal and Proceedings of the 

Asiatic Society of Bengal. 

JBBRAS Journal of the Bombay Branch of 

the Royal Asiatic Society. 

JBORS Journal of the Bihar and Orissa 

Research Society. 

JIMS Journal of the Indian Mathematical 


JRAS Journal of the Royal Asiatic 


Kaye, Indian Mathematics . . . G. E,. Kaye, Indian Mathematics. 

Calcutta, 1915. 

Kaye, Hindu Astronomy "Memoirs of the Archaeological 

Survey of India," No. 18. Cal- 
cutta, 1924. 

Khandakhddyaha By Brahmagupta. Edited by 

Babua Misra Jyotishacharyya. 
University of Calcutta, 1925. 

Lalla The Sisyadhwrddhida of Lalla. 

Edited by Sudhakara Dvivedin. 
Benares (no date). 

Mahasiddhanta By Aryafbhata. Edited by Sudha- 
kara Dvivedin in the "Benares 
Sanskrit Series." 1910. 


Marici The Ganitadhydya of Bhaskara's 

SiddhantaHromani with Vasana- 
bha§ya, Vasanavarttika, and 
Marici. Pandit (N.S.), Vols. 
XXX-XXXI. Benares, 1908-9. 

Paneasiddhdntika G. Thibaut and Sudhakara Dvi- 

vedl, The Pancasiddhantika. The 
Astronomical Work of Varaha 
Mihira. Benares, 1889. 

Sudhakara, Oanakatarangim. .BenaTes, 1892. 

Suryasiddhdnta Edited by F. E. Hall and Bapu 

Deva Sastrin in the Bibliotheca 
indica. Calcutta, 1859. 
Translated by Burgess and Whit- 
ney, Journal of the American 
Oriental Society, Vol. VI. 

Vasanavarttika The Ganitadhydya of Bhaskara's 

Siddhdntasiromani with Vasa- 
nabha§ya, Vasanavarttika, and 
Marici. Pandit (N.S.), Vols. 
XXX-XXXI. Benares, 1908-9. 

ZDMG Zeitschrift der Deutschen Morgero- 

Idndischen Gesellschaft. 
I, II, III, and rV refer to the four sections of the Aryabhatzya. 



A. Ha\ing paid reverence to Brahman, who is one (in causal- 
ity, as the creator of the universe, but) many (in his manifesta- 
tions), the true deity, the Supreme Spirit, Arj^abhata sets forth 
three things: mathematics [ganita], the reckoning of time 
[kalakriyd], and the sphere [gola]. 

Baidyanath suggests that satyd devatd may denote 
Sarasvati, the goddess of learning. For this I can 
find no support, and therefore follow the commen- 
tator Paramesvara in translating "the true deity," 
God in the highest sense of the word, as referring to 
Prajapati, Pitamaha, Svayambhu, the lower indi- 
viduahzed Brahman, who is so called as being the 
creator of the universe and above all the other gods. 
Then this lower Brahman is identified with the higher 
Brahman as being only an individuafized manifesta- 
tion of the latter. As Paramesvara remarks, the use 
of the word kam seems to indicate that Aryabhata 
based his work on the old Pitdmahasiddhdnta. Sup- 
port for this view is found in the concluding stanza 
of our text (IV, 50), drydbhatiyam ndmnd purvam 
svdyamhhuvam sadd sad yat However, as shown by 
Thibaut^ and Kharegat,^ there is a close connection 
between Aryabhata and the old Suryasiddhdnta. At 

1 Pancasiddhdntikd, pp. sviii, xxvij. 

2 JBBRAS, XIX, 129-31. 



present the evidence is too scanty to allow us to 
specify the sources from which Aryabhata drew. 

The stanza has been translated by Fleet.^ As 
pointed out first by Bhau Daji,^ a passage of Brahma- 
gupta (XII, 43), jdndty ekam api yato naryahhato 
ganitakalagoldnam, seems to refer to the Ganitapdda, 
the Kdlakriydpdda, and the Golapdda of our Arya-, 
bhatiya (see also Bibhutibhusan Datta).^ Since 
Brahmagupta (XI, 8) names the Dasagltika and the 
Arydstasata (108 stanzas) as works of Aryabhata, and 
since the three words of XI, 43 refer in order to the 
last three sections of the Aryabhatiya (which contain 
exactly 108 stanzas), their occurrence there in this 
order seems to be due to more than mere coincidence. 
As Fleet remarks,* Aryabhata here claims specifically 
as his work only three chapters. But Brahmagupta 
(628 A.D.) actually quotes at least three passages of 
our Dasagltika and ascribes it to Aryabhata. There 
is no good reason for refusing to accept it as part of 
Aryabhata's treatise. 

B. Beginning with, ka the varga letters (are to be used) in the 
varga places, and the avarga letters (are to be used) in the avarga 
places. Ya is equal to the sum of na and ma. The nine vowels 
(are to be used) in two nines of places varga and avarga. Navantya- 
varge vet. 

Aryabhata's system of expressing numbers by 
means of letters has been discussed by Whish,^ by 

1 JRAS, 1911, pp. 114^15. » BCMS, XVIII (1927), 16. 

» Ibid., 1865, p. 403. ** JRAS, 1911, pp. 115, 125. 

8 Transactions of the Literary Society of Madras, I (1827), 54, 
translated with additional notes by Jacquet, J A (1835), II, 118. 


Brockhaus,^ by Kern,^ by Barth,^ by Rodet,^ by 
Kaye,^ by Fleet,^ by Sarada Kanta Ganguly/ and by 
Sukumar Ranjan Das.* I have not had' access to the 
Prthimr Itihasa of Durgadas Lahiri.^ 

The words varga and avarga seem to refer to the 
Indian method of extracting the square root, which 
is described in detail by Rodet^" and by Avadhesh 
Narayan Singh.^^ I cannot agree with Kaye's state- 
ment^2 that the rules given by Aryabhata for the 
extraction of square and cube roots (II, 4-5) "are 
perfectly general (i.e., algebraical)" and apply to all 
arithmetical notations, nor with his criticism of the 
foregoing stanza: "Usually the texts give a verse 
explaining this notation, but this explanatory' verse 
is not Aryabhata's."!* Sufficient evidence has not been 
adduced by him to prove either assertion. 

The varga or "square" places are the first, third, 
fifth, etc., counting from the right. The avarga or 
"non-square" places are the second, fourth, sixth, 
etc., counting from the right. The words varga and 
avarga seem to be used in this sense in II, 4. There 
is no good reason for refusing to take them in the same 
sense here. As applied to the Sanskrit alphabet the 
varga letters referred to here are those from k to m, 

1 Zeitschrift fur die Kunde des Morgenlandes, IV, 81. 

2 JRAS, 1863, p. 380. » IHQ, III, 110. 
» CEiwres, III, 182. » III, 332 ff. 

< J A (1880), II, 440. " Op. at. (1879), I, 406-8. 

5 JASB, 1907, p. 478. • " BCMS, XVIII (1927), 128 

8 Op. di., 1911, p. 109. " Op. dt., 1908, p. 120. 

' BCMS, XVII (1926), 195. « Ibid., p. 118. 


■which are arranged in five groups of five letters each. 
The avarga letters are those from y to h, which are 
not so arranged in groups. The phrase "beginning 
with ka" is necessary because the vowels also are 
divided into vargas or "groups." 

Therefore the vowel a used in varga and avarga 
places with varga and avarga letters refers the varga 
letters fc to m to the first varga place, the unit place, 
multiplies them by 1. The vowel a used with the 
avarga letters y to h refers them to the first avarga 
place, the place of ten's, multiplies them by 10. In 
like manner the vowel i refers the letters A; to m to 
the second varga place, the place of hundred's, multi- 
plies them by 100. It refers the avarga letters y to h 
to the second avarga place, the place of thousand's, 
multiplies them by 1,000. And so on with the other 
seven vowels up to the ninth varga and avarga places. 
From Aryabhata's usage it is clear that the vowels to 
be employed are a, i, u, r, I, e, ai, o, and au. No 
distinction is made between long and short vowels. 

From Aryabhata's usage it is clear that the letters 
fc to m have the values of 1-25. The letters y to h 
would have the values of 3-10, but since a short a is 
regarded as inherent in a consonant when no other 
vowel sign is attached and when the virdma is not 
used, and since short a refers the avarga letters to the 
place of ten's, the signs ya, etc., really have the values 
of 30-100.^ The vowels themselves have no numerical 
values. They merely serve to" refer the consonants 
(which do have numerical values) to certain places. 

1 See Sarada Kanta Ganguly, op, eit., XVII (1926), 202. 


The last clause, wMch has been left untranslated, 
offers great difficulty. The commentator Paramesvara 
takes it as affording a method of expressing stiU 
higher numbers by attaching anusvdra or visarga to 
the vowels and using them in nine further varga (and 
avarga) places. It is doubtful whether the word 
avarga can be so supplied in the compound. Fleet 
would translate "in the varga place after the nine" 
as giving directions for referring a consonant to the 
nineteenth place. In view of the fact that the plural 
subject must carry over into this clause Fleet's in- 
terpretation seems to be impossible. Fleet suggests as 
an alternate interpretation the emendation of vd to 
hau. But, as explained above, au refers h to the 
eighteenth place. It would run to nineteen places only 
when expressed in digits. There is no reason why such 
a statement should be made in the rule. Rodet 
translates (without rendering the word nava), "(s6p- 
arement) ou a un groupe termini par un varga." That 
is to say, the clause has nothing to do with the ex- 
pression of numbers beyond the eighteenth place, 
but merely states that the vowels may be attached 
to the consonants singly as gara or to a group of con- 
sonants as gra, in which latter case it is to be under- 
stood as applying to each consonant in the group. So 
giri or gri and guru or gru. Such, indeed, is Arya- 
bhata's usage, and such a statement is really nec- 
essary in order to avoid ambiguity, but the words do 
not seem to warrant the translation given by Rodet. 
If the words can mean ''at the end of a group," and 
if nava can be taken with what precedes, Rodet's in- 


terpretation is acceptable. However, I know no other 
passage which, would warrant such a translation of 

Sarada Kanta Ganguly translates, "'[Those] nine 
[vowels] [should be used] in higher places in a similar 
manner." It is possible for vd to have the sense of 
"beliebig," "fakultativ," and for nava to be sepa- 
rated from antyavarge, but the regular meaning of 
antya is "the last." It has the sense of ''the following" 
only at the end of a compound, and the dictionary 
gives only one example of that usage. If navdntyavarge 
is to be taken as a compound, the translation "in the 
group following the nine" is all right. But Ganguly's 
translation of antyavarge can be maintained only if he 
produces evidence to prove that antya at the begin- 
ning of a compound can mean "the following." 

If nava is to be separated from antyavarge it is 
possible to take it with what precedes and to trans- 
late, "The vowels (are to be used) in two nine's of 
places, nine in varga places and nine in avarga places," 
but antyavarge vd remains enigmatical. 

The translation must remain uncertain until 
further evidence bearing on the meaning of antya 
can be produced. Whatever the meaning may be, the 
passage is of no consequence for the numbers actually 
dealt with by Aryabhata in this treatise. The largest 
number used by Aryabhata himself (1, 1) runs to only 
ten places. 

Rodet, Barth, and some others would translate "in 
the two nine's of zero's," instead of "in the two nine's 
of places." That is to say, each vowel would serve to 


add two zero's to the niimerical value of the con- 
sonant. This, of course, will work from the vowel i 
on, but the vowel a does not add two zero's. It adds 
no zero's or one zero depending on whether it is used 
•with varga or avarga letters. The fact that khadvi- 
navake is amplified by varge 'varge is an added difficulty 
to the translation "zero." It seems to me, therefore, 
preferable to take the word kha in the sense of "space" 
or better ^ 'place. "^ Later the word kha is one of the 
commonest words for ''zero," but it is still disputed 
whether a sjrmbol for zero was actually in use in 
Aryabhata's time. It is possible that computation 
may have been made on a board ruled into columns. 
Only nine symbols may have been in use and a blank 
column may have served to represent zero. 

There is no evidence to indicate the way in which 
the actual calculations were made, but it seems cer- 
tain to me that Aryabhata could write a number in 
signs which had no absolutely fixed values in them- 
selves but which had value depending on the places 
occupied by them (mounting by powers of 10). Com- 
pare II, 2, where in giving the names of classes of 
numbers he uses the expression sthdndt sthanam 
dasagunam syat, "from place to place each is ten times 
the preceding." ■ 

There is nothing to prove that the actual calcula- 
tion was made by means of these letters. It is prob- 
able that Aryabhata was not inventing a numerical 
notation to be used in calculation but was devising 
a system by means of which he might express large, 

1 Cf. Fleet, op. clL, 1911, p. 116. 


unwieldy numbers in verse in a very brief form.^ 
The alphabetical notation is employed only in the 
Dasagltika. In other parts of the treatise, where only 
a few numbers of small size occur, the ordinary words 
which denote the numbers are employed. 

As an illustration of Aryabhata's alphabetical 
notation take the number of the revolutions of the 
Moon LQ a yuga (I, 1), which is expressed by the word . 
cayagiyihusuchlr. Taken syllable by syllable this 
gives the numbers 6 and 30 and 300 and 3,000 and 
50,000 and 700,000 and 7,000,000 and 50,000,000. 
That is to say, 57,753,336. It happens here that the 
digits are given in order from right to left, but they 
may be given in reverse order or in any order which 
will make the syllables fit into the meter. It is hard 
to believe that such a descriptive alphabetical nota- 
tion was not based on a place-value notation. 

This stanza, as being a technical parihhdsa stanza 
which indicates the system of notation employed in 
the Dasagltika, is not counted. The invocation and 
the colophon are not counted. There is no good reason 
why the thirteen stanzas should not have been named 
Dasagltika (as they are named by Aryabhata himself 
in stanza C) from the ten central stanzas in Giti 
meter which give the astronomical elements of the 
system. The discrepancy offers no firm support to the 
contention of Kaye that this stanza is a later addition. 
The manuscript referred to by Kaye^ as containing 
fifteen instead of thirteen stanzas is doubtless com- 

» See J A (1880), II, 454, and BCMS, XVII (1926), 201. 
2 Op. cii., 1908, p. 111. 


parable to the one referred to by Bhau DajP as having 
two introductory stanzas "evidently an after-addi- 
tion, and not in the Arya metre." 

1. In a yuga the revolutions of the Sun are 4,320,000, of the 
Moon 57,753,336, of the Earth eastward 1,582,237,500, of Saturn 
146,564, of Jupiter 364,224, of IMars 2,296,824, of Mercury and 
Venus the same as those of the Sun. 

2. of the apsis of the IMoon 488,219, of (the conjunction of) 
Mercurj' 17,937,020, of (the conjunction of) Venus 7,022,388, of 
(the conjunctions of) the others the same as those of the Sun, of 
the node of the Moon westward 232,226 starting at the beginning 
of Mesa at sunrise on Wednesday at Lanka. 

The so-called revolutions of the Earth seem to 
refer to the rotation of the Earth on its axis. The 
number given corresponds to the number of sidereal 
days usually reckoned in a yuga. Paramesvara, who 
follows the normal tradition of Indian astronomy 
and believes that the Earth is stationary, tries to 
prove that here and in IV, 9 (which he quotes) 
Aryabhata does not really mean to say that the Earth 
rotates. His effort to bring Aryabhata into agreement 
with the views of most other Indian astronomers 
seems to be misguided ingenuity. There is no warrant 
for treating the revolutions of the Earth given here 
as based on false knowledge (jnithyajnana) , which 
causes the Earth to seem to move eastward because 
of the actual westward movement of the planets (see 
note to I, 4). 

In stanza 1 the syllable su in the phrase which 
gives the revolutions of the Earth is a misprint for 
bu as given correctly in the commentary.^ 

1 Ibid., 1865, p. 397. ^ See ibid., 1911, p. 122 n. 


Here and elsewhere in the Dasagltika words are 
used in their stem form without declensional endings. 

Lalla (Madhyamddhikdra, 3-6, 8) gives the same 
numbers for the revolutions of the planets, and differs 
only in giving "revolutions of the asterisms" instead 
of "revolutions of the Earth." 

The Suryasiddhanta (I, 29-34) shows slight varia- 
tions (see Pancasiddhdntikd, pp. xviii-xix, and 
Kharegat^ for the closer relationship of Aryabhata 
to the old Suryasiddhanta). 

Bibhutibhusan Datta,^ in criticism of the number 
of revolutions of the planets reported by Alberuni (II, 
16-19), remarks that the numbers given for the 
revolutions of Venus and Mercury really refer to the 
revolutions of their apsides. It would be more accu- 
rate to say "conjunctions." 

Alberuni (I, 370, 377) quotes from a book of 
Brahmagupta's which he calls Critical Research on the 
Basis of the Canons a number for the civil days accord- 
ing to Aryabhata. This corresponds to the number of 
sidereal days given above (cf . the number of sidereal 
days given by Brahmagupta [I, 22]). 

Compare the figures for the number of revolutions 

of the planets given by Brahmagupta (1, 15-21) which 

differ in detail and include figures for the revolutions 

of the apsides and nodes. Brahmagupta (I, 61) 

akrtaryabhatat sighragam induccam patam alpagam svagate^i | 
tithyantagrahananam gh.unak§aram tasya samvadah (| 

criticizes the numbers given by Aryabhata for the 
revolutions of the apsis and node of the Moon.^ 

1 JBBRAS, XVIII, 129-31. « BCMS, XVII (1926), 71. 

* See further Bragmagupta (V, 25) and Alberuni (I, 376). 


Brahmagupta (II, 46^7) remarks that according 
to Aryabhata all tlie planets were not at the first point 
of IMesa at the beginning of the yuga. I do not know 
on what evidence this criticism is based.^ 

Brahmagupta (XI, 8) remarks that according to 
the Arydstasata the nodes move while according to 
the Dasagitika the nodes (excepting that of the Moon) 
are fixed: 

arya§tasate pata bhramanti dasagitike stlurah pata^ | 
muktvendupatam apamandale bhramanti sthira nata^i. (| 

This refers to I, 2 and IV, 2. Aryabhata (I, 7) gives 
the location, at the time his work was composed, of 
the apsides and nodes of all the planets, and (I, 7 and 
IV, 2) implies a knowledge of their motion. But he 
gives figures only for the apsis and node of the Moon. 
This may be due to the fact that the numbers are so 
small that he thought them negligible for his purpose. 
Brahmagupta (XI, 5) quotes stanza 1 of our text : 

yugaravibhaganah khynghriti yat proktam tat tayor yugam 

spa^tam | 
trisati ravyudayanam tadantaram hetuna kena. ||'' 

1 See Suryasiddhanta, pp. 27-28, and JRAS, 1911, p. 494. 

2 Cf. JRAS, 1865, p. 401. This_ implies, as Sudhakara says, that 
Brahmagupta knew two works by Aryabhata each giving the revolu- 
tions of the Sun as 4,320,000 but one reckoning 300 savana days more 
than the other. Cf. Kharegat (op. ciL, XIX, 130). Is the reference to 
another book by the author of our treatise or was there another 
earlier Aryabhata? Brahmagupta (XI, 13-14) further implies that 
he knew two works by an author named Aryabhata in one of which 
the yuga began at sunrise, in the other at midnight (see JRAS, 
1863, p. 384; JBBRAS, XIX, 130-31; JRAS, 1911, p. 494;.7HQ, IV, 
506). At any rate, Brahmagupta does not imply knowledge of a 
second Aryabhata. For the whole problem of the two or three Arya- 
bhatas see Kaye {Bihl. math., X, 289) and Bibhutibhusan Datta 


3. There are 14 Manus in a day of Brahman [a kalpa], and 
72 y-ugas constitute the period of a Manu. Since the beginning 
of this kalpa up to the Thursday of the Bharata battle 6 Manus, 
27 yugas, and 3 yugapddas have elapsed. 

Tiie word yugapada seems to indicate that Arya- 
bhata divided the yuga into four equal quarters. 
There is no direct statement to this effect, but also 
there is no reference to the traditional method of 
dividing the yuga into four parts in the proportion of 
4, 3, 2, and 1. Brahmagupta and later tradition 
ascribes to Aryabhata the division of the yuga into 
four equal parts. For the traditional division see 
Suryasiddhdnta (I, 18-20, 22-23) and Brahmagupta 
(I, 7-8). For discussion of this and the supposed 
divisions of Aryabhata see Fleet.^ Compare III, 10, 
which gives data for the calculation of the date of 
the composition of Aryabhata' s treatise. It is clear 
that the fixed point was the beginning of Aryabhata' s 
fourth yugapada (the later Kaliyuga) at the time of 
the great Bharata battle in 3102 b.c. 

Compare Brahmagupta (I, 9) 

yugapadan aryabhatas catvari samSni kftasmgadini [ 
yad abhihitavan na te§ani smrtyuktasamanam ekam api |[ 

and XI, 4 

aryabhato yugapadarfis trin yatan aha kaliyugadau yat | 
tasya krtantar yasmat svayngadyantau na tat tasmat || 

(op. dt., XVII [1926], 60-74). The PancasiddhantiM also (XV, 20), 
"Aryabhata maintains that the beginning of the day is to be reckoned 
from midnight at Lanka; and the same 'teacher again says that the 
day begins from sunrise at Lanka," ascribes the two theories to one 

1 Op. at., 1911, pp. Ill, 486. 


with the commentary of Sudhakara. Brahmagupta 

(I, 12) quotes stanza I, 3, 

manusandhiiii jiigam iccliaty an-abhatas tanmanur yatati 

skhajTigah \ 
kalpas caturjoiganaih sahasram a§tadliikam tasj'a. ||^ 

Brahmagupta (I, 28) refers to the same matter, 
adhikat smrtj-uktamanor aryabhatoktas catunnagena manul;i | 
adhikam -v-imsamsajTitais tribhir joigais tasya kalpagatam. 1| 

Brahmagupta (XI, 11) criticizes Aryabhata for be- 
ginning the Kaliyuga with Thursday (see the com- 
mentary of Sudhakara). 

Bhau Daji^ first pointed out the parallels in 
Brahmagupta I, 9 and XI, 4 and XI, 11.® 

4. The revolutions of the Moon (in a yiiga) multiplied by 12 
are signs [rasi].* The signs multiphed by 30 are degrees. The 
degrees multiplied by 60 are minutes. The miautes multiplied 
by 10 are yojanas (of the circumference of the sky). The Earth 
moves one minute in a prdna.^ The circumference of the sky (in 
yojanas) di^-ided by the. revolutions of a planet in a yuga gives 
the yojanas of the planet's orbit. The orbit of the Sun is a sixtieth 
part of the circle of the asterisms. 

In translating the words sasirasayas tha cakram 
I have followed Paramesvara's interpretation sasinas 
cakram hhagand dvddasagunitd rdsayah. The Sanskrit 
construction is a harsh one, but there is no other way 
of making sense. Sasi (without declensional ending) 
is to be separated. 

Paramesvara explains the word grahajavo as fol- 

1 Cf. Ill, S. . 2 Op. ciL, 1865, pp. 400-401. 

' Cf . Alberuni, I, 370, 373-74. 

* A rail is a sign of the zodiac or one-twelfth of a circle. 
^ For jrrana see III, 2. 


lows : ekapanvrttau grahasya javo gatimdnam yojana- 
tmakam bhavati. 

The word yojanani must be taken as given a figure 
in yojanas for the circumference of the sky (akasa^ 
kaksya). It works out as 12,474,720,576,000, which is 
the exact figure given by Lalla {Madhyamadhikdra 
13) who was a follower of Aryabhata. Compare 
Suryasiddkdnta, XII, 80-82; Brahmagupta, XXI, 
11-12; Bhaskara, Golddhydya, Bhuvanakosa, 67-69 
and Ganitddhydya, Kaksddhydya, 1-5. 

The statement of Alberuni (I, 225) with regard to 
the followers of Aryabhata, 

It is sufficient for us to know the space which is reached by 
the solar rays. We do not want the space which is not reached 
by the solar rays, though it be in itself of an enormous extent. 
That which is not reached by the rays is not reached by the per- 
ception of the senses, and that which is not reached by per- 
ception is not knowable, 

may be based ultimately upon this passage. 

The reading bham of our text must be incorrect. 
It is a reading adopted by Paramesvara who was de- 
termined to prove that Aryabhata did not teach the 
rotation of the Earth. This passage could not be ex- 
plained away by recourse to false knowledge (mith- 
ydjndna) as could I, 1 and IV, 9 and therefore was 
changed. The true reading is bhuh, as is proved con- 
clusively by the quotation of Brahmagupta (XI, 17) : 
pra,nenaiti kalarh bhur yadi tarhi kuto vrajet kam adhvanam | 
avarttanam urvyas cen na patanti samucchraya^i kasmat. || 

Compare Brahmagupta (XXI, 59) and Alberuni (I, 
276-77, 280). 


5. A yojana consists of 8,000 times a nr [the height of a man]. 
The diameter of the Earth is 1,050 yojanas. The diameter of the 
Sun is 4,410 yojanas. The diameter of the Moon is 315 yojanas. 
Meru is one yojana. The diameters of Venus, Jupiter, Mercury, 
Saturn, and Mars are one-fifth, one-tenth, one-fifteenth, one- 
twentieth, and one-twenty-fifth of the diameter of the Moon. 
The years of a yuga are equal to the number of revolutions of the 
Sun in a yuga. 

As pointed out by Bhau Dajl,^ Brahmagupta (XI, 
15-16) seems to quote from this stanza in his criticism 
of the diameter of the Earth given by Aryabhata 

§odasagaviyojana paridhim pratibhuvyasam pulavadata | 
atmajnanaih khyapitam aniscayas tanikrtakanyat f[ 
bhuvyasasyajnanad vyartharh desantaram tadajnanat | 
sphutatithyantajnanaih tithinasad grahanayor nasalj. j| 

The text of Brahmagupta is corrupt and must be 
emended. See the commentary of Sudhakara, who 
suggests for the j5rst stanza 

nf§iyojanabhuparidhim prati bhm^asam punar nila vadata | 
atmajnanam khyapitam aniscayas tatkf tavyasati. 1 1 

Lalla (Madhyamddhikara, 56 and Candragrahand- 
dhikdra, 6) gives the same diameters for the Earth 
and the Sun but gives 320 as the diameter of the 
Moon, and {Grahayutyadhikara, 2) gives for the 
planets the same fractions of the diameter of the 

Alberuni (I, 168) quotes from Brahmagupta 
Aryabhata's diameter of the Earth, and a confused 

1 JRAS, 1865, p. 402. 

2Cf. Suryasiddhanta, I, 59; IV, 1; VII, 13-14; Brahmagupta, 
XXI, 32; Kharegat {op. cit, XIX, 132-34, discussing Suryasiddhanta, 
IX, 15-16). 


passage (I, 244-46) quotes Balabhadra on Arya- 
bhata's conception of Meru. Its height is said to be 
a yojana. The context of the foregoing stanza seems 
to imply that its diameter is a yojana, as Paramesvara 
takes it. It is probable that its height is to be taken 
as the same. 

If Paramesvara is correct in interpreting samarka- 
samdh as yugasamd yugdrkahhaganasamd, the nomi- 
native plural samdh has been contracted after sandhi. 

6. The greatest declination of the ecliptic is 24 degrees. The 
greatest deviation of the Moon from the ecliptic is 4| degrees, of 
Saturn 2 degrees, of Jupiter 1 degree, of Mars 1| degrees, of 
Mercury and Venus 2 degrees. Niaety-six angulas or 4 hastas 
make 1 nr. 

Paramesvara explains the words hhdpakramo 
grahdmsdh as follows: grahdndm bha amsds catur- 
vimsatihhdgd apakramah. paramdpakrama ity arthah. 
The construction is as strange as that of stanza 4 

7. The ascending nodes of Mercury, Venus, Mars, Jupiter, 
and Saturn having moved (are situated) at 20, 60, 40, 80, and 
100 degrees from the beginning of Me§a. The apsides of the Sun 
and of the above-mentioned planets (in the same order) (are 
situated) at 78, 210, 90, 118, 180, and 236 degrees from the 
beginning of Mesa. 

I have followed Paramesvara's explanation of 
gatvdmsakdn as uktdn etdn evdrhsakdn mesddito gatvd 

In view of IV, 2, "the Sun and the nodes of the 
planets and of the Moon move constantly along the 

^ Cf . SUryasiddhanta, I, 68-70 and II, 28; Brahmagupta, IX, 1 
and XXI, 52. 


ecliptic," and of I, 2, which gives the number of revo- 
lutions of the node of the Moon in a yuga, the word 
gatva ("having gone") seems to imply, as Parame- 
svara says, a knowledge of the revolution of the nodes 
of the planets and to indicate that Aryabhata in- 
tended merely to give their positions at the time his 
treatise was composed. The force of gatvd continues 
into the second line and indicates a knowledge of the 
revolutions of the apsides. 

Aryabhata gives figures for the revolutions of the 
apsis and node of the Moon. Other siddhdntas give 
figures for the revolutions of the nodes and apsides of 
all the planets. These seem to be based on theory 
rather than on observation since their motion (except 
in the case of the Moon) is so slow that it would take 
several thousand years for them to move so far that 
their motion could easily be detected by ordinary 
methods of observation.^ Aryabhata may have re- 
frained from giving figures for the revolutions of nodes 
and apsides (except in the case of the Moon) because 
he distrusted the figures given in earher books as 
based on theory rather than upon accurate observa- 
tion. Brahmagupta XI, 8 (quoted above to stanza 2) 
remarks in criticism of Aryabhata that in the Dasa- 
gltika the nodes are stationary while in the Aryd- 
stasata they move. This refers to I, 2 and IV, 2. In 
the Dasagitika only the revolutions of the nodes of 
the Moon are given; in the Arydsiasata the nodes and 
apsides are said expHcitly to move along the ecliptic. 
In the present stanza the word gatvd seems clearly to 

^ Cf. SUryasiddhdnta, pp. 27-28. 


indicate a knowledge of the motion of the nodes and 
apsides of the other planets too. If Aryabhata had 
intended to say merely that the nodes and apsides are 
situated at such-and-such places the word gatvd is 
superfluous. In a text of such studied brevity every 
word is used with a very definite purpose. It is true 
that Aryabhata regarded the movement of the nodes 
and apsides of the other planets as negligible for pur- 
poses of calculation, but Brahmagupta's criticism 
seems to be captious and unjustified (see also Bra- 
hmagupta, XI, 6-7, and the commentary of Sudha- 
kara to XI, 8). Earth's criticism^ is too severe. 

Lalla {Spastddhikdra, 9 and 28) gives the same 
positions for the apsides of the Sun and five planets 
(see also Pancasiddhdntikd, XVII, 2) . 

For the revolutions of the nodes and apsides see 
Brahmagupta, 1, 19-21, and Suryasiddhdnta, 1, 41-44, 
and note to I, 44. 

8. Divided by 4| the epicycles of the apsides of the Moon, 
the Sun, Mercury, Venus, Mars, Jupiter, and Saturn (in the first 
and third quadrants) are 7, 3, 7, 4, 14, 7, 9; the epicycles of the 
conjunctions of Saturn, Jupiter, Mars, Venus, and Mercury (in 
the first and third quadrants) are 9, 16, 53, 59, 31; 

9. the epicycles of the apsides of the planets Mercury, Venus, 
Mars, Jupiter, and Saturn in the second and fourth quadrants 
are 5, 2, 18, 8, 13; the epicycles of the conjunctions of the planets 
Saturn, Jupiter, Mars, Venus, and Mercury in the second and 
fourth quadrants are 8, 15, 51, 57, 29. The circumference within 
which the Earth-wind blows is 3,375 yojanas. 

The criticism of these stanzas made by Brahma-' 
gupta (II, 33 and XI, 18-21) is, as pointed out by 

1 Op. cit., Ill, 154. 


Sudhakara, not justifiable. For the dimensions of 
Brahmagupta's epicycles see II, 34-39). 

Lalla (Spastadhikdra, 28) agrees closely with 
stanza 8 and (Grahabhramana, 2) gives the same figure 
for the Earth-wind. Compare also Suryasiddhdnta, 
II, 34-37 and note, and Pancasiddhdntikd, XVII, 1, 3. 

10. The (twenty-four) sines reckoned in minutes of arc are 
225, 224, 222, 219, 215, 210, 205, 199, 191, 183, 174, 164, 154, 
143, 131, 119, 106, 93, 79, 65, 51, 37, 22, 7. 

In Indian mathematics the "half-chord" takes 
the place of our "sine." The sines are given in minutes 
(of which the radius contains 3,438) at intervals of 225 
minutes. The numbers given here are in reality not 
the values of the sines themselves but the differences 
between the sines. 

Compare Suryasiddhanta (II, 15-27) and Lalla 
{Spastadhikdra, 1-8) and Brahmagupta (II, 2-9). 
Bhaskara (Ganitddhydya, Spastadhikdra, Vdsanabhd- 
§ya to 3-9) refers to the Suryasiddhdnta and to 
Aryabhata as furnishing a precedent for the use of 
twenty-four sines.^ 

Krishnaswami Ayyangar^ furnishes a plausible 
explanation of the discrepancy between certain of the 
values given in the foregoing stanza and the values 
as calculated by II, 12.^ Some of the discrepancies 
may be due to bad readings of the manuscripts. Kern 

^ For discussion of the stanza see Bartb, ihid., Ill, 150 n., and 
JRAS, 1911, pp. 123-24. 

2 JIMS, XV (1923-24), 121-26. 

* See also Naraharayya, "Note on the Hindu Table of Sines," 
ihU., pp. 105-13 of "Notes and Questions." 


in a footnote to the stanza and Ayyangar (p. 125 n.) 
point out that the text-reading for the sixteenth and 
seventeenth sines \dolates the meter. This, however, 
may be remedied easily without changing the values.^ 
C. Whoever knows this Dasagltika Sutra which describes 
the movements of the Earth and the planets in the sphere of the 
asterisms passes through the paths of the planets and asterisms 
and goes to the higher Brahman. 

1 Cf. JRAS, 1910, pp. 752, 754, and lA, XX, 228. 


1. Having paid reverence to Brahman, the Earth, the Moon, 
Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the aster- 
isms, Aryabhata sets forth here [in this work] the science which 
is honored at Kusumapura.^ 

The translation "here at Kusumapura the revered 
science" is possible. At any rate, Aryabhata states 
the school to which he belongs. Kusumapura may or 
may not have been the place of his birth. 

2. The numbers eka [one], dasa [ten], sata [hundred], sahasra 
[thousand], ayuta [ten thousand], niyuta [hundred thousand], 
prayuta [million], koti [ten million], arbuda [hundred million], and 
vrnda [thousand million] are from place to place each ten times 
the preceding.^ 

The names for classes of numbers are given only 
to ten places, although I, B describes a notation 
which reaches at least to the eighteenth place. The 
highest number actually used by Aryabhata himseK 
runs to ten places. 

3. A square, the area of a square, and the product of two 
equal quantities are called varga. The product of three equal 
quantities, and a soUd which has twelve edges are called ghana} 

1 Translated by Ffeet, JRAS, 1911, p. 110. See Kern's Preface to 
his edition of the Brhat Samhita, p. 57, and BCMS, XVIII (1927), 7. 

2 See JRAS, 1911, p. 116; IHQ, III, 112; BCMS, XVII (1926), 
198. For the quotation in Alberuni (1, 176), which differs in the last 
two names, see the criticism in BCMS, XVII (1926), 71. 

5 Read dvdda§a§Tas with ParameSvara. For oiro in the sense of 
"edge" see Colebrooke, Algebra, pp. 2 n. and 280 n. The translations 
given by Rodet and Kaye are inaccurate. 



4. One should always divide the avarga by twice the (square) 
root of the (preceding) varga. After subtracting the square (of the 
quotient) from the varga the quotient will be the square root to 
the next place. 

Counting from right to left, the odd places axe 
called varga and the even places are called avarga. 
According to Paramesvara, the nearest square root 
to the number in the last odd place on the left is set 
down in a place apart, and after this are set down the 
successive quotients of the division performed. The 
number subtracted is the square of that figure in the 
root represented by the quotient of the preceding 
division. The divisor is the square of that part of the 
root which has already been found. If the last sub- 
traction leaves no remainder the square root is exact. 
"Always" indicates that if the divisor is larger than 
the number to be divided a zero is to be placed in the 
line (or a blank space left there). Sthandntare ("in an- 
other place") is equivalent to the pahkti ("line") of 
the later books. 

This process seems to be substantially correct, but 
there are several difficulties. Sthandntare may mean 
simply "to another place," that is to say, each 
division performed gives another figure of the root. 
Nityam ("always") may merely indicate that such is 
the regular way of performing the operation. 

All the translators except Saradakanta Ganguly 
translate vargdd varge suddhe with what precedes. I 
think he is correct in taking it with what follows. In 
that case the parallelism with the following rule is 
exact. Otherwise the first rule would give the opera- 


tion for the varga place and then that for the avarga 
place while the second rule would give first the opera- 
tions for the aghana places and then that for the 
ghana place. However, for purposes of description, it 
makes no difference whether the operations are given 
in one or the other of these orders. 

Parallelism with ghanasya mulavargena of the 
following rule seems to indicate that vargamulena is 
not to be translated "square root" but "root of the 
(preceding) varga." 

If the root is to contain more than two figures the 
varga of vargamulena is to be interpreted as applying 
to all the preceding figures up to and including the 
varga place which is being worked with. That is to 
say, the word mula would refer to the whole of that 
part of the root which had already been found.^ 

For discussion see Kaye,^ Avadhesh Narayan 
Singh,^ Saradakanta Ganguly.* I cannot agree with 
Ganguly's discussion of the words hhdgam hared 
avargdt. I see no reason to question the use of hhdgam 
harati with the ablative in the sense of "divide." 
Brahmagupta (XII, 7) in his description of the 
process of extracting the cube root has chedo 'ghandd 
dvitiyat, which means "the divisor of the second 

Kaye^ insists that this rule and the next are per- 
fectly general (i.e., algebraical) and apply to all 
arithmetical notations. He offers no proof and gives 

1 See Colebrooke, op. cit., p. 280 n. 

* JASB, 1907, pp. 493-94. ■« JBORS, XII, 78. 

» BCMS, XVIII (1927), 124. « Op. cit., 1908, p. 120. 


no example of the working of the rule according to 
his interpretation. To what do the words "square" 
and "non-square" of his translation refer? The words 
of Aryabhata exactly fit the method employed in later 
Indian mathematics. Although Brahmagupta does 
not give a rule for square root, his method for cube 
root is that described below, although the wording of 
his rule is different from that of Aryabhata's. I fail to 
see any similarity to the rule and method of Theon 
of Alexandria. 

In the following example the sign ° indicates the 
varga places, and the sign - indicates the avarga 

16129 (root =1 
Square of the root 1 

Twice the root 2)05(2= quotient (or next digit of root) 

(2X1) 4 


Square of the quotient 4 

Twice the root 24)72(3 = quotient (.or next digit of root) 

(2X12) 72 


Square of the quotient 9 

Square root is 1 2 3 

5. One should divide the second aghana by three times the 
square of the (cube) root of the (preceding) ghana. The square 
(of the quotient) multiplied by three times the purva (that part 
of the cube root already found) is to be subtracted from the first 


aghana, and the cube (of the quotient of the above di\dsion) is to 
be subtracted from the gliana. 

The translation given by Avadhesh Narayan 
Singh^ as a "correct literal rendering" is inaccurate. 
There is nothing in the Sanskrit which corresponds 
to "after ha\dng subtracted the cube (of the quo- 
tient) from the ghana place" or to "the quotient 
placed at the next place gives the root." The latter 
thought, of course, does carry over into this rule from 
the preceding rule. In the same article (p. 132) the 
Sanskrit of the rule is inaccurately printed with 
trighanasya for trigunena ghanasya.^ 

Kaye^ remarks that this rule is given by Brahma- 
gupta "word for word." As a matter of fact, the 
Sanskrit of the two rules is very different, although 
the content is exactly the same. 

Counting from right to left, the first, fourth, etc., 
places are named ghana (cubic); the second, fifth, 
etc., places are called the first aghana (non-cubic) 
places; and the third, sixth, etc., places are called the 
second aghana (non-cubic) places. The nearest cube 
root to the number in (or up to and including) the last 
ghana place on the left is the first figure of the cube 
root. After it are placed the quotients of the succes- 
sive divisions. If the last subtraction leaves no 
remainder the cube root is exact. 

1 BCMS, XVIII (1927), 134. 

2 The rule has been discussed in JBORS, XII, 80. Cf . Brah- 
magupta (XII, 7) and the translation and note of Colebrooke (op. 
cit, p. 280). 

' Op. cit, 1908, p. 119. 


In the following example the sign ° indicates the 
ghana places and the sign - indicates the aghana 

1860867 (root=l 
Cube of root 1 

Three times square of root 3) 08 (2 = quotient (or next digit of 
(3XP) 6 root) 


Square of quotient multiplied 12 

by three times the purva 

(2^X3X1) 140 

Cube of quotient 8 

Three times square of root 432) 1328(3 = quotient (or next digit 
(3X122) 1296 of root) 


Square of quotient multiplied 324 

by three times the pilwa 

(3*X3X12) 27 

Cube of quotient 27 

Cube root is 1 2 3 

6. The area of a triangle is the product of the perpendicular 
and half the base. Half the product of this area and the height is 
the volume of a solid which has six edges (pyramid). 

If samadalakoti can denote, as Paramesvara says, 

a perpendicular which is common to two triangles the 

rule refers to all triangles. If samadalakoti refers to a 

perpendicular which bisects the base it refers only to 

isosceles triangles.^ 

1 For asra or asri in the sense of "edge" see note to stanza II, 3. 
See JBORS, XII, 84-85, for discussion of the inaccurate value given 
in the second part of the rule. 



7. Half of the circumference multipKed by half the diameter 
is the area of a circle. This area multiplied by its own square root 
is the exact volume of a sphere.'- 

8. The two sides (separately) multiplied by the perpendicu- 
lar and di\'ided by their sum will give the perpendiculars (from 
the point where the two diagonals intersect) to the parallel sides. 

The area is to be kno-wn by multipl5ing half the sum of the 
two sides by the perpendicular. 





Area = 


The rule applies to any four-sided plane figure of 
which two sides are parallel, i.e., trapezium. The 
word translated "sides" refers to the two parallel 
sides. The perpendicular is the perpendicular be- 
tween the two parallel sides. 

In the example given above a and b are the 
parallel sides, c is the perpendicular between them, 
and d and e are the perpendiculars from the point 
"of intersection of the two diagonals to the sides a and 
b, respectively. 

9. The area of any plane figure is found by determining two 
sides and then multiplsdng them together. 

The chord of the sixth part of the circumference is equal to 
the radius. 

1 See ibid, and Bibl. maih., IX, 196, for discussion of the inac- 
curate value given in the second part of the rule. For a possible 
reference to this passage by Bhaskara, Goladhydya, Bhuvanakosa, 
stanza 61 {Vdsanabhd^a) (not stanza 52 as stated), see BCMS, 
XVm (1927), 10. 


The very general rule given in the first half of this 
stanza seems to mean, as Paramesvara explains in 
some detail, that the mathematician is to use his in- 
genuity in determining two sides which will represent 
the average length and the average breadth of the 
figure. Their product will be the area. Methods to 
be employed with various kinds of figures were doubt- 
less handed down by oral tradition. 

Rodet thinks that the rule directs that the figure be 
broken up into a number of trapeziums. It is doubtful 
whether the words can bear that interpretation. 

10. Add 4 to 100, multiply by 8, and add 62,000. The 
result is approximately the circumference of a circle of which the 
diameter is 20,000. 

The circumference is 62,832. The diameter is 

By this rule the relation of circumference to 
diameter is 3.1416.^ 

Bhaskara, Golddhydya, Bhuvanakosa (stanza 52), 
Vasandbkdsya, refers to this rule of Aryabhata. 

11.. One should divide a quarter of the circumference of a 
circle (into as many equal parts as are desired). From the tri- 
angles and quadrilaterals (which are formed) one wiU have on the 
radius as many sines of equal arcs as are desired.'^ 

The exact method of working out the table is not 
known. It is uncertain what is intended by the 
triangle and the quadrilateral constructed from each 
point marked on the quadrant.^ 

1 See JBORS, XII, 82; JRAS, 1910, pp. 752, 754. 

* See the table given in I, 10 of the differences between the sines. 
Twenty-four sines taken at intervals of 225 miautes of arc are regu- 
larly given in the Indian tables. 

* Note the methods suggested by Kaye and Rodet and of. JIMS, 
XV (1923-24), 122 and 108-9 of "Notes and Questions." 


12. By what number the second sine is less than the first 
sine, and by the quotient obtained by di\dding the sum of the 
preceding sines by the first sine, by the sum of these two quanti- 
ties the following sines are less than the first sine. 

The last phrase may be translated "the sine- 
differences are less than the first sine."^ 

This rule describes how the table of sine-differ- 
ences given in I, 10 may be calculated from the first 
one (225). The first sine means always this first sine 
225. The second sine means any particular sine with 
which one is working in order to calculate the follow- 
ing sine. 

Subtract 225 from 225 and the remainder is 0. Di- 
\'ide 225 by 225 and the quotient is 1 . The sum of and 
1 is subtracted from 225 to obtain the second sine 224. 
Subtract 224 from 225 and the remainder is 1. 
Divide 225 plus 224 by 225 and the nearest quotient 
is 2. Add 2 and 1 and subtract from 225. The third 
sine will be 222. Proceed in like manner for the fol- 
lowing sines. 

If this method is followed strictly there results 
several shght divergences from the values given in I, 
10. It is possible to reconcile most of these by assum- 
ing, as Klrishnaswami Ayyangar does, that from time 
to time the neglected fractions were distributed 
among the sines. But of this there is no indication in 
the rule as given. 

1 For discussion of the Indian sines see the notes of Rodet and 
Kaye; Pancasiddkaniikd,, chap, iv; SuryasiddMMa, II, 15-27; Lalla, 
p. 12; Brahmagupta, II, 2-10; JRAS, 1910, pp. 752, 754; lA, XX, 
228; Brennand, Hindu Astronomy, pp. 210-13; JIMS, XV (1923-24), 
121-26, with attempted explanation of the variation of several of 
the values given in the table from the values calculated by means of 
this rule, and ifnd., pp. 106-13 of "Notes and Questions." 


How Kaye gets "If the first and second be bisected 

in succession the sine of the half -chord is obtained" 

is a puzzle to me. It is impossible as a translation 

of the Sanskrit. 

13. The circle is made by turning, and the triangle and the 
quadrilateral by means of a harm; the horizontal is determined 
by water, and the perpendicular by the plumb-line. 

Tnhhuja denotes triangle in general and catur- 
bhuja denotes quadrilateral in general. The word 
,karna regularly denotes the hjrpotenuse of a right- 
angle triangle and the diagonal of a square or rec- 
tangle. I am not sure whether the restricted sense of 
karria limits trihhuja and caturhhuja to the right-angle 
triangle and to the square and rectangle or whether 
the general sense of tribhuja and caturhhuja general- 
izes the meaning of kariia to that of one chosen side of 
a triangle and to that of the diagonal of any quadri- 
lateral. At any rate, the context shows that the 
rule deals with the actual construction of plane 

Paramesvara interprets it as referring to the con- 
struction of a triangle of which the three sides are 
known and of a quadrilateral of which the four sides 
and one diagonal are known. One side of the triangle 
is taken as the karna. Two sticks of the length of the 
other two sides, one touching one end and the other 
the other end of the karna, are brought to such a posi- 
tion that their tips join. The quadrilateral is made 
by constructing two triangles, one on each side of the 


The circle is made by the turning of the karkata 
or compass.^ 

14. Add the square of the height of the gnomon to the square 
of its shadow. The square root of this sum is the radius of the 

The text reads khavrtta ("sky-circle"). Para- 
mesvara reads svavrtta ("its circle")- I do not know 
which is correct. 

Kaye remarks that in order "to mark out the 
hour angles on an ordinary sun-dial, it is necessary to , 
describe two circles, one of which has its radius equal 
to the vertical gnomon and the other with radius 
equal to the hypotenuse of the triangle formed by the 
equinoctial shadow and the gnomon." It may be that 
this second circle is the one referred to here. Para- 
mesvara has chayagramadhyam sankusirahprdpi yan 
mandalam urdhvadhahsthitam tat svavrttam ity ucyate, 
"the circle which has its centre at the extremity of 
the shadow and which touches the top of the gnomon 
is called the svavrtta.'' As Eodet remarks, it is diffi- 
cult to see for what purpose such a circle could serve. 

15. Multiply the length of the gnomon by the distance be- 
tween the gnomon and the bhuja and divide by the difference 
between the length of the gnomon and the length of the bhuja. 
The quotient will be the length of the shadow measured from 
the base of the gnomon.^ 

1 For parallels to the stanza see Lalla {Yantradhyaya, 2) and 
Brahmagupta, XXII, 7. See BCMS, XVIII (1927), 68-69, which is 
too emphatic in its assertion that karoa must mean "diagonal" and 
not "hjrpotenuse." 

2 See Brahmagupta, XII, 53; Colebrooke, op. dt., p. 317; 
Brennand, op. dt., p. 166. 



Because of the use of the word kotl in the following 
rule Rodet is inclined to think that the gnomon and 
the hhujd were not perpendicular but projected hori- 
zontally from a waU. Bhujd denotes any side of a 
triangle, but kotl usually refers to an upright. It is 
possible, however, for kotl to denote any perpendicu- 
lar to the bhuja whether horizontal or upright. 


BA is the bhuja which holds the light, 
DE is the gnomon, 



16. The distance between the ends of the two shadows multi- 
plied by the length of the shadow and divided by the difference 
in length of the two shadows gives the kotl. The kotl multiplied 
by the length of the gnomon and divided by the length of the 
shadow gives the length of the bhuja. 

The literal translation of chdydgunitam chdyd- 
gravivaram unena bhdjitd kotl seems to be "The dis- 
tance between the ends of the two shadows multiplied 
by the length of the shadow is equal to the kotl 
divided by the difference in length of the two shad- 
ows." This is equivalent to the translation given 



AB is the bkujd, 
AE is the koii, 

CD is the gnomon in its first position, 
CD' is the gnomon in its second position, 
CE and C'E' are the first and second shadows, 

AE = 

AB = 


The length of the hhujd which holds the light and 
the distance between the end of the shadow and the 
base of the hhujd are unknown. In order to find them 
the gnomon is placed in another position so as to give 
a second shadow. 

The length of the shadow is its length when the 
gnomon is in its first position. The koti is the dis- 
tance between the end of the shadow when the gno- 
mon is in its first position and the base of the hhujd. 

The word koti means perpendicular (or upright) 
and the rule might be interpreted, as Rodet takes it, 
as meaning that the hhujd and the gnomon extend 
horizontally from a perpendicular wall. But the 
words hhujd and kopi also refer to the sides of a right- 
angle triangle without much regard as to which is 
horizontal and which is upright. 



Or the first position of the gnomon may be CD' 
and the second CD. To find AE' and AB} 

17. The square of the bhuja plus the square of the hotl is the 
square of the karna. 

In a circle the product of two saras is the square of the half- 
chord of the two arcs. 

The hhujd and koti are the sides of a right-angle 
triangle. The karna is the hypotenuse. 

The saras or ''arrows'^ are the segments of a 
diameter which bisects any chord.^ 

where c is the half-chord. 

18. (The diameters of) two circles (separately) minus the 
grdsa, multiplied by the grasa, and divided separately by the sum 
of (the diameters of) the two circles after the grasa has been sub- 
tracted from each, will give respectively the sampatasaras of the 
two circles. 

When two circles intersect the word grdsa ("the 
bite") denotes that part of the common diameter of 
the two circles which is cut off by the intersecting 
chords of the two circles. 

^ See Brahmagupta, XII, 54; Colebrooke, op. cit., p. 318; Bren- 
nand, op. cit., p. 166. 

2 Cf. Brahmagupta, XII, 41. See BCMS, XVIII (1927), 11, 71, 
with discussion of the quotation given by Colebrooke, op. cit., p. 309, 
from Prthudakasvaml's commentary to Brahmagupta. 



AE = 


EB = 

AB is the grasa, 
AE and BE are the 


D+d-2AB' D+d-2AB ' 

where d and D are the diameters of the two circles. 

The sampdtasaras are the two distances (within 
the grdsa), on the common diameter, from the cir- 
cumferences of the two circles to the point of inter- 
section of this common diameter with the chord con- 
necting the two points where the circumferences 
intersect. 1 

19. The desired number of terms minus one, halved, plus the 
number of terms which precedes, multiplied by the common 
difference between the terms, plus the first term, is the middle 
term. This multiplied by the number of terms desired is the sum 
of the desired number of terms. 

Or the sum of the first and last terms is multiplied by half the 
number of terms. 

This rule tells how to find the sum of any desired 
number of terms taken anywhere within an arith- 
metical progression. Let n be the number of terms 
extending from the (p-l-l)th to the {p-\-n)tla. terms 
in an arithmetical progression, let d be the common 
difference between the terms, let a be the first term 
of the progression, and I the last term. 

^ Cf . Brahmagupta, XII, 43; Colebrooke, op. cU., p. 311. 


The second part of the rule applies only to the sum 
of the whole progression beginning with the first 


As Paramesvara says, samukhamadhyam must be 
taken as equivalent to samukham madhyam. 

Whether Paramesvara is correct in his statement 
hahusutrdrtha'pradarsakam etat sutram. ato hahudhd 
yojana karyd and subsequent exposition seems very 

Brahmagupta, XII, 17 has only the second part of 
the rule.^ 

20. Multiply the sum of the progression by eight times the 
common difference, add the square of the difference between twice 
the first term and the common difference, take the square root of 
this, subtract twice the first term, divide by the common differ- 
ence, add one, divide by two. The result will be the number of 

^_l[ V8dS+(d-2ay-2a , ,-\ , 
''-2L d """^J • 

As Rodet says, the development of this formula 
from the one in the preceding rule seems to indicate 
knowledge of the solution of quadratic equations in 
the form ax^-\-hx+c = 0.^ 

1 Cf . Colebrooke, op. dt., p. 290. 

2 See Brahmagupta, XII, 18; Colebrooke, op, dt., p. 291. 
' See also J A (1878), I, 28, 77, and JBORS, XII, 86-87. 


21. In the case of an wpaciti which has one for the first term 
and one for the common difference between the terms the product 
of three terms ha\'ing the number of terms for the first term and 
one as the common difference, di\dded by six, is the dtighana. Or 
the cube of the number of terms plus one, minus the cube root of 
this cube, di\-ided bj' six. 

Form an arithmetical progression 12 3 4 5, etc. 
Form the series 1 3 6 10 15, etc., by taking for the 
terms the sum of the terms of the first series. The 
rule gives the smn of this series. 

It also gives the cubic contents of a pile of balls 
which has a triangular base. The wording of the rule 
would seem to imply that it was intended especially 
for this second case. Citighana means ''cubic con- 
tents of the pile," and wpaciti ("pile") refers to the 
base (or one side) of the pile, i.e., 12 3 4 5, etc.^ 

As Rodet remarks, it is curious that in the face of 
this rule the rule given above (stanza 6) for the 
volume of a pyramid is incorrect. 

n{n+l){n+2) {n+\f-{n+l) 

6 6 • 

22. The sixth part of the product of three quantities con- 
sisting of the number of terms, the number of terms plus one, 
and twice the number of terms plus one is the sum of the squares. 
The square of the sum of the (original) series is the sum of the 

From the series 12 3 4, etc., form the series 
1 4 9 16, etc., and 1 8 27 64, etc., consisting of the 

1 Cf. Brahmagupta, XII, 19; Colebrooke, op. cit., pp. 292-93. 
Brahmagupta (XII, 20) directs that the summation of certain series 
be illustrated by means of piles of roimd balls. 


squares and cubes of the terms of the first series. The 
rule tells how to find the sums of the second and 
third series.^ 

The rule for finding the sum of the first series was 
given above in stanza 19. 

The sum of the squares is 


23, One should subtract the sum of the squares of two factors 
from the square of their simi. Half the result is the product of 
the two factors. 

ab= . 

24. Multiply the product (of two factors) by the square of 
two (4), add the square of the difference between the two factors, 
take the square root, add and subtract the difference between the 
two factors, and divide the result by two. The results wiU be the 
two factors. 

l/4ab+(a-6)^±(a-6) .,, . ,, 
^^ — ^ ^^ will give a and o. 

25. Multiply the sum of the interest on the principal and 
the interest on this interest by the time and by the principal. 
Add to this result the square of half the principal. Take the 
square root of this. Subtract half the principal and divide the 
remainder by the time. The result will be the interest on the 

1 Cf. Brahmagupta, XII, 20; Colebrooke, op. cit., p. 293; BCMS, 
XVIII (1927), 70. 

' Cf. the somewhat similar problem in Brahmagupta, XII, 15; 
Colebrooke, op. cit, pp. 287-28, and see the discussion of Kaye. 


The fonnula involves the solution of a quadratic 
equation in the form of ax^+2bx = c.^ 

A sum of money is loaned. After a certain unit of 
time the interest received is loaned for a known num- 
ber of units of time at the same interest. What is 
known is the amount of the interest on the principal 
plus the interest on this interest. Call this B. Let the 
principal be A. Let t be the time. 

x = 



The following example is given by Paramesvara. 
The sum of 100 is loaned for one month. Then the 
interest received is loaned for six months. At that 
time the original interest plus the interest on this 
interest amounts to 16. 

f = 100 "t^l6Xl0QX6+25Q0-50 __^Q 

The interest received on 100 in one month was 10. 

26. In the rule of three multiply the fruit by the desire and 
divide by the measure. The result will be the fruit of the desire. 

The rule of three corresponds to proportion. 
In the proportion a is to b as c is to a; the measure 
is a, the fruit is 6, the desire is c, and the fruit of the 

desire is x. 


1 See JBOBS, XII, 87. 

* See Brahmagupta, XII, 10 and Colebrooke, op. dt., p. 283. 

x=- 2 


27. The denominators of multipliers and divisors are multi- 
plied together. Multiply numerators and denominators by the 
other denominators in order to reduce fractions to a common 

For the first part of the rule I have given what 
seems to be the most likely hteral translation. The 
exact sense is uncertain. Kaye (agreeing with Rodet) 
translates, "The denominators are multiplied by one 
another in multiplication and division." If that is the 
correct translation the genitive plural is curious. 
Paramesvara explains gunakara as gunaguriyayor 
ahatir air a gunakdrasahdena vivaksitd. hdrya ity arthah 
and then seems to take bhdgahdra as referring to a 
fractional divisor of this product. Can the words bear 
that construction? In either case the inversion of 
numerator and denominator of the divisor would be 
taken for granted. 

It is tempting to take gunakdrahhdgahdra as mean- 
ing "fraction" and to translate, "The denominators 
of fractions are multiplied together." But for that 
interpretation I can find no authority. 

28. Multipliers become divisors and divisors become multi- 
pliers, addition becomes subtraction and subtraction becomes 
addition in the inverse method. 

The inverse method consists in beginning at the 
end and working backward. As, for instance, in the 
question, "What number multiphed by 3, divided by 
5, plus 6, minus 1, will give 5?" 

29. If you know the results obtained by subtracting suc- 
cessively from a sum of quantities each one of these quantities 
set these results down separately. Add them all together and 
divide by the number of terms less one. The result will be the 
sum of all the quantities. 


The translation given by Kaye is incorrect. The 
revised translation given in his Indian Mathematics, 
page 47, is not an improvement.^ 

According to the rule 3a+3&+3c+3(i 

X— d=a+6+c 

, , , J since 4x=4a+4&+4c+4d. 

x—c = a+o+a 

30. One should di\-ide the difference between the pieces of 
money possessed by two men by the difference between the ob- 
jects possessed by them. The quotient ■ndll be the value of one of 
the objects if the wealth of the two men is equal. 

Two men possess each a certain number of pieces 
of money (such as rupees) and a certain number of 
objects of merchandise (such as cows). 

Let a and 6 be the number of rupees possessed by 
two men, and let m and p be the number of cows 
possessed by them. 

h—a . , , , 

x= smce mx-\-a = v^+o . 


If one man has 100 rupees and 6 cows and the 
other man has 60 rupees and 8 cows the value of a 
cow is 20 rupees provided the wealth of the two men 
is equal. 

31. The two distances between two planets moving in 6ppo- 
site directions is divided by the sum of their daily motions. The 
two distances between two planets moving in the same direction 
is divided by the difference of their daily motions. The two results 
(in each case) wiU give the time of meeting of the two in the past 
and in the future. 

1 Cf. JBORS, XII, 88-90. 


In each case there will be two distances between 
the planets, namely, that between the one which is 
behind and the one which is ahead, and, measiu-ing 
in the same direction, the distance between the one 
which is ahead and the one which is behind. This 
seems to be the only adequate interpretation of the 
word dve. The translations of Rodet and Kaye fail 
to do full justice to the word dve.^ 

The next two stanzas give a method for the solu- 
tion of indeterminate equations of the first degree; 
but no help for the interpretation of the process in- 
tended, which is only sketchily presented in Arya- 
bhata, is to be found in Mahavira, Bhaskara, or the 
second Aryabhata. The closest parallel is foimd in 
Brahmagupta, XVIII, 3-5.2 The verbal expression is 
very similar to that of Aryabhata, but with one im- 
portant exception. In place of the enigmatic state- 
ment matigunam agrdntare ksiptam, ''(The last re- 
mainder) is multipUed by an assumed number and 
added to the difference between the agras," Brahma- 
gupta has, "The residue (of thfe reciprocal division) is 
multiplied by an assumed number such that the 
product having added to it the difference of the 
remainders may be exactly divisible (by the residue's 
divisor). That multipher is to be set down (under- 
neath) and the quotient last." It is possible that this 
same process is to be understood in Aryabhata. 

^ Cf. Paramesvara, dve Hi vacanam antarasya dvaividhyat. 
iighragalihino mandagatir antaram bhavaii. numdagatihlnas slghragatis 
cdntaram hhavati. Hi dvaividhyam and his further interpretation of the 

Cf. Brahmagupta, IX, 5-6 and Bhaskara, Gav-itSLdhydya, Gra- 
hayviyadhikara, 3-4, and Vasanabha^ya, and see J A (1878), 1, 28. 

* Colebrooke, op. cit., p. 325. 


First I shall explain the stanza on the basis of 
Paramjesvara's interpretation and of Brahmagupta's 
method : 

32-33. Divide the dmsor which gives the greater agra by the 
divisor -which gives the smaller agra. The remainder is reciprocal- 
ly di^'ided (that is to say, the remainder becomes the di\isor of 
the original di\'isor, and the remainder of this second di\-ision 
becomes the di-\dsor of the second di-visor, etc.)- (The quotients 
are placed below each other in the so-called chain.) (The last 
remainder) is multiplied by an assumed number and added to the 
difference between the agras. Multiply the penultimate number 
by the number above it and add the number which is below it. 
(Continue this process to the top of the chain.) Divide (the top 
number) by the di\'isor which gives the smaller agra. Multiply 
the remainder by the di\'isor which gives the greater agra. Add 
this product to the greater agra. The result is the number which 
will satisfy both divisors and both agras. 

In this the sentence, "(The last remainder) is 
multiplied by an assumed number and added to the 
difference between the agras," is to be understood as 
equivalent to the quotation from Brahmagupta given 

The word agra denotes the remainders which con- 
stitute the provisional values of x, that is to say, 
values one of which will satisfy one condition, one of 
which will satisfy the second condition of the prob- 
lem. The word dviccheddgra denotes the value of x 
which will satisfy both conditions. 

I cannot agree with the translation given by Kaye 
(and followed by Mazumdar, BCMS, III, 11) or 
accept the method given by Kaye. Kaye's transla- 
tion of matigunam agrdntare ksiptam, "An assum.ed 
number together with the original difference is thrown 
in," is an impossible translation, and any method 


based on that translation is bound to be incorrect. It 
omits altogether the important word gunam (".multi- 
plied"). Since the preceding phrase dealt with the 
remainders of the reciprocal division, the natural 
word to supply with matigunam seems to be sesam 
("remainder"). Something has to be supplied, and 
Brahmagupta's method offers a possible interpreta- 
tion. A second possible interpretation, which will be 
given below, would supply "quotient" instead of 

The following example is given by Paramesvara. 

Sa; 17x 

-TTT gives a remainder of 4 -rz- gives a remainder of 7. 

These are equivalent to ^^ =y and — 7^ = 2 or 8a;— 

29 45 

292/ =4 and l7x-45«=7 

where y and 2 are the quotients of the division (y and 2 to be 
whole numbers). 

1. First process. To find a value of x which will satisfy the first 





Take an assumed number such that multiplied by 
1 (the last remainder of the reciprocal division) and 


plus or minus 4 (the original remainder) it Tvill be 
exactly di\'isible by 2 (the last divisor of the recip- 
rocal division). 

6 is taken because —^ — 1. 

Therefore 6 and 1 are to be added to the quotients 

to form the chain. 

3 73 29)73(2 

1 20 58 

1 13 15 

1 7 This remainder 15 is the agra, that is to say, a value of x 

6 which win satisfy the equation. 


2. Second process. To find a value of x which ^sill satisfy the 
second equation: 





Take an assumed number such that multiplied 
by 1 (the last remainder of the reciprocal division) 
and plus or minus 7 (the original remainder) it will be 
exactly divisible by 5 (the last divisor of the reciprocal 
division) . 

3 is taken because —zr- =2. 


Therefore 3 and 2 are to be added to the quotients to 
form the chain. 

2 34 34)45(1 
1 13 34 

1 8 — 
15 11 

3 This remainder 11 is the agra, that is to say, a Talue of x 

2 which mil satisfy the equation. 

These numbers 15 and 11 are the agras mentioned 
at the beginning of the rule. The corresponding 
divisors are 29 and 45. The difference between the 
agras is 4, i.e., 15 — 11. 

3. Third process. To find a value of x which wiQ satisfy both 





Take an assumed number such that multiphed by 
1 (the last remainder of the reciprocal division) and 
plus or minus 4 (the difference between the agras) it 
will be exactly divisible by 3 (the last divisor of the 
reciprocal division). 

2 is taken because -~— =2. 


Therefore 2 and 2 are to be added to the quotients 
to form the chain. 













Therefore 34 is the remainder, 


Then in accordance with the rule 34X29 = 986 
and 986+15 = 1001 

This number 1001 is the smallest number which 
will satisfy both equations. 

Strictly speaking, the rule applies only to the third 
process given above. The solution of the single inde- 
terminate equation is taken for granted and is not 
given in full. There is nothing to indicate how far 
the reciprocal division was to be carried. Must it be 
carried to the point where the last remainder is 1? 
Must the number of quotients taken to make the 
chain be even in number? 

On page 50 of Kern's edition a 1 has been omitted 
by mistake (twice) as the fourth member of the chains 

The following method was partially worked out by 
Mazimadar/ who was misled in some details by fol- 
lowing Kaye's translation, and by Sen Gupta,^ and 
fully worked out by Sarada Kanta Ganguly.^ 

1 BCMS, III, 11-19. 

* Journal of the Department of Letters (Calcutta University), XVI, 

3 BCMS, XIX (1928), 170-76. 


According to Gangulj^'s interpretation, the trans- 
lation would be : 

32-33. Di\-icle the di\isor corresponding to the greater re- 
mamder by the di\-isor corresponding to the smaller remainder. 
The remainder (and the divisor) are reciprocally divided. (This 
process is continued until the last remainder is 0.) (The quotients 
are placed below each other in the so-called chain.) Multiply 
any assumed number by the last quotient of the reciprocal 
division and add it to the difference between the two remainders. 
(Interpreted as meaning that this product and this difference are 
placed in the chain beneath the quotients.) Multiply the penulti- 
mate number by the number above it and add the number which 
is below it. (Continue this process to the top of the chaia.) 
Divide (the lower of the two top numbers) by the di-visor cor- 
responding to the smaller remainder. Multiply the remainder 
by the di-\dsor corresponding to the greater remainder. Add 
the product to the greater remainder. The result is the (least 
number) which will satisfy the two divisors and the two re- 

He remarks : 

The impUcation is that the least number satisfying the given 
conditions can also be obtained by multiplying the remainder, 
obtained as the result of division of the upper number by the 
divisor corresponding to the greater given remainder, by the 
di-visor corresponding to the smaller given remainder and then 
adding the smaller remainder to the product. 

From this point of view the problem would be 
that of finding a number which will leave given 
remainders when divided by given positive integers. 

For example, to take a simple case: What num- 
ber divided by 3 and 7 will leave as remainders 2 
and 1? 

Following the rule literally, even though a smaller 


number has to be divided by a larger number we get 
the following: 




Multiply the last quotient (3) by an assumed 
number (for instance, 3) and set this product and the 
difference between the remainders 2 and 1, i.e., (1) 
down below the quotients to form the chain. 

28 7)65(9 

2 65 63 

3 28 — 
9 2 

Then 2X3 = 6 and 6+2 = 8. 

or 3)28(9 



Then 1X7=7 and 7+1=8. 

Therefore 8 is the number desired. 

The two methods attach different significations to 
the word agra and supply different words with bhajite 
in the third line ("remainder," ia one case; "quo- 
tient," in the other). They differ fundamentally in 
their interpretations of the words matigunam agra- 


ntare k^iptam. In the first method it is necessary to 
supply much to fill out the meaning, but the transla- 
tion of these words themselves is a more natural one. 
In the second method it is not necessary to supply 
anything except "quotient" with matigunam (in the 
first method it is necessary to supply "remainder"). 
But if the intention was that of stating that the 
product of the quotient and an assumed number, and 
the difference between the remainders, are to be 
added below the quotients to form a chain the 
thought is expressed in a very curious way. Gangtdy 
finds justification for this interpretation (p. 172) in 
his formulas, but I cannot help feeling that the San- 
skrit is stretched in order to make it fit the formula. 
The general method of solution by reciprocal 
division and formation of a chain is clear, but some 
of the details are uncertain and we do not know to 
what sort of problems Aryabhata appHed it. 



1. A year consists of twelve months. A month consists of 
thirtj' days. A day consists of sixty nad'is. A m<fi consists of 
sixty vinddikas} 

2. Sixty long letters or six prdnas make a sidereal vinadikd. 
This is the division of time. In like manner the di\ision of space 
beginning with a revolution.^ 

3. The difference between the number of revolutions of two 
planets in a yuga is the number of their conjunctions. 

Twice the sum of the revolutions of the Sun and Moon is the 
number of vyatipdtas.^ 

This is a yoga of the Sun and Moon when they are 
in different ayanas, have the same dechnation, and 
the sum of their longitudes is 180 degrees. 

4. The difference between the number of revolutions of a 
planet and the number of revolutions of its ucca is the number of 
revolutions of its epicycle. 

The number of revolutions of Jupiter multipUed by 12 are 
the years of Jupiter beginning with Asvayuja.* 

1 Cf. SuryasiddMnta, I, 11-13; Albenmi, I, 335; Bhattotpala, p. 

2 Cf. Suryasiddharda, I, 11, 28; Bhattotpala, p. 24; Pancasi- 
ddhdntika, XIV, 32, for the first part, of 2; Brahmagupta, I, 5-6, and 
Blmskara, Ganitadhaya, Kdlamanddhydya, 16-18, for both stanzas. 

^ See Lalla, Madhyamddhik6ra, 11; Brahmagupta, XIII, 42, for 
the first part. For vyatlpata see Suryasiddhdnta, XI, 2; Pancad- 
ddhantika, III, 22; Lalla, Mahapatddhikdra, 1; Brahmagupta, XIV, 
37, 39. 

*For the first part see Lalla, Madhyamddhikara, 11; Brahma- 
gupta, XIII, 42; Bhaskara, Gai^itadhyaya, Bhaganadhyaya, 14. For 
the second part see JRAS, 1863, p. 378; iUd., 1865, p. 404; 
Suryasiddharda, I, 55; Bhattotpala, p. 182. 



The vrord ucca refers both to mandocca ("apsis") 
and sighrocca ("conjunction"). 

Paramesvara explains that the number of revolu- 
tions of the epicycle of the apsis of the Moon is equal 
to the difference between the number of revolutions 
of the Moon and the revolutions of its apsis; that since 
the apsides of the six others are stationary, the number 
of revolutions of the epicycles of their apsides is 
equal to the number of revolutions of the planets; 
and that the number of revolutions of the epicycles 
of the conjunctions of Mercury, Venus, Mars, 
Jupiter, and Saturn is equal to the difference between 
the revolutions of the planets and the revolutions of 
their conjunctions. 

As pointed out in the note to I, 7, the apsides were 
not regarded by Aryabhata as being stationary in the 
absolute sense. They were regarded by him as sta- 
tionary for purposes of calculation at the time when 
his treatise was composed since their movements were 
very slow. 

5. The revolutions of the Sun are solar years. The conjunc- 
tions of the Sun and Moon are lunar months. The conjunctions 
of the Sun and the Earth are [civil] days. The revolutions of the 
asterisms are sidereal days. 

The word yoga applied to the Sun and the Earth 
(instead of bhagana or avarta) seems clearly to indi- 
cate that Aryabhata believed in a rotation of the 
Earth (see IV, 48). Paramesvara's explanation, ravi- 
bhuyogasahdena raver bhuparibhramaiiam abhihitam, 
seems to be impossible. 

6. Subtract the solar months in a yitga from the lunar months 
in a yxiga. The result will be the number of intercalary months in 


a yuga. Subtract the natural [chil] days in a yiiga from the lunar 
days in a yu^a. The result ■will be the number of omitted lunar 
days in a yuga.^ 

7. A solar year is a year of men. Thirty of these make a year 
of the Fathers. Twelve years of the Fathers make a year of the 

S. Twelve thousand years of the gods make a yuga of all the 
planets. A thousand and eight yiigas of the planets make a day 
of Brahman.'*- 

9. Thefirsthalf of a jTiga is called wfsorpim [ascending]. The 
latter half is called avasarpinl [descending]. The middle part of a 
yuga is called summa. The beginning and the end are called 
dus^ama. Because of the apsis of the Moon. 

Alberuni (I, 370-71) remarks: 

Aryabhata of Kusumapura, who belongs to the school of the 
elder Aryabhata, says in a small book of his on Al-nij (?), that 
"1,008 caturyugas are one day of Brahman. The first half of 504 
caiuryugas is called utsarpim, during which the sun is ascending, 
and the second half is called avasarpinl, during which the sun is 
descending. The midst of this period is called sama, i.e., equality, 
for it is the midst of the day, and the two ends are called durtama 


"This is so far correct, as the comparison between day and 

kalpa goes, but the remark about the sun's ascending and 
descending is not correct. If he meant the sun who makes our 
day, it was his duty to explain of what kind that ascending and 
descending of the sim is; but if he meant a sun who specially be- 
longs to the day of Brahman, it was his duty to show or to 
describe him to us. I almost think that the author meant by these 
two expressions the progressive, increasing development of 
things during the first half of this period, and the retrograde, 
decreasing development in the second haK." 

1 Cf. Suryasiddhdnta, I, 35-36; Lalla, MadhyamMhikdra, 10; 
Brahmagupta, I, 24 and XIII, 26. 

»Cf. Suryasiddhanta, I, 13-15; I, 20. Brahmagupta (I, 12) 
criticizes Aryabhata's figure of 1,008 yugas instead of 1,000 yugas. Cf . 
JBAS, 1866, p. 400. Cf. also I, 3 and see JBAS, 1911, p. 486. 


The reference is to the foregoing stanza. The mid- 
dle of the yuga seems to be called susamd (''even") 
because good and bad are evenly mixed. The begin- 
ning and the end are called dussamd, ("uneven") be- 
cause in one case goodness and in the other case 
badness predominates. 

Paramesvara remarks that the vydkhyakara has 
given no explanation. Then he quotes from the 
Bhataprakdsikd a statement to the effect that our 
text refers to the increase and decrease of men's Hves 
in the course of a yuga and a criticism (asydrtho 
'bhiyuktair nirupya vaktavyah) of the last phrase of 
the stanza. He then continues by saying that he does 
not see what meaning can be intended by the word 
induccat, and adds that the word has nothing to do 
with the matter under discussion, has no significance 
for the calculation of the places of the planets. Then 
he adds two forced explanations. The meaning of 
induccdt is quite uncertain. 

Sudhakara (Ganakatarangini, p. 7) suggests the 
emendation to agnyamsdt. 

The terminology is distinctively Jaina.^ 

10. When three yugapddas and sixty times sixty years had 
elapsed (from the beginning of the ytiga) then twenty-three years 
of my hfe had passed." 

If Aryabhata began the KaHynga at 3102 b.c. as 
later astronomers did, and if his fourth yugapdda 

^See Hemacandra, Abhidhdnacintamazd, 128-35; Glasenapp, 
Der Jainismus, pp. 244-45; Kirfel, Kosmographie, p. 339; ZDMG, 
LX, 320-21 ; Stevenson, The Heart of Jainism, pp. 272 ff . See also 
Hardy, Manual of Buddhism, p. 7. 

2 See JRAS, 1863, p. 387; ibid., 1865, p. 405; Kern, Brhat 
Samhita, Preface, p. 57; JRAS, 1911, pp. 111-12. 


began mth the beginning of the Kaliyuga, we arrive 
at the date 499 a.d. It is natural to take this as the 
date of composition of the treatise. Paramesvara 
quotes the Prakdsikakdra to the effect that this is to 
be taken as the date at which the calculations of the 
true places of the planets made by it would be correct, 
and that for later times a correction would have to be 

The word iha may mean "here" or "now." 
Paramesvara takes it as referring to this present 
twenty-eighth caturyuga. 

11. The yuga, the year, the month, and the day began all 
together at the beginning of the bright fortnight of Caitra. Time, 
which has no beginning and no end, is measured by (the move- 
ments of) the planets and the asterisms on the sphere. 

Bhau Daji^ first pointed out the criticism made 
of this stanza by Brahmagupta (XI, 6) : 
jTigavarsadin vadata caitrasitadeh samam pra\'rttan yat| 
tad asad yata^^ sphutayugam tat sthairyan mandapatanam.|| 

Compare Brahmagupta, I, 4, and Bhaskara, Ganitd- 
dhydya, Kdlamdnddhydya, 15, who refers to an earlier 
commentary in which time is called endless.^ 

12. The planets moving equally (traversing the same distance 
in yojanas each day) in their orbits complete the circle of the 
asterisms in sixty solar years, and the circle of the sky in a divine 
age [caturyv^a]. 

In sixty years they move a distance in yojanas 
equal to the circle of the asterisms. In a caturyuga 
they move a distance in yojanas equal to the cirexun- 
ference of the sky (akdsakaksyd) (cf. I, 4). 

1 JBAS, 1865, p. 401. 

s For discussion of the stanza see Fleet, ibid., 1911, pp. 489-90; 
cf. I, 2. 


The planets really all move at the same speed. 
The nearer ones seem to move more rapidly than the 
more distant ones because their orbits are smaller. ^ 

13. The Moon, being below, completes its small orbit in a 
short time. Saturn, being above all the others, completes its large 
orbit in a long time.^ 

14. The zodiacal signs (a twelfth of the circle) are to be 
known as small in a small circle and large in a large circle. Like- 
\sise the degrees and minutes are the same in number in the 
various orbits.^ 

15. Beneath the asterisms are Saturn, Jupiter, Mars, the Sun, 
Venus, Mercury, and the Moon, and beneath these is the Earth 
situated in the center of space like a hitching-post.* 

16. These seven lords of the hours, Saturn and the others, 
are in order swifter than the preceding one, and coimting suc- 
cessively the fourth in the order of their swiftness they become 
the Lords of the days from sumise. 

They are called "swifter than the preceding" be- 
cause their orbits being successively smaller they 
complete their revolutions in less time (traverse a 
given number of degrees in less time). The order of 
the planets is Saturn, Jupiter, Mars, the Sun, Venus, 
Mercury, and the Moon. Therefore they become 
rulers of the days of the week as follows: 

1 Cf. Suryasiddhanta, I, 27 and note; Brahmagupta, XXI, 12; 
Pancasiddhantika, XIII, 39; Bhaskara, Goladhydya, Bhuvanakosa, 
69; JBAS, 1911, p. 112. 

» Cf. JRAS, 1863, p. 375; Suryasiddhanta, XII, 76-77; Pancasi- 
ddhantika, XIII, 41; Brahmagupta, XXI, 14; Bhattotpala, p. 45. 

» Cf. JRAS, 1863, p. 375; Pancasiddhantika, XIII, 40; Suryasi- 
ddhanta, XII, 75; Brahmagupta, XXI, 14; Bhattotpala, p. 45. 

* Cf. JRAS, 1863, p. 375; Pancasiddhantika, XIII, 39; Lalla, 
MadhyarmdhikSra, 12; Brahmagupta, XXI, 2; Bhattotpala, p. 44. 


Saturday — Saturn Wednesday — Mercury 

Sunday — Sun Thursday — ^Jupiter 

Monday — Moon Friday — ^\'enus 
Tuesday — jSIars 

For the first part see Brahmagnpta, XXI, 13; 
Suryasiddhdnta, XII, 78.^ 

Bhau Dajl^ first pointed out tlie criticism of this 
stanza made by Brahmagupta (XI, 12) : 
stiryadayas caturtlia dinavara yad uvaca tad asad arj-abhatahj 
lankodaye yato 'rkasyastamayam praha siddhapure. || 

As Sudhakara shows, the criticism is a futile one. 

17. AH the planets move by their (mean) motion on their 
orbits and their eccentric circles from the apsis eastward and 
from the conjunction westward.^ 

The mean planet moves with, its mean motion on 
its orbit the center of which is the center of the Earth. 
The true planet moves with its (mean) motion on an 
eccentric circle the center of which does not coincide 
with the center of the Earth. 

Kaksyd in this passage stands for kaJcsydmandala, 
the orbit on which the mean planet moves. The pra- 
timandala is the eccentric circle on which the true 
planet moves. Because of the eccentricity of this 
second circle the planet .is sometimes seen ahead of 
and sometimes back of its mean place.'* 

1 See Barth {CEuvres, III, 151) concerning this as the only refer- 
ence to astrology in Aryabhata's treatise. The reference to vyailpata 
(III, 5) should be added. 

2 JRAS, 1865, p. 401. 

= See Lalla, ChedyaJcadhilcara, 12-13; Brahmagupta, XIV, 11 and 
XXI, 24. 

*See Brennand, Hindu Astronomy, pp. 224 fl.; Surijasiddh&nUi, 
p. 64. 


18. The eccentric circle of each planet is equal to its hah^ya- 
mai), [the orbit on -vrhich the mean planet moves]. The center 
of the eccentric circle is outside the center of the solid Earth. 

The kaksydmandala is determined by I, 4. 

19. The distance between the center of the Earth and the 
center of the eccentric circle is equal to the radius of the epicycle. 
The planets move -n-ith their mean motions on their epicycles.^ 

Brahmagupta, XI, 52, has 

nIcocca\Tttamadhyasya golabahyena nama krtam uccam | 
tatstho na bhavati ucco yatas tato vetti noccam api. || 

If this really refers to Aryabhata the criticism is futile 
since Aryabhata does not call the center of the epi- 
cycle ucca. As Caturvedacarya says, vdgbalam etat 

20. The planet in its swift motion from its ucca has a pra- 
tiloma motion on its epicycle. In its slow motion from its iicca it 
has an anuloma motion on its epicycle. 

The exact meaning of this is not clear to me. It 
can hardly mean that the planet moves on its epicycle 
pratiloma from its slghrocca and anuloma from its 
mandocca. On the epicycle of the apsis the motion 
should be exactly the reverse of these.^ 

Anulomajnesuas "eastward" or "ahead" ; Pratiloma 
means ''westward" or "behind." 

Paramesvara remarks that anuloma and pratiloma 
refer to the planet's position with reference to the 
mean planet as ahead of it or behind it. He also re- 
marks that the planet is sighragati in the six signs 

' Cf. Lalla, Chedyakadhikara, 8-9; Brahmagupta, XIV, 10 and 
XXI, 24r-26. 

' See Brahmagupta, XXI, 25-26 and Suryasiddhanta, pp. 63-64, 


which are above, and mandagati in the six signs which 
are below the ucca. When pratiloma the true planet 
is below the mean planet. When anuloma the true 
planet is above the mean planet.^ 

madhyamakaksa^Ttte madhyama3'a gacchati graho gatya | 
uparisthat tallaghvya tadadMkagatya tv adhahsthah syat. jj 

Paramesvara sums up the content of the stanza with 
madhyamdt sphutasya pratilomdnulomagatitvam uktam. 
The meaning of the stanza seems to be that during 
half of its revolution on its epicycle the planet is 
ahead of the mean planet and during half of its 
revolution is behind the mean planet. 

21. The epicycles move eastward from the apsis and west- 
ward from the conjunctions. The mean planet, situated on its 
orbit, appears at the center of its epicycle.^ 

The next three stanzas state briefly the method of 
calculating the true places of the planets from their 
mean places. 

Paramesvara explains the method as follows: 

For the Sun and Moon only one process of cor- 
rection is required, that for the apsis. 

For Mars, Jupiter, and Saturn four processes are 
necessary: (1) From the mean place the mandaphala 
is calculated and (half of it is) applied to the mean 
place. (2) From this corrected place the sighraphala 
is calculated and half of it is applied to the corrected 
place. (3) From this result the mandaphala is again 
calculated and applied to the mean place. (4) From 
this result the sighraphala is again calculated and 
applied to the place obtained in the third process. 

1 Cf . Lalla, Bhuvanakosa, 38. » Cf . Brahmagupta, XXI, 25. 


For Venus and Mercury three processes are nec- 
essary: (1) From the mean place the sighraphala is 
calculated and half of it is applied vyastam (in reverse 
order) to the mandocca (apsis). (2) This corrected 
mandocca is subtracted from the mean place, the 
mandaphala is calculated from this and applied to the 
mean place. (3) From this corrected place the sighra- 
phala is calculated and applied to the place obtained 
in the second process. 

22-23. (The corrections) from the apsis are minus, plus, plus, 
minus (in the four quadrants). (The corrections) from the con- 
junctions are just the reverse. 

In the case of Saturn, Jupiter, and Mars in the first process 
half of the mandaphala obtained from the apsis is minus and plus 
to the mean planet. Half (the correction) from the conjunction is 
minus and plus to the manda planets. (By applying the correc- 
tion) from the apsis they become sphutamadhya. (By applying 
the correction) from the conjunction they become sphuta. 

24. Half (the correction) from the conjunction is to be ap- 
plied minus and plus to the apsis. (By applying the correction) 
from the manda [apsis] thus obtained Venus and Mercury become 
sphutamadhya. They become sphuta (by applying the correction 
from the conjunction). 

The first half of stanza 22 gives the general rule as 
to whether the equations of anomaly and of com- 
mutation (mandaphala and sighraphala) are to be 
added or subtracted in each of the four quadrants. 
The equation from the apsis is minus in the half of 
the orbit beginning with Mesa, plus in the half of the 
orbit beginning with Tula. The equation from the 
conjunction is plus in the half of the orbit beginning 
with Mesa, minus in the half of the orbit beginning 
with Tula. 


The planet is called manda after the first correc- 
tion from the apsis has been apphed to the mean 
place. Sphuta means "true." 

In stanza 24 Paramesvara gives no explanation of 
the two last words, sphutau hhavatah. It would be 
natural to take these words as summing up what 
precedes and to understand that only two processes 
are involved. But Paramesvara's detailed description 
of the process in his commentary to stanza 21 indi- 
cates that three processes are involved, that sphutau 
hhavatah indicates a further application of the equa- 
tion from the conjunction. The commentary to 
stanza 24 gives in detail the process of calculating the 
equations for apsis and conjunction.^ 

Brahmagupta (II, 19, 33, 46-47) criticizes Arya- 
bhata for the inaccuracy of his method of calculating 
the true places. 

25. The product of its hypotenuses di\ided by the radius wHl 
give the distance between the planet and the Earth. 

The planet has the same speed on its epicycle that it has on 
its orbit. 

Paramesvara explains that the karnas referred to 
are the slghrakarv^ and the mandaharna employed 
in the last and the next to the last processes for 
calculating the true places of the planets. 

The second half of the stanza is uncertain. This 
same statement was made in unmistakable terms in 

^See Pancasiddkantikd, XVII, 4-10; SHryasiddhanta, II, -43-45; 
Brahmagupta, II, 34r-40; Lalla, Spa§iadhiMra, 31-36; Bhaskara, 
Gavitadhyaya, Spa^tadhik&ra, 34-36 and GolSdhyaya, Chedyaka- 
dhikara, 10 ff.; JRAS, 1863, pp. 353-59; Brennand, op. dt., pp. 214- 
28; Kaye, Hindu Astronomy, pp. 87-89. 


III, 19. Paramesvara quotes the author of the earlier 
Prakasika, hhutaragrahavivaravyasdrdhaviracAtaydrh 
kdksyayarh yo grahasya javas sa mandanlcocce bhavati. 
tdvatpramandydm kaksydydm graho mandasphutagatyd 
gacchatlty arthah. ity aha. asmdn kirn tv etan nopa- 
pannam iti pratibhdti. Then he explains that the 
meaning may be that the radius of the epicycle is 
equal to the greatest distance by which the mean 
orbit lies inside or outside of the eccentric circle. 

Grahavegah is reminiscent of grahajavah in I, 4 
but the meaning can hardly be the same. 

Karna ("hypotenuse") is the distance between the 
center of the Earth and the planet.^ 

1 Cf. Brahmagupta, XXI, 31; Bhaskara, Gariitadhyaya, Candra- 
graharicLdhikara, 4—5; SUryasiddhanta, p. 69. 


1. From the beginning of Me§a to the end of Kanya is the 
northern half of the ecUptic. The other half from the beginning 
of Taulya to the end of Mina is the southern half of the echptic. 
Both deviate equally from the Equator. 

Therefore the greatest declinations north and 
south are equal, and the declinations of the first three 
signs in each half are equal to the declinations of the 
last three signs taken in reverse order.^ 

2. The Sun, the nodes of the planets, and the node of the 
Moon move constantly along the echptic. The shadow of the 
Earth moves along the echptic at a distance of 180 degrees from 
the Sun. 

Bhau Daji'- first pointed out the reference to this 
passage made by Brahmagupta, XI, 8.^ 

Barth^ questions the stanza, but without good 

3. The Moon, from its nodes, moves northward and south- 
ward of the ecliptic. Likewise Jupiter, Mars, and Saturn. Venus 
and Mercury do the same from their conjunctions.* 

4. When the Moon has no dechnation it is visible when 12 
degrees from the Sun. Venus when 9 degrees. The other planets 

1 Cf. JBAS, 1863, p. 374; Bhattotpala, p. 45. 
2Ji2A/S, 1865,p. 401. 

» Cf. I, 7 and note; Brahmagupta, XXI, 53; SuryasiddhdrUa, 
IV, 6. 

* (Euvres, III, 154. 

5 Cf . SHrya^ddhanta, I, 68-69 and AryabhaUya, I, 6. 



in succession according to their decreasing sizes when at 9 
degrees increased by two's. 

Compare Brahmagupta, VI, 6; Suryasiddhanta, 

IX, 6-9 and X, 1; Paficasiddhantikd, XVII, 12 and 

XVIII, 58. Bhau Dajl^ first pointed out the criticism 

of this stanza made by Brahmagupta, VI, 12 : 

aryabhatah ksetramsair dfsyadrsyan yad nktavams tad asat | 
drggaioitavisamvadad drgganitaikyam svakalamsaih. || 

5. Half of the spheres of the Earth, the planets, and the 
asterisms is darkened by their shadows, and haK, being turned 
toward the Sun, is light (being small or large) according to their 

6. The sphere of the Earth, being quite round, situated in the 
center of space, in the middle of the circle of asterisms, surrounded 
by the orbits of the planets, consists of water, earth, fire, and air.' 

7. Just as a ball formed by a Kadamba flower is surrounded 
on all sides by blossoms just so the Earth is surrounded on all 
sides by all creatures terrestrial and aquatic* 

8. During a day of Brahman the sphere of the Earth increases 
a yojana in size all around. During a night of Brahman, which is 
equal in length to a day of Brahman, there is a decrease by the 
same amount of the Earth which has been increased by Earth.* 

9. As a man in a boat going forward sees a stationary object 
moving backward just so at Lanka a man sees the stationary 
asterisms moving backward (westward) in a straight line. 

The natural interpretation of this stanza seems 

1 JRAS, 1865, p. 401. 

2 Cf. Lalla, MadhyagaUvasamM, 40-41; Bhattotpala, p. 100; PaTV- 
casiddhantika, XIII, 35, for the Moon. 

'Cf. Ill, 15. Cf. Lalla, BhUgoladhyaya, 1; PancasiddhantiM, 
XIII, 1; Bhattotpala, p. 58 (and see JRAS, 1863, pp. 373-74); 
Alberuni, I, 268. 

* Cf. Lalla, BhUgoladhyaya, 6; Bhattotpala, p. 58 (and see JRAS, 
1863, pp. 373-74); Bhaskara, Goladhyaya, BhuvanakoSa, 3. 

^Cf. Lalla, GrahabhramasaThsthadhydya, 20; Bhaskaxa, Gola- 
dhyaya, Bhuvanako§a, 62. 


to be that an observer at the Equator of the Earth, 
which , rotates toward the East, sees the stationary 
celestial objects as though moving westward. But 
Paramesvara explains that whereas the Earth does 
not really move, it appears to move tow^ard the east 
because of the westward movement of the asterisms. 
He is forced to take the w^ords anuloma and viloma, 
which regularly mean "ahead," "eastward," and 
"backward," "westward," in exactly the opposite 
senses. He explains that persons on the asterisms, 
which move toward the west, would seem to see sta- 
tionary objects on the Earth moving eastward. As 
Barth^ points out, this explanation is quite unac- 
ceptable. It seems that Paramesvara completely mis- 
represents the opinion of Aryabhata, as clearly stated 
in several places in the text, and as described by 
Brahmagupta and other critics of Aryabhata. 

There is nothing to indicate that this stanza repre- 
sents a state of affairs caused by mithydjndna ("false 

Bhattotpala (pp. 58-59) quotes this stanza and 
then refutes it by quoting the Pancadddhantikd, 
XIII, 6-8, Pauhsa, Brahmagupta, and strangely 
enough Aryabhata himself (the following stanza, IV, 
10). It is curious that Aryabhata should be quoted 
against himself, and that Bhattotpala should not 
indicate clearly which view really represents Arya- 
bhata's own opinion. It looks as though Bhattotpala 
regarded the first stanza as containing a purvapaksa 
or erroneous view.^ 

1 Op. cit, III, 158 n. 2 cf . jrj^js^ iges, pp. 375-77. 


For criticisms of the rotation of the Earth see 
Alberuni (I, 276-77, 280); Lalla, MithyajMnd- 
dhydya, 42-43; Sripati as reported in the Lucknow 
edition of Bhaskara's Golddhyaya, page 83; see also 

Colebrooke^ quotes Prthtidaka the commentator 
on Brahmagupta as follows : 

bhapafijarati sthiro bhur evavftyavrtya pratidaivasikau | 

udayastamayau sampadayati nak|atragrahanam. ||' 

The Vdsandvdrttika to Bhaskara's Grahaganita, 
page 113,* quotes the foregoing stanza and remarks 
that according to Aryabhata the planets move toward 
the east, the asterisms are stationary, and the Earth 
rotates eastward. 

10. The cause of their rising and setting is due to the fact 
that the circle of the asterisms, together with the planets, driven 
by the provector wind, constantly moves straight westward at 

Bhattotpala (p. 59) quotes this stanza to disprove 
the preceding stanza which he quoted on page 58 (cf . 
JRAS, 1863, p. 377). 6 

The Marici (p. 43) to Bhaskara's Grahagardta^ 
quotes this stanza. 

1 Op. cit., Ill, 158. * Essays, II, 392. 

3 See JRAS, 1865, pp. 403-4; IHQ, I, 666 (the words given as a 
direct quotation from Aryabhata are incorrect) ; BCMS, XVII (1926), 
175. The author of the last article remarks that it is not clear 
whether Aryabhata had in mind the geocentric or the heliocentric 
motion of the Earth. The latter is out of the question. Cf . Ill, 15, 
bkumir medhlbhuta khamadhyastfia, and IV, 6, khamadhyagaiah,. 

* Pandit, Vol. XXXI. 

*Cf. Suryasiddhanta, II, 3: LaUa, Madhyamadhikdra, 12. 

8 See Earth, op. dt., Ill, 158. ' Pandit, Vol. XXX. 


The Vdsandvarttika to Bhaskara's Grahaganita (p. 
118)^ quotes this stanza apparently without seeing in 
it anything contradictory to the preceding stanza 
whichjvas quoted on page 113, and with the remark 
that Aryabhata is here following the opinion of 

If the readings of our text are correct it is difficult 
to see how the two stanzas can be brought into agree- 
ment. The ninth stanza states unequivocally that 
the asterisms are stationary and implies the rotation 
of the Earth. The tenth stanza seems to state that the 
asterisms, together with the planets, are driven by 
the pro vector wind. This would imply the ordinary- 
point of view of most Indian astronomers that the 
Earth was stationary. Paramesvara avoids the diffi- 
culty by assuming that stanza 9 describes a state of 
mind brought about by mithydjndna ("false knowl- 
edge"). But since several other stanzas (I, 1; I, 4; 
III, 5 ; IV, 48) and the testimony of later writers who 
quote Aryabhata prove that Aryabhata believed in 
the rotation of the Earth, it is impossible to follow 
Paramesvara. We might understand in stanza 10 the 
phrase "they seem to move" as stating a purvapaksa 
(the erroneous view), but in the absence of any word 
to suggest this interpretation it is a doubtful expedi- 
ent. Stanza 10 cannot be regarded as an interpolation 
(unless one stanza has been dropped out in order to 
make room for it) because the last three sections of 
Aryabhata's work were known to Brahmagupta as 
"The Hundred and Eight Stanzas" (and our text con- 
tains 108 stanzas). 

1 Ibid., Vol. XXXI. 


11. In the center of the Nandana forest is Mount Mem, a 
yojana in measure (diameter and height), shining, surrounded by 
the Himavat Moimtains, made of jewels, quite round.^ 

12. Heaven and Meru are at the center of the land, Hell and 
Vadavamukha are at the center of the water. The gods and the 
dwellers in Hell both think constantly that the others are beneath 

Quoted by Bhattotpala, page 58.' 

13. Sunrise at Laiika is sunset at Siddhapura, midday at 
Yavakoti, and midnight at Romaka.* 

Brahmagupta's criticism (XI, 12) 

suryadayas caturtha dinavara yad uvaca tad asad aryabhatat | 
lankodaye yato 'rkasyastamayam praha siddhapure || 

is incorrect, as pointed out by Sudhakara in his com- 

14. Lanka is 90 degrees from the centers of the land and 
water [north and south poles]. Ujjain is straight north of Laiika 
by 22§ degrees.^ 

15. From a level place half of the sphere of the asterisms 

1 Cf. I, 5. Cf. also Suryasiddhdnta, XII, 34; Lalla, Bhuvanako&a, 
18-19; Bhaskara, Goladhydya, Bhuvanakoia, 31; Alberuni (I, 244, 
246). Quoted by Bhattotpala, p. 58 (cf. JRAS, 1863, p. 373). 

2 Cf. Paficasiddhaniika, XIII, 2-3; SuryasiMhmta, XII, 35-36, 
53; Brahmagupta, XXI, 3; Lalla, BhugolSdhyaya, 3-4; Bhaskara, 
Goladhyaya, Bhuvanako&a, 17-20, 31. 

' Cf. JRAS, 1863, p. 373. 

* Cf. Kem, Brhat Samhita, Preface, p. 57; Suryasiddhanta, XII, 
38r-41; Panca^iddhdntika, XV, 23; Lalla, Bhugoladhyaya, 12; Bhas- 
kara, Goladhyaya, Bhmanako&a, 17, 44; Alberuni I, 267-68; JRAS, 
1865, p. 402. 

* Cf. SUryasiddMnta, I, 62; PancasiddhdnHka, XIII, 17; Lalla, 
MadhyamOdhikdra, 55, and Bhuvanakoia, 41; Brahmagupta, XXI, 
9; Bhaskara, Goladhydya, Bhuvanakoia, 50 {Vdsandhhd^a), and 
Madhyagati, 24; Alberuni, I, 316 (cf. BCMS, XVII [1926], 71). 


decreased by the radius of the Earth is \'isible. The other half, 
plus the radius of the Earth, is cut off by the Earth.^ 

16. The gods, who dwell in the north on Mem, see the 
northern half of the sphere of the asterisms mo\'ing from left to 
right. The Pretas, who dwell in the south at Vadavamukha, see 
the southern half of the sphere of the asterisms moving from 
right to left.2 

Quoted by Bhattotpala, page 324.' 

17. The gods and the Pretas see the Sun after it has risen for 
half a solar year. The Fathers who dwell in the Moon see it for 
half a Ixinar month. Here men see it for half a natural [ci'S'il] day.* 

Referred to by Albemni, I, 330. 

18. There is a circle east and west (the prime vertical) and 
another north and south (the meridian) both passing through 
zenith and nadir. There is a horizontal circle, the horizon, on 
which the heavenly bodies rise and set.* 

19. The circle which intersects the east and west points and 
two points on the meridian which are above and below the horizon 
by the amount of the observer's latitude is called the unmar),4ala. 
On it the increase and decrease of day and night are measured. 

The unmandala is the east and west hour-circle 
which passes through the poles. It is also called "the 
horizon of Lanka."^ 

1 Cf. LaUa, Bhuvanahosa, 36; Brahmagupta, XXI, 64; Bhaskara, 
GolSdhyaya, Tripra&nwasana, 38. 

* Cf. Suryasiddhdnta, XII, 55; Pancasiddhantika, XIII, 9; 
Brahmagupta, XXI, 6-7; Lalla, GrahabhramasaThsthadhySya, 3-5; 
Bhaskara, GolSdhyaya, Bhuvanako§a, 51. 

» Cf. JRAS, 1863, p. 378. 

* Cf. Suryadddhanta, XII, 74 and XIV, 14; Lalla, Gralvahhror 
masamsthadyaya, 14; Brahmagupta, XXI, 8; Pancasiddh&ntika, 
XIII, 27, 38. 

* Cf. Lalla, GolabandkSdhikdra, 1-2; Brahmagupta, XXI, 49. 

6 Cf. Lalla, Golabandkadhikara, 3; Brahmagupta, XXI, 50. 


20. The east and west line and the north and south line and 
the perpendicular from zenith to nadir intersect in the place where 
the observer is. 

21. The vertical circle which passes through the place where 
the observer is and the planet is the drhmandala. There is also 
the drkk§epamandala which passes through the nonagesimal 

The nonagesimal or central-ecliptic point is the 
point on the ecliptic which is 90 degrees from the 
point of the ecliptic which is on the horizon. 

These two circles are used in calculating the 
parallax in longitude in eclipses. 

22. A light wooden sphere should be made, round, and of 
equal weight in every part. By ingenuity one should cause it to 
revolve so as to keep pace with the progress of time by means of 
quicksilver, oil, or water.^ 

Sukumar Ranjan Das^ remarks that two instru- 
ments are named in this stanza (the gola and the 
cakra). I can see no reference to the cakra. 

23. On the visible half of the sphere one should depict half 
of the sphere of the asterisms by means of sines. 

The equinoctial sine is the sine of latitude. The sine of co- 
latitude is its kolti. 

The sine of the distance between the Sun and the 
zenith at midday of the equinoctial day is the equi- 
noctial sine. This is the same as the equinoctial 
shadow and equals the sine of latitude. It is the base. 

^ Cf. SuryasiddharUa, V, 6-7 n.; Kaye, Hindu Astronomy, p. 76. 

2 Cf. Suryasiddhanta, XIII, 3 ff.; Lalla, Yantradhyaya, 1 ff.; IHQ, 
IV, 265 ff . 

8/HO, IV, 259, 262. 


The sine of co-latitude is the koti (the side perpen- 
dicular to the base) or sanku (gnomon).^ 

24. Subtract the square of the sine of the given declination 
from the square of the radius. The square root of the remainder 
■will be the radius of the day-circle north or south of the Equator. 

The day-circle is the diurnal circle of revolution 
described by a planet at any given declination from 
the Equator. So these day-circles are small circles 
parallel to the Equator.^ 

• 25. Multiply the day-radius of the circle of greatest declina- 
tion (24 degrees) by the sine of the desired sign of the zodiac 
and divide by the radius of the day-circle of the desired sign of 
the zodiac. The result -will be the equivalent in right ascension of 
the desired sign beginning with Me§a. 

To determine the right ascension of the signs of 
the zodiac, that is to say, the time which each sign 
of the ecliptic will take to rise above the horizon at 
the Equator.3 

26. The sine of latitude multiplied by the sine of the given 
declination and divided by the sine of co-latitude is the earth- 
sine, which, being situated in the plane of one's day-circle, is the 
sine of the increase of day and night. 

The earth-sine is the distance in the plane of the 
day-circle between the observer's horizon and the 

1 Cf. Brahmagupta, III, 7-8; Lalla, SamanyagolabandJm, 9-10; 
BhSskara, Ganitadhydya, Triprasnadhikara, 12-13. 

2 Cf . Lalla, Spa^iadhikara, 18; Pancasiddhantiha, IV, 23; Surya- 
siddhanta, II, 60; Brahmagupta, II, 56; Bhaskara, Gav-itSdhyaya, 
Spa§tadhikara, 48 {Vasanabha^a); Kaye, op. cii., p. 73. 

3 Cf. Lalla, Triprainadhikara, 8; Brahmagupta, II, 57-58; 
Suryasiddhanta, III, 42-43 and note; PaficasiddhantiM, IV, 29-30; 
Bhaskara, Gamiadhyaya, Spa§tadhikara, 57; Kaye, op. cit., pp. 79-80. 


horizon of Lanka (the unmandala). When trans- 
formed to the plane of a great circle it becorrfes the 
ascensional difference.^ 

27. The first and fourth quadrants of the ecliptic rise in a 
quarter of a day (15 ghatikds) minus the ascensional difference; 
The second and third quadrants rise in a quarter of a day plus 
the ascensional difference, with regular increase and decrease. 

The last phrase means that the values for signs 
1, 2, 3 are equal, respectively, to those of signs 6, 5, 4 
and that the values of 7, 8, 9 are equal, respectively, 
to those of 12, 11, 10. They increase in the first 
quadrant, decrease in the second, increase in the 
third, and decrease in the fourth. There are, there- 
fore, only three numerical values involved, those cal- 
culated for the first three signs. See the table given in 
Siiryasiddhdnta, III, 42-45 n.^ 

28. The sine of the Sun at any given point from the horizon 
on its day-circle multiplied by the sine of co-latitude and divided 
by the radius is the sahku when any given part of the day has 
elapsed or remains. 

The sanku is the sine of the altitude of the Sun at 
any time on the vertical circle from the zenith pas- 
sing through the Sun. Cf. Brahmagupta, XXI, 63, 
drgmandale natdrhsajya drgjya sankur unnatarhsajyd, 

* Cf. Suryasiddhanla, 11, 61-63; Lalla, Spa^tadhikdra, 17, and 
Samanyagolabandha, 4; Brahmagupta, II, 57-60; Paricasiddhantika, 
IV, 26 and note; Bhaskara, Qaiyitadhydya, Spa§tadhikara, 48; Kaye, 
op. dt., p. 73. 

^ Cf. Lalla, Madhyagativasand, 15; Bhaskara, Gardiddhydya, 
Spa^tddhikdra, 65 {Vdsan&bha^ya) who names Aryabhata in connec- 
tion with this rule. 


and Bhaskara, Golddhydya, Tri'prasnavdsand, 36, 
sankuv unnatalavajyakd hhavet} 

Paramesvara remarks: uttaragole gatagantavyd- 
suhhyas caradaldsun visodhya jivdm dddya svdhord- 
trdrdhena nihatya trijyayd vibhajya lahdhe hhujydm 
praksipet. sd ksitijdd utpannd svdhoratrestajyd bhavati. 
This corresponds to the so-called cheda of Brahma- 

29. Multiply the given sine of altitude of the Sun by the sine 
of latitude (the equinoctial sine) and di%"ide by the sine of co- 
latitude. The result mil be the base of the sahku of the Sun south 
of the rising and setting line. 

Sankvagra is the same as sankutala {"the base of 
the sanku") and denotes the distance of the base of 
the sanku from the rising and setting line.^ 

30. The sine of the greatest decUnation multiplied by the 
given basersine„pf„the»Sun and divided by the sine of co-latitude 
is the Sun's agra on the east and west horizons. 

The agrd is the Sun's amplitude or the sine of the 
degrees of difference between the day-circle and the 
east and west points on the horizon.^ 

The proportions employed are those given in 
Suryasiddhdnta, V, 3 n. 

1 Cf. SHryasiddhanta, III, 35-39 and note; Brahmagupta, III, 
25-26; BCMS, XVIII (1927), 25. 

2 Cf. Brahmagupta, III, 65 and XXI, 63; Bhaskara, Goladhyaya, 
Triprainavasana, 40-42 (and Vdsandbhdsya) and Ganitadhyaya, Tri- 
pra§nMhikara, 73 (and Vasanabha§ya) ; Lalla, Triprainadhikara, 49. 

*See S-uryasiddhanta, III, 7 n.; Brahmagupta, XXI, 61; Bhas- 
kara, Goladhy&ya, Tri-praknavdsana, 39 and GavitadhySya, Tripraina- 
dhikara, 17 (yssanabhd§ya). 


31. The measure of the Sun's amplitude north of the Equator 
[i.e., when the Sun is in the Northern hemisphere], if less than the 
sine of latitude, multipUed by the sine of co-latitude and divided 
by the sine of latitude gives the sine of the altitude of the Sun 
on the prime vertical.^ 

Bhau Dajl^ first pointed out that Brahmagupta 
(XI, 22) contains a criticism of stanzas 30-31. 

uttaragole 'grayam vi§uvajjyato yad uktam unayam | 
samamandalagas tad asat krantijyayam yato bhavati || 

Paramesvara remarks: visuvajjyond cet. visuvaj- 
jyonayd krdntya sddhitd ced ity arthah. visuvaj- 
jyonakrdntisiddhd sodaggatdrkdgrd. 

32. The sine of the degrees by which the Sim at midday has 
risen above the horizon will be the sine of altitude of the Sim at 
midday. The stae of the degrees by which the Swa is below the 
zenith at midday will be the midday shadow. 

33. Multiply the meridian-sine by the orient-sine and divide 
by the radius. The square root of the difference between the 
squares of this result and of the meridian-sine will be the sine of 
the ecliptic zenith-distance. 

The madhyajyd or "meridian-sine" is the sine of 
the zenith-distance of the meridian ecliptic point. 

The udayajyd or "orient-sine" is the sine of the 
amplitude of that point of the ecliptic which is on 
the horizon. 

The sine of the ecliptic zenith-distance of that 
point of the ecliptic which has the greatest altitude 
(nonagesimal point) is called the drkksepajyd.^ 

^Cf. Suryasiddhanta, III, 25-26 n.; Brahmagupta, III, 52; 
Pancasiddhdntika, IV, 32-3, 35 n. 

2 JRAS, 1865, p. 402. 

^ Cf . Suryasiddhanta, V, 4-6; Pancasiddhdntika, IX, 19-20 and 
note; Lalla, Suryagrahanadhik&ra, 5-6; Kaye, op. cit., pp. 76-77; 
BCMS, XIX (1928), 36. , 


Brahmagupta (XI, 29-30) criticizes this stanza as 
follows : 

vitriblialagne dfkk^epamandalam tadapamandalaj-utau jya | 
madhya dykk^epajya naxj^abhatoktanaya tulya \\ 
drkk§epajyato 'sat taimasad avanater nasalj I 
avanatinasad grasasyonadhikata ra\igraha;ie. |i 

34. The square root of the difference of the squares of the 
sines of the ecliptic zenith-distance and ofjthe zenith-distance is 
the sine of the echptic-altitude. *^2-'>* ^1 ^7 A 
kuvasat k§itije sva drk chaya bhuvj'asardham nabhomadhyat. 

The sine of the altitude of the nonagesimal point 
of the ecliptic is called the drggatijyd. 

Drk is equivalent to drgjyd the sine of the zenith- 
distance of any planet.^ 

This stanza is criticized by Brahmagupta (XI, 

drkk§epajya bahur drgjya l/arno 'nayoh krti%-isesat | 
miilam drgnatijiva samsthanam ayuktam etad api. 1| 

The construction of the second part of the stanza 
and the exact meaning of drk and chdyd are not clear 
to me. It seems to mean that when the sine of the 
zenith-distance is equal to the radius the greatest 
parallax (horizontal parallax) is equal to the radius 
of the Earth. Kuvasat (''because of the Earth") 
seems to indicate that parallax is due to the fact 
that we are situated on its surface and not at its 
center, and that parallax, therefore, is the difference 
between the positions of an object as seen from the 
center and from the surface of the Earth. 

1 Cf. Suryasiddhanta, V, 6; Lalla, Suryagrahapa, 6; Paficasi- 
ddhardika, IX, 21 and p. 60; Bhaskara, Ga^iMdhyaya, Suryagrahaxia, 
II, 5-6 (and Vasanabha$ya); BCMS, XIX (1928), 36-37. 


Paramesvara's explanation is as follows: 

Dfgbhedahetubhuta svacchaya drgjya va svadrggatijya va 
drkk§epajya vety arthah. sa yadi ksitije bhavati nabhomadhyat 
k?itijanta bhavati vyasardhatulya bhavatity arthas tada 
kuvasad bbumivasan nifpanno drgbhedo vyasardham bhavati 
bhuvyasardhatulyam dj-gbhedayojanam ity arthajt. antarale 
'nupatat kalpyam. 

Sukumar Ranjan Das^ states that there is no 
reference to parallax in Aryabhata. If Paramesvara is 
correct in interpreting the second part of the rule as 
giving yojanas of drghheda (parallax), we must ascribe 
to Aryabhata the knowledge of parallax, even though 
no rules are given for its calculation at intermediate 
points. It is hard to see what else the "radius of the 
Earth" can refer to when given immediately after 
rules for finding the drkksepajyd and the drggati- 
jya (cf. Brahmagupta, XXI, 64-65, and Bhaskara, 
Golddhydya, Grahanavdsand, 11-17), especially since 
parallax was well known to the old Suryasiddhdnta 
which antedated Aryabhata.^ 

It seems to me that the passage is probably to be 

interpreted in the light of Brahmagupta, XXI, 64-65 : 

drsyadfsyam drggolardham bhuvyasadalavihinayutam [ 
dra§ta bhugolopari yatas tato lambanavanati || 
k§itije bhudalaliptah kak§ayam drnnatir nabhomadhyat | 
avanatilipta yamyottarS ravigrahavad anyatra. | p 

35. The sine of latitude multiplied by the sine of celestial 
latitude and divided by the sine of co-latitude is minus and plus 
to the Moon when it is north of the ecliptic depending on 

1 "ParaUax in Hindu Astronomy," BCMS, XIX (1928), 29-42. 

* Cf. Pancasiddkantika, p. 60. 

' See also LaUa, Madhyagativasana, 23-28. 


whether it is in the Eastern or Western hemisphere, plus and 
minus when it is south of the ecliptic under the same circum- 

This stanza and the next give the calculation 
called drkkarman, an operation for determining the 
point on the ecliptic to which a planet ha\'ing a given 
latitude will be referred by a secondary to the prime 
vertical It has been called "operation for apparent 
longitude" and falls into two parts, namely, the 
"operation for latitude" (ahsadrkkannan) treated in 
this stanza and the "operation for ecliptic-de^dation" 
(dyanadrkkarman) treated in the following stanza.^ 

The stanza is criticized by Brahmagupta (XI, 34) : 

vikiepagunaksajya lambakabhakta grahe dhanam rnam yat | 
uktam udayastamayayor na pratighatikam yatas tad asat. |i 

Brahmagupta, X, 13-14 gives a general criticism 
of Aryabhata's drkkarman, followed by an exposition 
of his own method. 

36. Multiply the versed sine (of the Moon) by the celestial 
latitude and by the (greatest) declination, and divide by the 
square of the radius. The result is minus or plus to the Moon 
when it is in the northern ayana depending on whether its celestial 
latitude is north or south, and plus or minus when it is in the 
southern ayana under the same conditions. 

Paramesvara explains utkramanam by kotyd 

The ayanas are the northern and southern paths 
of the Sun from solstice to solstice.^ 

^ Cf. Suryasiddhanta, VII, S-9 n.; Kaye, op. cU., pp. 78-79. 
* Cf. Suryasiddhdnta, VII, 10 n.; Lalla, MadhyagativasanS,, 


Criticized by Brahmagupta (XI, 35) : 

trijyakftibhakta vik^epapakramagunotkramajyendoti | 
ayanante yad rnadhanam tat tasyadau tato 'sat tat. 1 1 

37. The Moon consists of water, the Sun of fire, the Earth 
of earth, and the Earth's shadow of darkness. The Moon ob- 
scures the Sun and the great shadow of the Earth obscures the 

Brahmagupta (XI, 9) remarks: 

aryabhato janati grahagtagatim yad uktavaihs tad asat | 
rahukrtam na grahanam tatpato nagtamo rahuh. 1 1 

There is no such statement in our text and Brahma- 
gupta himself (XXI, 43-48) ascribes eclipses to Rahu. 

38. When at the end of the true lunar month the Moon, being 
near the node, enters the Sun, or when at the end of the half- 
month the Moon enters the shadow of the Earth that is the 
middle of the eclipse which occurs sometimes before and some- 
times after the exact end of the lunar month or half-month. 

Paremesvara remarks, sphutasasimdsdnte lamha- 
nasaihskrte 'mdvdsydntakdle. He also takes the words 
adhikonam as meaning "middle of the eclipse which 
lasts for a longer or shorter time," but gives as an 
alternate explanation offered by some the foregoing 

39. Multiply the distance between the Earth and the Sun by 
the diameter of the Earth and divide by the difference between 
the diameters of the Earth and the Sun. The result will be the 
length of the shadow of the Earth (measured) from the diameter 
of the Earth. 

* Cf. SuryaMddhanta, IV, 9; Lalla, Madhyagativasana, 29, 34. 

* Cf. Suryasiddhanta, TV, 6, 16; Lalla, Candragrahatia, 10; Albe- 
runi, II, 111. 


The last clause seems to indicate that the measure- 
ment is to be reckoned from the center of the Earth.^ 

40. The difference between the length of the Earth's shadow- 
and the distance of the Moon from the Earth multiplied by the 
diameter of the Earth and di^^ded by the length of the Earth's 
shadow is the diameter of the Earth's shadow (in the orbit of the 
Moon) .=2 

41. Subtract the square of the celestial latitude of the iloon 
from the square of half the sum (of the diameters of the Sun and 
Moon or of the Moon and the shadow). The square root of the 
remainder is known as the sthityardha. From this the time is 
calculated by means of the daily motions of the Sun and ^Nloon. 

The sthityardha is half of the time from first to 
last contact.^ 

42. Subtract the radius of the Moon from the radius of the 
Earth's shadow. Subtract from the square of the remainder the 
square of the celestial latitude. The square root of this remainder 
will be the vimarddrdha. 

The vimarddrdha denotes half of the time of total 

43. Subtract the radius of the Moon from the radius of the 
Earth's shadow. Subtract tliis remainder from the celestial 
latitude. The remaiader is the part of the Moon which is not 

44. Subtract the given time from half of the duration of the 
obscuration. Add this to the square of the celestial latitude. Take 

1 Cf . Brahmagupta, XXIII, 8. 

2 Cf. Brahmagupta, XXIII, 9. 

' Cf. Pancaaiddhdntikd, VI, 3 and X, 26-3; Swryasiddhanta, W, 
12-13; Brahmagupta, IV, 8. 

*Cf. SuryasMdhanta, IV, 13; Pancasiddhantikd, X, 7; Brah- 
magupta, IV, 8. 


the square root. Subtract tliis from half the sum of the diameters. 
The remainder will be the obscuration at the given time.^ 

The first sentence ought to be: "Subtract the 
koti of the given time from the koti of the sthityardha. 
Square this." 

45. The sine of the latitude multiplied by the sine of the 
hour-angle and divided by the radius is the deflection due to 
latitude. It is south. 

sthityardhac carkendos trirasisahitayanat sparse. 

For the difficulty of the stanza and the gap in the 
commentary of Paramesvara see the Preface to 
Kern's edition (pp. v-vi) with the references to 

"Hour-angle" is expressed by madhydhnat krama 
(gunitah). "Deflection due to latitude" seems to be 
the meaning of dik. 

The first part deals with the dksavalana or "deflec- 
tion due to latitude." According to Paramesvara, it 
is south in the Eastern and north in the Western 
hemisphere. The other books give just the opposite. 

Paramesvara remarks, etad aksavalanam sthitya- 
rdhdc ca. sihityardhasabdena tanmulahhuto viksepa 

Paramesvara also remarks, ayanasahdendpakrama 
ucyate. trirdsisahitdd arkdc candrdc ca nispanno 
'pakramo 'pi tayor arkendor valanam bhavati. 

Paramesvara explains sparse as sparsa iti grahana 
ity evdrthatah. 

However the second part of the stanza is to be 

^ Cf. Suryasiddhanta, IV, 18-20; Pancasiddhantika, X, 5-6; 
Brahmagupta, IV, 11-12. 


translated it must deal with the so-called dyanavalana 
or "deflection due to the deviation of the ecliptic 
from the equator." 

Both valanas ("deflection of the ecliptic") were 
employed in the projection of eclipses.^ 

46. At the beginning of an eclipse the Moon is dhumra, when 
half obscured it is kr§na, when completely obscured it is kapila, 
at the middle of an eclipse it is hrsnatamra."^ 

47. When the Moon eclipses the Sun even though an eighth 
part of the Sun is covered this is not preceptible because of the 
brightness of the Sun and the transparency of the Moon.' 

48. The Sun has been calculated from the conjunction of the 
Earth and the Sun, the Moon from the conjunction of the Sun and 
Moon, and all the other planets from the conjunctions of the 
planets and the Moon.* 

49. By the grace of God the precious sunken jewel of true 
knowledge has been rescued by me, by means of the boat of my 
own knowledge, from the ocean which consists of true and false 

50. He who disparages this universally true science of 
astronomy, which formerly was revealed by Svayambhu, and is 
now described by me in this Aryabhatiya, loses his good deeds 
and his long life.^ 

Read pratikuncuko. 

1 Cf. Bralunagupta, IV, 16-17 and XXI, 66; Lalla, Candroffraha- 
Xtadhikara, 23, 25; Suryasiddhdnta, IV, 24-25: "From the position of 
the eclipsed body increased by three signs calculate the degrees of 

See Brennand, Hindu Astronomy, pp. 280-83; Kaye, Hindu 
Astronomy, pp. 77-78. 

* Cf. SHryasiddhanta, VI, 23; Lalla, Candragrahavadhikara, 36; 
Brahmagupta, IV, 19. 

' Cf . Suryasiddhanta, VI, 13. 

* Cf. BCMS, XII (1920-21), 183. 

s Cf. iUd., p. 187. « a. JBAS, 1911, p. 114. 


[Including the most important Sanskrit proper names] 

Alberuni, 10, 14, 15, 53, 69 

Alphabet, letters of, used with 
numerical value, 2-9 

Altitude of Sun, sine of, 72, 73 

at midday, 74 

on prime vertical, 74 

Amplitude, of Sun, 73, 74 

Anomaly, equation of, 60 

Apparent longitude, 77 

Apsides of planets, epicycles, 18 

motion, 16-18, 52 
position, 16 

Apsis of Moon, 53, 54 

epicycle, 18 
revolutions, 9 

Apsis of Sun, epicycle, 18 
motion, 16-18, 52 
position, 16 

Area, any plane figure, 27 

circle, 27 

square, 21 

trapezium, 27 

triangle, 26 
Ascensional difference, 72 
Asterisms, 55, 56 

half-dark, half-light, 64 

revolutions, 52 

stationary, xiv, 64-67 
Asterisms, circle of, 55 

driven by provector wind, 66 

sixty times orbit of Sun, 13 

surrounds Earth as center, 64 

Asterisms, sphere of, half, de- 
picted on a sphere, 70 
half, minus radius of Earth 
visible to men, 68-69 

\-isible to Gods, half to 
Pretas, 69 

Balabhadra, 16 

Balls, pile of, with triangular 

base, 37 
Base of sanku, 73 
Base of triangle, 26, 33, 70 
Base-sine of Sun, 73 
Bharata battle, 12 
Bhaskara, 14, 19, 27 n., 28, 55, 

66, 67, 73, 76, 80 

Bhattotpala, 65, 68, 69 

Brahmagupta, 2, 10, 11, 12, 13, 
14, 15, 17, 18, 55, 57, 58, 64, 
68, 74, 75, 76, 78 

Brahman, day of. See Day 

night of, 64 

Central ecHptic-point. See Non- 

agesimal point 
Chain, in indeterminate equa- 
tions, 43, 45-50 
Circle, area, 27 

chord of one-sixth circumfer- 
ence, 27 
construction, 30 
quadrant, in constructing 

sines, 28 
relation of circumference to 

diameter, 28 
saras, 33 
sampatasaras, 34r-35 

avil day, 52, 69 

Co-latitude, sme of, 70, 71, 72, 

73, 74, 76 
Commutation, equation of, 60 
Compass, 31 




Conjunctions, • Earth and Sun, 
52, SI 

!Moon and Sim, 52, 81 
■\Ioon and planets, 81 
planets with one another 
number in a yuga, 51 
past and future, calculated 
from distance apart, 
Conjunctions, of planets, epi- 
cycles, 18 
revolutions, 9 

Venus and Mercury cross 
ecliptic at, 63 
Cube, defined, 21 
Cube root, 24-26 

Day, 51 

civil, 52, 69 
sidereal, 52 

Day-circle, 71, 72 

radius, 71 
Day of Brahman, 12 

increase of Earth during, 64 

measurement of, 53 

part which has elapsed, 12 

Day-radius, 71 

Declination, 71 

Declination, greatest : of ecliptic, 

day-radius, 71 
sine of, 73, 77 

Deflection, in eclipses, 80-81 

Degrees, 13, 56 

Deviation, of ecliptic from Equa- 
tor, 63 

of Moon and planets from 
ecliptic, 16 

Diameter, of circle, relation to 
circumference, 28 
of Earth, Sun, Moon, and 
planets, 15 

Earth, compared to round Ka- 
damba flower, 64 

conjunction with Sun, 52, 81 
constitution, 64, 78 
diameter, 15 
half dark, half light, 64 
increase and decrease in size, 

located in center of space, 56, 

moves one minute in a prdna, 

revolutions eastward, 9 
rotation, 9, 14, 65-66 
shadow. See Shadow 
simile of man moving in a 

boat, 64 

Earth-sine, 71 

Earth--wind, 18 

East and west hour-circle, 69 

Eccentric circle, equal in size to 
orbit, 58 

location of center, 58 
movement of planet on, 57 

Eclipses, 78-81 

causes and time, 78 

color of Moon, 81 

deflection, 80-81 

half-duration, 79 

middle, 78 

obscuration at given time, 

of Sun not perceptible if less 

than one-eighth obsciured, 

part of Moon not eclipsed, 79 
shadow of Earth, 78-79 
total obscuration, 79 

Ecliptic, deviates equally from 
Equator, 63 

deviation of Moon and plan- 
ets from, 16 

greatest declination, 16 

northern and southern halves, 

quadrants, 72 

Sun, nodes of Moon and plan- 
ets, and shadow of Earth 
move along, 63 



Ecliptic-altitude, sine of, 75 
EcUptic-deviation, 77 

Ecliptic zenith-distance, sine of, 

Epicycles, dimensions, 18, 58 

mean planets at centers, 59 
movement of, 59 

planets on, 58, 59, 61 
number of revolutions, 51 

Equator, celestial, 63 
Equator, terrestrial. See Lanka 
Equinoctial shadow, 70 
Equinoctial sine, 70, 73 

Factors, problems relating to 
two, 38 

Fathers, dwell in Moon, 69 
see Sun for half a month, 69 
year of, 53 

Fractions, 40 

Gnomon, 31-33, 71 
Gods, dwell on Meru, 69 

see northern half of stellar 

sphere, 69 
see Sun for half a year, 69 
think dwellers in Hell are be- 
neath them, 68 
year of, 53 

Half-duration, of eclipse, 79 
Heaven, at center of land, 68 
Hell, at center of ocean, 68 
Himavat Mountains, 68 
Horizon, 69 

Horizon of Lanka, 69, 72 
Horizontal, how determined, 30 
Hour-angle, 80 
Hypotenxise, of a planet, 61 

relation to side of right-angle 
triangle, 31, 33 

Indeterminate equations of first 
degree Qeuttaha), xiv, 42-60 

Intercalarj' months, 52 
Interest, 3S-39 
Inverse method, 40 

Jupiter. (See Planets 
years of, 51 

Kusumapura, 21 

Lalla, 10, 14, 15, IS, 19, 59, 66 

Lanka (terrestrial Equator), 9. 
64, 66, 6S 

horizon of, 69, 72 
90 degrees from poles, 68 
Latitude, celestial, 76, 77, 79 
Latitude, sine of, 70, 71, 73, 74, 

Long letter, as measure of time, 

Longitude, apparent, 77 

Lunar days, omitted, 53 

Lunar month, 52, 69 

Mahasiddhanta, xvi 

Manu, period of, 12 

Marici, 66 

Mars. See Planets 

Mean planet, 57, 58, 59, 60 

Mean motion, of planets, 57, 58 

Mercury. See Planets 

Meridian, 69 

Meridian-sine, 74 

Meru, at center of land, 68 

description, 68 

dimensions, 15, 68 

home of the gods, 68 

Midday shadow, 74 

Midnight school, 11 n. 

Minutes, 13, 56 

Month, 51 

intercalary, 52 
lunar, 52, 69 

Moon, calculated from conjunc- 
tion of Sun and Moon, 81 



causes eclipse of Sun, 78 

crosses ecliptic at node, 63 

diameter, 15 

distance from Sun at which 
\'isible, 63 

eclipse. See Eclipses 

epicycle of apsis, 18 

greatest deviation from eclip- 
tic, 16 

half dark, half light, 64 

home of the Fathers, 69 

Lord of Monday, 56-57 

made of water, 78 

nearest to Earth, 56 

node moves along ecliptic, 63 

revolutions, 9 
of apsis, 9 
of node, 9 

true place, calculation of, 59 
Munisvara, xv 

Nandana forest, 68 

Night of Brahman, decrease of 

Earth during, 64 
Node of Moon, moves along 

ecliptic, 63 

revolutions, 9 
Nodes of planets 

Jupiter, Mars, and Saturn 
cross ecliptic at nodes, 63 

move along ecliptic, 16-18, 63 

position of ascending nodes, 
Nonagesimal point, 70 

altitude, 75 
zenith-distanee, 74 
Numbers, classes of, enumerated 
by powers of ten, 21 

Oblique ascension, equivalents 
in, of quadrants of ecliptic, 72 

Omitted lunar days, 53 

Orbits, of planets, in yojanas, 13 
movement of planets on, 57 
surroimd Earth as center, 56, 

Orient-siae, 74 

Pancasiddhantika, 10, 12 n 18 
19, 64, 65 

Parallax, 75-76 

Perpendicular, how determined 

Perpendiculars, from intersection 

of diagonals of trapezium, 27 
Plane figure, area of any, 27 

Planets, as lords of the days. 

calculated from conjunctions 
with the Moon, 81 

conjunctions past and future 
calculated from distance 
apart, 41 
number in a yuga, 51 

cross ecliptic at nodes or con- 
junctions, 63 

diameters, 15 

distance from Earth at which 
visible, 63-64 

driven by provector wind, 66 

epicycles, 18, 51, 58-59, 61 

greatest deviation from eclip- 
tic, 16 

half dark, half light, 64 

mean, 57, 58, 59, 60 
motion, 57, 68 

move with equal speed, 55 

movement of apsides, 16-18, 

on orbits and eccentric cir- 
cles, 57 

nodes move along ecliptic, 63 

orbits of, in yojanas, 13 

periods of revolution differ 
because orbits differ in 
size, 56 

positions of apsides and as- 
cending nodes, 16 

relative position with refer- 
ence to Earth as center, 56 

revolutions, 9 

of conjunctions, 9 

time in which they traverse 
distance equal to circle of 
asterisms and of sky, 55 

true distance from Earth, 61 



true places, calculation of, 

yugdoi, 53 

Plumb-line, 30 

Pretas, dwell at Vadavamukha, 

see southern half of stellar 

sphere, 69 
see Sun for half a year, 69 
Prime vertical, 69 

altitude of Sun on, 74 

Progression, arithmetical, num- 
ber of terms, 36 
sum, 35-36 

of any number of terms 
taken any^vhere within, 
of series made by taking 
sum of terms, 37 
sums of series made by taking 
squares and cubes of 
terms, 37-38 
Proportion, 39 
Provector wind, 66 
Prthudaka, quoted, 66 
Pjoramid, volume, 26 

Quadrilateral, construction, 30 

formed in quadrant of circle, 

Radius, 61, 72, 74, 80 

equals chord of one-sixth cir- 

cmaoference, 27 
square of, 71, 77 

Reciprocal division, 42-46, 48- 

Revolutions, of apsis of Moon, 9 

of asterisms are sidereal days, 

of conjunctions of planets, 9 
of Earth eastward, 9 
of epicycles, 51 
of node of Moon, 9 
of Sun, are solar years, 52 
Moon, and planets, 9 

in Tjvga equal years of 
yuga, 15 
time and place from which 
calculated, 9 
Right ascension, equivalents in, 

of signs of zodiac, 71 
Romaka, 68 
Rule of three, 39 

Saturn. See Planets 
farthest from Earth, 56 

Series, made by taking sum of 
terms of arithmetical progres- 
sion, 37 

made by taking squares and 
cubes of terms of arith- 
metical progression, 37-38 
Shadow, midday, 74 
Shadow of Earth, causes eclipse 
of Moon, 78 

diameter of, in orbit of Moon, 

length, 78 

made of darkness, 78 

moves along ecliptic, 63 
Shadow of gnomon, 31-33 
Sidereal day, 52 
Sidereal mid4ika, 51 
Siddhapura, 68 
Signs of zodiac, 13, 56 

day-circle of, 71 

right ascension, equivalents 

in, 71 
sine of, 71 

Sine of Sun, 72 

Sines, construction of, on radius 
of quadrant, 28 

table of sine-differences, 19 
calculation of, 29 

Sky, circumference of, 13, 14, 55 
Solid with twelve edges, 21 
Solai year, 52, 53, 69 
Space, measurement of, 51 
Sphere, volume, 27 



wooden, made to revolve, 70 
half of stellar sphere de- 
picted on, 70 
Square, defined, 21 
Square root, 22-24 
Sripati, quoted, 66 

Sun, amplitude, 73, 74 
apsis, position of, 16 
base-sme, 73 
calculated from conjunction 

of Earth and Sun, 81 
diameter, 15 
eclipse. iSee Eclipses 
epicycle of apsis, 18 
Lord of Sunday, 56—67 
made of fire, 78 
moves along ecUiptic, 63 
orbit is one-sixtieth circle of 

asterisms, 13 
relative position among plan- 
ets, 56 
revolutions, 9, 52 

in yiiga equal years of 
yuga, 15 
sine of, 72 

altitude. See Altitude 
true place, calculation of, 59 
visible to gods and Pretas for 

half a year, 69 

to Fathers for half a 
month, 69 

to men for haK a day, 69 
Sunrise school, 11 n. 
Suryasiddh&nta, 10, 12, 14, 18, 
19, 57, 64, 72, 73 

Three, nile of, 39 

Time, beginningless and end- 
less, 55 
measurement, 51 

Total obscuration, in eclipse, 79 
Trapezium, area, 27 

perpendicular from intersec- 
tion of diagonals, 27 
Triangle, area, 26 
construction, 30 
formed in quadrant of circle, 

hypotenuse of right-angle, 31, 
True places of planets, 59-61 

Ujjain, 22^ degrees north of 
Lanka, 68 

Vadavamukha, 68, 69 
Vdsandvarttika, 66, 67 
Venus. See Planets 
Versed sine, of Moon, 77 
Volume, pyramid, 26 

solid with twelve edges, 21 

sphere, 27 

Yavakoti, 68 
Year, 51 

of gods, 53 

of the Fathers, 53 

of men, 53 

solar, 53, 69 

Zenith-distance, 75 

Zero, 6-7 

Zodiacal signs. See Signs 


Agra, 43, 45-46, 49 

Agra, 73 

Aghana, 24-26 

Angula, 16 

Adhikonam, 79 

Anuloma, 58, 65 

Antyavarga, 2, 6 

Ay ana, 51, 77 

Avarga, letters and places, 2-5 

in square root, 22-24 
Avasarjiixfl,, 53 
^sra, 21 n., 26 n. 
Akvayuja, 51 
Akasakaksya, 14, 55 
Ak§adrkkarman, 77 
Ak§avalana, 80 
Ayqnadrkkarman, 77 
Ayanavalana, 81 

Induccdt, 54 

CTcco, 51, 52, 58 
Utkramav,am, 77 
Utsarpizil, 53 
Udayajyd, 74 
Unmai^dala, 69, 72 
Upaciti, 37 

Kak$yS, 57 

Kak§yamari4ala, 57, 58 

Kanya, 63 

Kam, 1 

Karkata, 31 

iTarna, 30, 33, 61, 62 

ITdZpa, 12 

Kuiiaka, xiv 

Kuvaiat, 75 

iToii, 70, 71 

iCofi, 32, 33 
iiT/ia. 7 

Kkadvinavaka, 7 
Khavrtta, 31 

Gunakarabhdgahdra, 40 
GoZa, 70 

Grahajava, 13, 62 
Grahavega, 62 
Grdsa, 34 
Ghatikd, 72 
Ghana, defioed, 21 
in cube root, 24-26 

Cakro, 70 
Caiurbhuja, 30 
Caturyuga, 55 
Citighana, 37 
Caitra, 55 
Chat/a, 75 
C/ierfa, 73 

Tuia, 60 
Taulya, 63 
Tribhuja, 30 

Difc, 80 

Du$§aina, 53-54 
DrA;, 75 
Drkkarman, 77 
Dfkkqepajya, 74, 76 
DTrkk^epamav-^la, 70 
Drggatijya, 75, 76 
Drgjyd, 75 
Drgbheda, 76 
Drhmazidala, 70 
Dvdda§d^a, 21 
Dvicckedagra, 43 




A'oi^I, 51 

iVr, measurement of, 16 
number in a yojana, 15 

Pankti, 22 
ParibhcLia, S 
Purva, 24, 26 
Purvapak§a, 65, 67 
PTaiimai}.iala, 57 
Pratiloma, 58 
Prar}.a, 13 

number in a vina^ikd, 51 

Bham, 14 
Bhuja, 31-33 

Madhyajya, 74 
Madhyahnat krama, 80 
Mando, 60, 61 
Mandakarzia, 61 
Mandagati, 59 
Mandaphala, 59-60 
Mandocca, 52, 58, 60 
Mithydjnana, 9, 14, 65, 67 
Mma, 63 
Mesa, 9, 11, 16, 60, 63, 71 

-3^*,"Feginmng of, 9, 55 
at midnight, 11 n. 
at sunrise, 9, 11 n. 
division into four equal parts, 

measurement of, 53 
names for parts of, 53 
number in period of a Manu, 

revolutions of Earthj Sim, 

Moon, and planets m, 9 
years of, equal revolutions of 
Sun, 15 
Yitgapada, 12, 54 
Yoga, of Sun and Earth, 52, 81 
Yojana, measure of increase and 


decrease of Earth, 64 
measiurement of, 15 
Yojanas, in drcumferencfe of sky, 

in planet's orbit, 13 
of parallax, 76 

same number traversed by 
each planet in a day, 55 

RaH, 13 

Varga, defined, 21 

in square root, 22-24 
letters and places, 2-7 

Valana, 80-81 

Vina,4ika, 51 

Vimardardha, 79 

Viloma, 65 

Vyatipata, 51 

Vyastam, 60 

Sanhu, 71, 72, 73 
Sankutala, 73 
Sankvagra, 73 
Sara, 33 
iSlghrakarria, 61 
Slghragaii, 58 
Sighraphala, 59-60 
Sighrocca, 52, 58 
Satya devaia, 1 
Samadalakotl, 26 
Sampdta&ara, 34-35 
Svsama, 53-54 
Sth&nantare, 22 
Sthityardka, 79 
Sphuta, 60, 61 
Sphutamadhya, 60 
Svayamhhu, 1, 81 
Svavrtta, 31 

Hasta, 16