
TRANSLATION 

OP THE 


SIDDHANTA SIROMANI, 




CONTENTS, 


Page 

Chapter I. — In praise of the advantages of the study of the 

Spheric, 105 

Chapter II. — Questions on tho general view of the Sphere, ... 107 

Chapter III. — Called Bhuvana-kos'a or Cosmography, 112 

Chapter IV. — Called Madhya -gati-vasanfi ; on tho principles 
. of the Buies for finding the mean places of tho Planets,. .. 127 

Chapter V. — On the principles on which tho Iiules for finding 

the true places of tho Planets are grounded, 125 

Chapter VI. — Called Golahandha ; on the construction of an 

Armillary Sphere, x J51 

Chapter VII. — Called Tripras'na-vfisana ; on the principles of 
the Buies resolving the questions on time, Bpacc, and 

directions, ICO 

Chapter VIII. — Called Grahana-vdsana ; in explanation of tho 

canse of Eclipses of the Sun and Moon, 170 . 

Chapter IX. — Called Drikkarama-vfisanft ; on the principles of 
the Buies for finding the times of the rising and setting 

of the heavenly bodies, 100 

Chapter X. — Called S'ringonnati-vasanfi ; in explanation of . 

the cause of tho Phases of the Moon, 200 

Chapter XI. — Called Yantrudhy&ya ; on the use of astronomical 

instruments, 209 1 

Chapter XII. — Description of the Seasons, 228 

Chapter XIII. — Containing useful questions called Pras'nfi- 

dbyaya, 23 L 




TRANSLATION OF THE GOLADHYAtA OF T1IE 
SIDDHANTA-STROMANI. 


CHAPTER I. 


Invocation. 


Tit, of the ailrnntoyes of the ettuhj of the Spheric, 

Salutation to Ganesx ! 

1. Having saluted that God, who 
when called upon brings all under- 
takings to a successful issue, and also that GtHdess, through 
whose benign favour the tongues of poets, gifted with a flow of 
words over new and with elegance, sweetness and playfulness, 
sport in their mouths as in a place of recreation, as dancing- 
girls adorned with beauty disport themselves in the dance with 
elegance and with every variety of step, I proceed to indite 
this work on the Sphere. It haB been freed from all error, 
and rendered intelligible to the lowest capacity, 

2. Inasmuch as no calculator can 

Object of the work. _ A , . , - 

hope to acqrnro in the assemblage ot 

the learned a distinguished reputation as an Astronomer, with- 
out a clear understanding of the principles upon which all the 
calculations of the mean and other places of the planets are 
founded, and to remove the doubts which may arise in his 
own mind, I therefore proceed to treat of the sphere, in such 
a manner as to make the reasons of all ray calculations 
manifest. On inspecting the Globe they become clear and 
manifest as if submitted to the eye, and are as completely 
at command, as the wild apple (AiiwlA) held in the palm ot' 
the hand. 



106 


Traiutlatio)i of the* [I. g, 

3. A a a fcaat #witli abundance of 
of Si!wric. an ,6norunce all things but without clarified butter, 

* and as a kingdom without a king, and 

an assemblage without eloquent speakers have little to recom- 
mend them ; so the Astronomer who has no knowledge of the 
spheric, commands no consideration. 

4. As a foolish impudent disputant, who ignorant of gram- 
mar (rudely) enters into the company of the learned and vainly 
prates, is brought to ridicule, and put to shame by the frowns 
and ironical remarks of even children of any smartness, so he, 
who is ignorant of the spheric, is exposed in an assemblage of 
the Astronomers, by the various questions of really accom- 
plished Astronomers. 

Object of the Amillary 5. ,Tho Armillary sphere is said, by 
Bp k flro ' the wise, to be a representation of tho 

celestial sphere, for the purpose of ascertaining the proofs of 
the positions of the Earth, tho stars, and the planets : this is a 
Bpceios of figure, and hence it is deemed by tho wise to be an 
object of mathematical calculation. 

•» f .. .. 6. It is said by ancient nstrono- 

In praise of mathematics. J 

mors that the purpose of the science 
is judicial astrology, and this indeed depends upon the influence 
of tho horoscopo, and this on the true places of the planets : 
these (true places) can be found only by a perfect knowledge 
of the spheric. A knowledge of the spheric is not to bo 
attained without mathematical calculation. How then can a 
man, ignorant of mathematics, comprehend the doctrine of 
the sphere &c. ? 

Who u likely to under- 7 * Mathematical calculations are 
take the etudy with effoct. 0 f two kinds, Arithmetical and Alge- 
braical : he who has mastered both forms, is qualified if he have 
previously acquired (a perfect knowledge of) the Grammar (of 
the Sanskrit Language,) to undertake the study of the various 
branches of Astronomy. Otherwise he may acquire tho name 
(but never tho substantial knowledge) of on Astronomer. 



8 it Ih a nta-8 iro mani. 


107 


II. 3.] 


• 8. Ho who has acquired a perfect 

In praise of Grammar. knowledge of Grammar, which has been 

termed Vedavadana i. e. the rnoiith of the Vedas and domi- 
cile of Saraswat^ may acquire a knowledge of every other 
science — nay of the Vedas themselves. For this reason it is 
that none, but he who has acquired a thorough knowledge of 
Grammar, is qualified to undertake the study of other sciences. 
The opinion of others on 9. 0 learned man ; if you intend 

lh*v of°cxtendingthe W Btudy 8 P^ er ^ c J study the Troatise 

of it. of BuXskara, it is neither too concise 

nor idly diffuse : it contains every essential principle of tho 
science, and is of easy comprehension ; it is moreover written 
in an eloquent style, is made interesting with questions ; it im- 
parts to all who study it that manner of correct expression in 
learned assemblages, approved of by accomplished scholars. 
End of Chapter 1. 


CHAPTER II. 

Questions on the Geneml view of tho Spit nr. 

Qaortioa. regarding the 1. Thus Earth being encircled l.y 
E» r th. the revolving planets, remains sta- 

tionary in tho heavens, within the orbits of all the revolving : 
fixed stars j tell mo by whom or by what is it supported, that it ^ 
falls not downwards (in space) ? 

2. Tell me also, after a full examination of all the various 
opinions on the subject, its figure and magnitude, how its prin- 
cipal islands mountains and seas are situated in it ? 

3. Tell me, 0 my father, why tho 
JtthZ TSfctZ place of a planet found out from 'well 

taining planet#’ true places calculated Ahargana (or enumeration 
aiid their causes. „ r J . “ , i . „ 

of mean terrestrial days, elapsed from , 



108 Tratifilatum of the • [II. 3. 

the commencement of the Kalpa)* by applying the rule of pro- 

* [A Kalra is that portion of time, which intervenes between one conjunction 
of gll the planets at the Horizon of LankA (that place at the terrestrial equator, 
where the longitude is 76* K., reckoned from Greenwich) at the first point of 
Aries, and a subsequent similar conjunction. A Kalpa consists of 14 MANUg 
and their 15 sandhis"; each manu lying between 2 sandhis. Each mane 

contains 71 togas ; each yoga is divided into 4 yugAnghbis viz., Krita, 
Theta', Dwapara and Katj, the length of each of these is as the numben 
4, 8, 2 and 1. The beginning and end of each YUG^'RoirRis being each one 12th 
part of it are respectively called its bandhyA and Sandhya'nba. The number 
of sidereal years contained in each YUOAfNGHU, Ac. are shewn below ; 

Kali, 432,000, 

Hwa'para, 864,000, 

Tbbta 1,206,000, 

Krita, 1,728,000, 

Yooa, 4,320,000, 

71 X Yuga = manu, 4,294,080,000, 

14 Manu 306,720,000, 

15 Manu sandhis each equal to a Krita YuoAnghm, 25,920,000, 

Kalpa, 4,320,000,000, 

Of the present Kalpa 6 manus with their 7 sandhis, 27 yugas and thcii 
three yuga'noiiki i. e. Krita, Theta, and Hwa'para, and 3179 sidereal years o 
tho fourth yuoa'noubi of the 28th Yuga of the 7th manu, that is to say 
1,972,947,179 sidorcal years have elapsed from the beginning of tho presem 
Kalpa to tho commencement of the Sa'uwa'itana ora. Now wo can easily find ou 
the numbor of years that have elapsed from tho beginning of the present Kalpj 
to any time we like. 

By astronomical observations tho number of terrestrial and synodic lunar day 
ill any given number of years can be ascertained and then, with the result found 
their number in a Kalpa or Yuga can be calculated by the rule of proportion. 

By this method anoient Astronomers found out tho number of lunar and ter 
restriAl days in a Kalpa as given below. 

. 1,602,999,000,000 (synodic) lunar days ) . . 4 

and 1,577,916,450,000 terrestrial days ) 111 a * ALrA ’ 

With tho foregoing results and a knowledge of the number of sidereal year 
contained in a Kalpa as well of those tint, have passed, wo can find out tli 
number of mean terrestrial days from the beginning of a Kalpa to any givei 
day. Tliii number is called Auargana and the method of finding it ia given ii 
GanitAdiiyAya by Bjia'sxara'cha'bya. 

By the daily mean motions of the planets, ascertained by aftronomical observe 
tions, the numbers of their revolutions in a Kalpa are known and are given ii 
works on Astronomy. 

To flud the place of a planet by the number of its revolutions, tho number c 
* days contained in a Kalpa and the Ahabga^a to a given day, the following pro 
portion is used. 

As the terrestrial days in a Kalpa, 

: the number of revolutions of a planet in a Kalpa 

: the AharGANA : 

: the number of revolutions and signs Ac. of the planet in the Ababga#a. 

By leaving out the number of revolutions, contained in the result found, th 
remaining signs Ac. indicate the place of the planet. 

Now, the inteution of the querist isthis,»wliy should not this be the tru 
place of a planet P In the GakitAdbtAya. 1 BhAskabAcha'kya has stated tli 
revolutions in a Kalpa, but he has here mentioned the revolutions in a TUG 
on account of his constant study of the S'^nrA-DHivRiDDBiDA-TANTRA, a Tm 
tieo on Astronomy by Lula who has stated in it the revolutions in a Yuoa.- 
1 B. D.l 



Siilhanta-siromani. 


i 


100 


II. 4.] 


portion to tho revglutions in the Yoga* &c. is not a tme ono ? 
( i. e. why is jt only. a mean and not the trap place), and why 
the rules for finding the true plades of the different planets uro 
nofoftEe same kind ? What are the Desant 4 ka, Udayantaba, 
Bhujantara, andCHARA corrections?t What is thoMANDoenoHAj: 
(slow or 1st Apogee) and S'IchrochchaJ (quick or 2nd Apogee)? 
What is the node ? 

4. What is the Kendra || and that which arises frdm it (i. o. 
the sine, cosine, &c. of it) ? Wlmt is tho Mandai»tiala|| (the 
first equation) and S'fon raphala^[ (the 2nd equation) which 
depend on the sine of tho Kendra ? Why does tho place of 
a planet become true, when tho Manuaphala or S'/ghrapiiala 

* [It may be proper to give notes explaining concisely tho technical terms 
occurring in these questions, which have no corresponding terms in English, in 
order tlmt the English Astronomer mty at once apprehend these questions with- 
out waiting for the explanation of them which tho Author gives in tho sequel.— 
B. D.] • 

+ [To find the place of a planot at the time of sun-rise at a given place, the 
several important corrections, i. c. tho jJpAYAVTAK A.PuuJ AVrAR A. IWAn taiu. 
and CtfAlU are t o bo applied to the mean pluce of the planet found out from tho 
Ail ABO ana by tho fact of the mean place being found from the Ahaboa^a for the 
time when a fictitious body, which is supposed to movo uniformly in the Equi- 
noctial, and to perform a complete revolution in the same time as the Sun, reaches 
the horizon of Lanka'. We now proceed to explain the corrections. 

Tho Udaya'ntaua and Bhijja'ntaba corrections are to be applied to the mean 
place of a planet found from the Ahabgana for finding the pluce of the planet at 
the true time whon the Sun comes to the horizon of Lanka' arising from llioso 
two portions of tho equation of time respectively, one due to thejucliuation 
of the ecliptic to the equinoctial aud the other to the uncquaTinotion of the 
Sun In tho ecliptic. 

Tho TTmV'ntara and Citaba corrections arc to be applied to the mean place of 
a planet applied with tho U data's tara aud Buuja'ntaha corrections, for finding 
tho place of the plunet at the time of sun rise at a given place. 

Tho DbbYntaba correction duo to the longitude of the place reckoned from 
tho meridian' of Xanka' and the Chaba correction to the ascentional differ- 
enc o. B . ]).] 

J [MANDocncHA is equivalent to the higher Apsis. The Sun's and Moon i 
MandochchaB (higher Apsides) arc the same as their Apogees, while the other 
planets' .M ANDocHCgAB aro equivalent to their Aphelions. B. 1).] 

§ [SVonBoenCHAisthat point of the orbit of each of the primary planets (i. e. 
Mars, Mercury, Jupiter, Venus and Saturn) which is furthest from the Earth. 
B.DJ 

|| [Kbndra is of two binds, one called M an da-kesdra corresponds with the 
anomaly and the other called SYqhba-kbndra is equivalent to the commutation 
added to or subtracted from ISO's as the Si&ba-knndha is greater or less then 
180° B. DJ 

f [Manda-phala is the same as tho equation of the centre of a planet and 
SYqhba-phala is equivalent to the annual parallax of the superior planet j and 
the elongation of the inferior planets. B. D.J 



no 


Translation of tho • 



are (at one time) added to and (at another) subtracted from it ? 
What is.jbhe twofold correction called Drikkarma* which 
lertrned astronomers have applied (to the true place of a pla- 
net) at the rising and setting of the planet ? Answer me all 
these questions plainly, if you have a thorough knowledge of 
the sphere. 

Questions regarding tlie 5. Tell me, 0 you acute. astrono- 
length of tW) day ind night. ^ why> whcn t ,, e g „ n jg „„ tho 

northern hemisphere, is the day long and the night short, and 
the day short and the night long when the Sun is on the south- 
ern hemisphere ? 

6. How is it that tho day and 

Questions regarding tho , * 

length of tho day and night night of the bods and their enemies 

Haityas correspond in length with 

the Rolar years ? How is it that the 

night and day of tho Pitrih is equal in length to a (synodic) 

lunar month, and how is it that tho day and night of Brahma 

is 2000 YUOAsf in length ? 

Question. Kpnlin* 7> Why> 0 Asfcr ™ omcr > 

periods of risings of tho the 12 signs of the Zodiac which aro 
i signs of tho ZodiftCs 

8 aU of equal length, rise in unequal 

times (even at the Equator,) and why arc not those periods of 
rising tho same in all countries ? 

Questions as to the places of 8. Shew mo, 0 learned one, tho 
the DrujyX, the Kujya , &c. p] aces 0 f the DyujyX (the radius of 

the diurnal circle), the KujyI (tho sine of that part of the 
arc of the diurnal circle intercepted between tho horizon and 
the six o’clock line, i. e. of the ascensional difference in terms 

• [Drixkabma is the correction requisite to 1 be applied fb the place of a pla- 
net, for find ingtfib point of tTie’Wiptio on tho horizon when the planet reaches 

it, This correction is to be applied to the place of a planet by means of its 

two portions, one called tbe AtanA'D^ixkarka and the other the Aksha-drik* 
xakxa. The plaoo of a planet with the Axana-dbixkahma applied, gives the 
point of tlie ecliptic on the six o'clock line when the planet arrive* at it : and 
this corrected place of the planet, again with the AxshA'D^ikxabxa applied, 
gives the point of the ecliptic on the horizon when the planet comes to it. B. D.J 

t The K*ita, Testa', Dwa'para and Kali are usually called Yogas : but 
the four together form only one Yoga, according to the Siddha'nta system, 
each of these four being held to be individually but a Yuga'itghbi. L. W. 



SUlhdnta-tiit'omani. 


Ill 


II. 10.] 

of a small circle), and show me also the places of the declination, 
Sama-sXnku,* Agra (the sine of amplitude), latitude and 
co-latitudo &c. in this Annilltiry sphere as these places are* in • 
the heavens. * » 

If the middlo of a lunar Eclipse 
uto Tirmuc^fTth" 8 ^ takes placo at the end of tho Tmii 
Mptei* 0 * solaran ‘ llunar (at the full moon), why deep not tho 
middle of the solar Eclipse take place 
in like manner at the change ? Why iH tho Eastern limb of 
the Moon in a lunar Eclipse first involved in obscurity, 
and the western limb of the Sun first eclipsed in a solar 
Eclipse ?f 

Questions regarding the 0. What, 0 most intelligent one, 
parallaxes. j s tho LamhanaJ and what is tho 

Nati? why is the Lam ban a applied to the Trmi and tho 
Nati applied to the latitude (of tho Moon)? and why are 
these corrections settled hy means (of tho radius) of tho 
Earth? 

Questions reding the 10. Ah ! why, after being full, decs 
phases of tho Moon. the Moon, having lost her pure bright-, 

ness, lose her circularity, as it were, by her too close associa- 
tion, caused by her diurnal revolution with tho night : and why 
again after having arrived in the same sign as tho bun, does 
she thenceforth, by successive augmentation of her puro 

• rSAVA'Bi/tfKU is the sine of the Sun’s altitude when it comes to the prime 

vertical. B. 1).] .. A . 

f TAn Eclipse of the Moon is caused by her entering into the Earths shadow 
and as the place of the Earth’s shadow and that of the Moon is the same at the 
full moon, the conjunction of tho Earth’s shadow and tho Moon must 
happen at the same time ; and an Eclipse of the Sun is caused by the nitcrpoM* 
tion of the Moon between the Earth aud tho Suu, and the conjunction of the 
Sun and Moon in like manner must happen at the new moon, aa then 
the place of the Sun and Moon is the same. A* this is the case with the eclipses 
of both of them (i. e. both the Sun and Moon) the querist asks, “ If the middle 
of a lunar eclipse Ac." It is scarcely necessary to add that the assumption 
that the middle of a lunar eclipse'takes plaqe exactly at the lull moon, is only 
approximately correct. B. D ] 

t [The LAMBAirAis equivalent tojhe Moon’s parallax in longitude from the 
Sun reduced into time by meant of the Moon’s motion from the Sun: and tha 
Nati is the same as the Moon’s parallax in latitude from the Sun. B. JJ.J 



112 


Tramhdion of the * [III. 1. 

brightness, as .from association with tho Surf, attain her circu- 
lar form V * 

• » 

End of the second Chapter. 


CHAPTER III. 

Culled Rhumna-lma or Gosnioyraphy. 

Tiie excellence of the The Supreme Boing Para Brah- 

Supremo Being. ma tho first principlo, excels eternally. 

From the soul (Purusha) and nature (Prakriti,) when excited 
by the first principlo, arose the first Great Intelligence called 
the Mahattattwa or Buddhitattwa : from it sprung self-con- 
sciousness (AhankXra :) from it were produced tho Ether, Air, 
Fire, Water, and Earth ; and by tho combination of these was 
mado tho universe BrahmInpa, in the centre of which is the 
Earth : and from BiiahmX Ciiatur^nana, residing on tho sur- 
face of the Earth, sprung all animato and inanimate things. 

' Decoription of the Mb. G1 ° b ° ° f the )iBrth 

od of (the five elementary principles) 

Earth, Air, Water, the Ether, and Fire, is perfectly round, and 
encompassed by the orbits of the Moon, Morcury, Venus, the 
Sun, Mars, Jupiter, and Saturn, and by the constellations. It 
has no (material) supporter \ but stands firmly in the expanse 
. of hcavon by its own inherent force. On its surface through- 
out subsist (in security) all animate and inanimate objects, 
Danujas and human beings, Gods and Daityas. 

• This t«w« has a double meaning, all the native writers, however grave tho 
subject, being much addicted to conceits. The second interpretation of this 
verse is as ibUows % * 

Ah ! why does the most learned of Brahnpns, though distinguished by his 
immaculate conduct, lose his puro honour arid influence as it were from his mis- 
conduct caused by derangement ? It is no wonder that the said Brahman after 
having met with a Brahman skilled in th* Vidas, and by having recourse to 
him, thenceforth beoomes distinguished for his eminent good conduct by gradual 
augmentation of his illustriousness. L. W. 



III. 6.] 


Sidluhita-siromanL 


113 


3. It is covcrtd on all sides with multitudes of mountains, 
groves, towns and sacred edifices, as is the bulb of tho Nauclca’s 
globular flower with its multitude of anthers. * 

Refutation of the fln^poti- 4 ' If the Enrlli were snpporteilby 
tion tlmt the Eurth has sue- any material substanco or living m*a- 
ct'ssivu supporters. ^ tore, then that would roqui re a second 

supporter, and for that second a third would bo required. 
Here wo have tho absurdity of an interminable series. If 
the last of tho series be supposed to remain firm by its own 
inherent power, then why may not tho same power bo supposed 
to exist iu the first, that is in tho Knrili ? For is not tho Earth 
one of tho forms of tho cight-fohl divinity i. e. of S'iva. 

Refutation of tho objec 8 * M >’™ tis "" inlim ' nt P ro F rt y 
tion, us to how the Karth 0 f tliQ Run and of Fire, as cold of tho 
bus its own inherent power. _ , n , 

Moon, fluidity ot water, and hardness 
of stones, and as the Air is volatile, so the earth is naturally 
immoveable. For oh ! tho properties existing in things aim 
wonderful. 

G. The* property of attraction, is inherent in tho Earth. 
By this property the Earth attacts any unsupported heavy ^ 
thing towards it : Tho thing appears to bo falling [but it is in 
a stuto of being drawn to the Earth] . Tho etherial expanse 
being equally outspread all around, where can tho Earth fall ? 

, L _ 7. Observing the revolution of the 

Opinion of the Baud- 7 _ . . 

deas. constellations, tho Bauddhas thought 

that tho Earth had no support, and as no heavy body is seen 

stationary in the air, they asserted that tho earthf goes eternal- ’ 


ly downwards in space. 
Opinion of the Jaixas. 


8. Tho Jajnas and others main- 
tain that there are two Suns and two 


* ["It is manifest from this that neither can tho Karth by any means fall 
downwards, nor tho men situated at the distances of a fourth part of the circum- 
ference from us or in the opposite hemisphere. B. D.] 
f [Ho who resides on the Karth, js not conscious of tho motion of it down- 
wards in space, as a niau sitting on a moving ship does not perceive its motion, 
B. D.J 

C 



m 


Translation of the • [III. 9. 


Moons, and also two sots of constellations, iwhich rise in con- 
stant alternation. To them I give this appropriate answer. 

ite.ti.rn of th. opinion iK ' 0bscrvin K y on 0 Ba ™- 
of the Bacdubas. f dha, that every hca/.y body projected 

into the air, comes back again to, and overtakes the Earth, 
how then can you idly maintain that tho Earth is hilling down 
in space? [If true, tho Earth being the heavier body, would, 
ho imagines* — perpetually gain on the higher projeetilo and 
never allow its overtaking it.] 


Refutation of tho opinion 10. But what shall \ say to thy 
of the Jainas. folly, 0 Jaina, who without object or 

uso supposcst a doublo set of constellations, two Suns and two 
Moons ? Dost thou not sco that the visible circumpolar con- 
stellations take a whole day to complete their revolutions ? 

Refutation of tho supposi- 11. If this blessed Earth were level, 
turn that tho Earth & level. ]^ 0 u pl„ ll0 mirror, then why is not tho 

gun, revolving above at a distance from the Earth, visible to 
men as well as to the Gods ? (on tho I’aukanika hypothesis, 
that it is always revolving ^bout Mkku, above and horizontally 
to the Eurth. 

12. If the Golden mountain (Mkrv) is the cause of night, 
then why is it not visible when it intervenes between us and 
tbo Suii? And Menu being admitted (by the Eaithaxikas) to 
lie to tho North, how comes it to pass that the Sun rises (for 
half tho year) to the South ? 


Reason of the false ep- 
* pearanee of tho plane form 
ot the Earth. 


13. As the one-hundredth part of 
tho circumference of a circle is (scarce- 
ly different from) a plane, and as 
the Earth is an excessively large body, and a mini exceedingly 
small (in comparison,) the whole visible portion of the Earth 
consequently appears to a man on its surface to be perfectly 
plane. , * 


• [This was Bbabkara’b own notion s—hut evon on tho more correct principle, 
, that ell bodio* loll with equal rapidity, the argument holds good. B. J).j 



III. 18.] 


Siflhd ntu-it'i romnni. 




115 


Proof of 1 ho corrcctfioss of 
alh'gfd rircumfercm*o of Ihc 
Earth. 


1 4. That tlio correct dimensions of 
tint circumference of the JSarth havo 
boon slated may ho proved by < the 


simple Rule of proportion in this mode: (ascertain thedijfer- 


ence in Yujanas between two towns in an exact north and 


south line, and ascertain also tin* difference of the latitudes of 
those towns : then say) if the difference of latitude gives this 
distance in Yujanas, what will the whole eircuinferehco of 300 
degrees give ? 

To confirm the mum cir- 15. As it is ascertained by caleula- 
cumfcrmee of the iSurtli. t ;„ n ttlat ( . ity of UwAYINf is 

situated at a distance from the equator equal to the one- sixteenth 
part of the whole circumference : this distance, therefore, 
multiplied by 10 will be the measure of the Earth’s cir- 
cumference. What reason then is there in attributing (as the 
PauhXnikas do) such an immense magnitude tl) the earth ? 

1 0. For the position of the moon’s cusps, the conjunction 
of the planets, eclipses, the time of the risings and settings 
of the planets, the lengths of the shadows of the gnomon, 
Sic., aro all consistent with this (estimate of the extent of tho) 
circumference, and not with any other ; therefore it is declared 
that the correctness of tho aforesaid measurement of the earth 
is proved both directly and iudirectly, — (directly, by its 
agreeing with the phenomena ; — indirectly, by no other estimate 
agreeing with the phenomena). 

17. LankX is situated in the middle of the Earth : Yama- 
koti is situated to tli6 East of Lanka', and Romakapattana to 
the west. The city of Siddhapura lies underneath LankL 
Sumeku is situated to the North (under tho North Polo,) and 
YadavXnala to the South of LankX (under the south Pole) : 

18. These six places aro situated at a distance of one-fourth 
part of the Earth’s circumference each from its adjoining one. 
So those who have a knowledge of Geography maintain. At 
Meru reside the Gods and the Siddhas, whilst at VadavXnaia 
are situated all the hells and the Daityas. 

c 2 



116 


Translation of tho , [III, 19 . 

19. A man on whatever part of the plobo he may be, 
thinks the purth to bo under his feet, and that he is standing 
np eight upon it : but two individuals placed at 90° from each 
othyr, fancy each that the other is standing iu a horizontal 
line, as it were at right angles to himself. 

20. Those who are placed at the distance of half the 
Earths circumference from each other aro mutually antipodes, 
as a man oh the bank of a river und his shadow reiiected iu 
the water : Hut as well those who aro situated at the distance 
of 90* as those who aro situated at that of 180* from you, main- 
tain their positions without difficulty. They stand with tko 
same ease os we do here in our position. 

Positions of the Dwipab 2 1 . Most learned astronoinors have 
* lldSew * stated that Jam Bunwf pa embraces the 

whole northern hemisphere lying to the north of the salt soa : 
and that the other six Dwi'fas and the (seven) Seas viz. 
those of salt, milk, &c. aro all situated iu tko southern hemis- 
phere. 

22. To tho south of the equator lies the salt sea, and to 
the south of it tho sea of milk, whence sprung tho nectar, 
tho Moon and tho Goddess LAKSiinf, and wlioro tho Omni- 
present VXsumovA, to whoso Lotus-feet GkaumI and all the 
Gods bow in reverence, holds his favorite residonco. 

23. ileyoud tho sea of milk lie in succession tho seas 
of curds, clarified butter, sugar-cane-juice, and wine: and, 
last of all, that of swoot Water, which surrounds Vadavanat.a. 
.The Patau Lukas or infernal regions, form tho concave 
strata of tho Earth. 

24. In those lower regions dwell the race of serpents (who 
live) in tho light shed by the rays issuing from the multitude 
of tho brilliant jewels of their crests, together with tho multi- 
tude of Asuuas; and them the Sujduas enjoy themselves 
with tho pleasing persons of beautiful females resembling the 
finest gold iu purity. 

* 25. The S'aka, S almau, Kavs'a, KbIuncha, Gomedaka, and 



117 


III. 31.] • Siilhinta-siromani 

PrsnKARA DwIpas /ire situated [in the intervals of tlio above 
mentioned seas] in regular alternation : each Dwipa lying, 
it is said, between two of these sc'as. 

Positions of the Mpun- 2G. To the North of Lank* lies 

tains 111 Jambu DWIPA and jj, m / lava mountain, and beyond 

its mno KiiaNiias parts * 

caused by the mountains., that the IIbMXkC'TA liiountaiil and 

beyond tliat agiiin tlio Nisilydha mountain. These threo 
Mountains stretch from sea to sea. In liko manner to 
the north of Siddiiapura lie in succession the S'^inuavXn 
S'ukla and NIla mountains. To tlio valleys lying between 
these mountains the wise have given the name of Vakshas. 

27. This valley which we inhabit is called the Bhaiiata- 

VAitsif a j to the North of it lies the Kinnabavabsua, and 
beyond it again the Harivabsha, and know that tlio north 
of StDDHAi’UBA in liko manner are situated tlio Kuuu, Hibav- 
maya and Ramyaka Varsiias. • 

28. To the north of Vamakoji lies the MalyavXm mountain, 
and to the north of Romakapattana tlio Gandhamadana 
mountain. Theso two mountains arc terminated by tlio Nila 
and Nisiiaoiia mountains, and the space between tlieso two 
is called the Ilavrita Vaksiia. 

29. Tho country lying between the Maf.yavan mountain 
and tho sea, is called the Biiadba's'wa-vaimiia by tho learned ; 
and geographers have denominated the country between tlio 
GandiiahIdana ami tho sea, the Kktumala-vakhha. 

80. Tlio Ilav^ita-varsiia, which is bounded by tho 
Nisiiadha, NfLA, Gasbiiamadaxa and Malyavan mountains, is • 
distinguished by a peculiar splendour. It is a laud rendered 
brilliant by its shining gold, and thickly covered with tho 
bowers of the immortal Gods. 

Position of thfl mountain dl. Ill tllC middle of tllC ILIVJJITA 
Mibu in Ilavrita. Vaksua stands the mountain Mkku, 

which is composed of gold and of precious stones, the abode 
of the immortal Gods. Expounders of tho Puranas have 
further described this Meru to be tho pericarp of the earth- * 
lotus whence Brahma had his birth. 



118 


Translation of the* [III. 82 . 

82. Tlio four mountains Mandara, Suc^andha, Vipula and 
SupXrs'w^ servo as buttresses to support this Meru, and upon 
theso four hills grow severtilly the Kadamba, Jamb£, Va^a 
and Pippala trees which are as banners on tjiose four hills. 

88. From tho clear juice which flows from the fruit of the 
*Tamb<7 springs tho jAMuu-NAnf ; from contact with this juico 
earth becomes gold : and it is from this fact that gold is 
called Jibuti nada : [this juice is of so exquisite a flavour that] 
tho multitude of tho immortal (lods and Siddttas, turning 
with distaste from nectar, delight to quaff this delicious 
beverage. 

8 1. And it is well known that upon those four hills [the 
buttresses of Mf.ru] are four gardens, (1*/) Oiiaitrakatha 
of varied brilliancy [sacred to Kuiieka], (2 ml) Nandana 
which is tho delight of tho Apsakas, (3rd) the Dn$m which 
gives ru fresh mi* lit to tho (lods, and (l//i) the resplendent 
VaibhrAja. 

35. And in those gardens are beautified four reservoirs, 
viz. the A run a, the Manama, the Maiiaurada* and the S'weta- 
jala, in duo order : and theso are the lakes in the waters 
* of which tho celestial spirits, when fatigued with their 
dalliauco with the fair Goddesses, lovo to disport themselves. 

30. Merit divided itself into three peaks, upon which are 
situated the three cities sacred to Vishnu, Brahma and S'iya 
[ denominated Vaikuntha, Brahma pur a, and KailAsa] , and 
beneath them are the eight cities sacred to Indka, Acini, Yama, 
. Nair^ita, Varuna, VXyu, S' as f, and fs'a, [i. e. tho regents 
of the eight Diks or directions,* viz., tho east sacred to 

9 [As the point; where tho equator cuts tho horizon is the east, the sun 
therefore rises duo east At time of I ho equinoxes but on thio ground, we 
oanuot determine the direction at Mkru [tho north pole] because there the 
equator coincides with the horizon and consequently the sun moves at MkuV 
under the horizon tho whole day of the equinox* Yet the ancient astronomers 
maintained that tho direction in which thtf iahakoji lies from AIkbu is tho 
east, because, according to their opinion, the inhabitants of Mjcbu saw tho 
sun rising towards the yimakoti at tho beginning of the kalpa. In the same 
manner, the direction in which Lanka lies front mount AInbu is south, that 
* in which Roaukapattana lies, is west, and the direction in which Siddha- 



» Sulhiinta-x'iromani. 


i 


119 


III. 40.] 


Indra, the south-east sacred to Aani, tlio south sacred to 
Yana, tho south-west sacred to Natrrita, tho west .sacred to 
Varuna, the north-west sucrod to Vayu, the north sacred to 
SWf and the north-cast sacred to IVa.] # 

Some peculiarity. :57 - 11*0 sacmUiangex, sprinRing 

from the Foot. of Vishnu, falls upon 
mount Mkru, and thence separating itself into four streams 
descends through tlio heavens down upon the four Visukam- 
riias or buttress hills, and thus falls into tho four reservoirs 
[above described] . 

U8. [Of tlio four streams above mentioned], tho first 
called SrrA, went to JIttadk.Vs'wa-vaiisua, tlio second, called 
AlakanandX, to BhIkata-varsua, the third, called Ciiakshu, 
to KktumXla-varsua, and the fpurtli, called Jill ADR A to Uttaka 
K UR it [or North Kuku]. . 

119. And this sacred river lias so rare an 1 efficacy that if 
her name be listened to, if sho bo sought to be seen, if seen, 
touched or bathed in, if her waters be tasted, if her mime 
be uttered, or brought to mind, and her virtues be celebrated, 
she purifies in many ways thousands of sinful moil [from 
their sins]. 

40. And if a man make a pilgrimage to this sacred stream, 
tho whole line of his progenitors, bursting tho bands [imposed 
on them by Yana], bound away in liberty, nml dance with 
joy ; nay oven, by a mail’s approach to its banks they rcpulso 
the slaves of Yama [who kept guard over thciu], and, escaping 
from Nauaka [tho infernal regions], secure an abode in the 
happy regions of Heaven. 

riTRA lio« from Meru is north. The hullrcoses of Merit, Mathura, Sitgaedua, 
Ac. aro situated in tho cnbt, south Ac. from Meru re*|icctivi'ly. H. D.l 

Note on i orae# from 21 to *13 :—U( 1 askara'cha'uva lias exercised his ingenuity 
in giving a locality on tlio earth to the poetical iniagiiialions of Vva'sa, at tho 
tamo tune that ho lias preserved his own principles in regard to tho form ami 
dimensions of the Eartlu But he 'himself attached no credit to what ho ‘has 
described in these verses for he ooncludos his recital iu his commentary with 
the words. , 

**** ycntrfsnou 

** What is stated here rests all on the authority of the Pura>'as." 

As much as to say V oredat JucUeus.” L. W. 



120 


Translation of the « [III, 4l f 


Tho ft .ml 7 41 ' Hcre in th » Bn/tHATA-VAKWA 

KiTiXcDALA^ of Uha'hata- arc embraced the following nine kuan- 
V ^ MIL4 ’ Pas Xportinns] viz. Aindha, Kas'bjw, 

Tamraparxa, (taiuustimat, Kumaktka, Naua* Saumya, VXbuna, 
and lastly (JXndharva. 

42. In the Kitm(uika alono is found tlio subdivision of 
inon iuto castes ; in the remaining khan das aro found all tlio 
tribes of Antyajas or outcusto tribes of men. In this region 
[Uiiauata-varsiia] aro also seven Kri vciialas, viz. the mahkn- 
I)ua, S ciKrr, Malaya, Kiksuaka, PXkiyatka, the Sahya, and 
Vinduya hills. 


Arrangement of the seven 43. The COimtiy to tlio south of 
Lokas worms. tlio e(|iialor is called tho Btiukloka, 

that to the north the Bhuvaia>ka mid Mkhw [the third] is 
called the Swarloka, next is the Mauarloka in the Heavens 
beyond this is*' the Janaloka, then the Tapoloka and lust of 
all tho Satyaloka. These lokas are gradually attained by 
ineroasiug religious merits. 

41. When it is sunrise nt Lanka, it is then midday at 
Yamakoji (1)0° cast of LankX), sunset at ►Siddhapuha and 
midnight at Bomakapattana. 

Point, of tho comp... 45 ' AfiaUmfl ttl0 P oint of 
vliy M*av is duo north of horizon at which the sun rises as 
all places. 

the east point, and that at which ho 
sots as the west point, and then determine the other two 
points, i. o., the north and south through the mathya* elFectcd 
by tho east and west points. Tlio line connecting the north 
and south points will bo a meridian lino and this lino in 
whatever place it is drawn will fall upon the north point : 
hence Mkku lies due north of all places. 

A curious fact is rehearsed. 40. Only YAMAKOfl lies due Oust 
(tagmpUoa Ana**,. from at tllc distance 0 f 90" 


* [From the oast and west points, as cflitres, with a common radius describe 
i two arcs, intersecting each other in two points, tho place contained by the 
arcs is oallcd Matsya “ a Ash” ami the iutunovtiug points are the north and 

south points. 13. D.j 





III. 51.] t Sidhdntas'irommL ■ 121 

'from it: but Lanka and not UjjayinI lios duo west from 
, | Yamakoti. 

: 47. The same is the case everywhere ; no place can lie west qf 
' that which is to its east except on the equator, so that east 
; and west are strangely related.* 1 

, 48. A man situated on tho enua- 

V Right sphere. 1 

tor sees both the north and south 

polos touching [the north and south points of] tho ^horizon, 
and the celestial sphere resting (as it were) Upon tho two 
polos as centres of motion and revolving vertically over his 
head in the heavens, as tho Persian water-wheel. 

, 49. As a man proceeds north 

Oblique sphere. r 

from the equator, ho observes the 

constellations [that revolve vertically over his head when 
seen from tho equator] to revolvo obliquely, being deflected 
from his vortical point : and tho north polo elevated above 
his horizon. Tho dogroos between the polo and tho horizon 
are tho degrees of latitudo [at tho place]. These degrees 
are caused by tho Yojanas [between tho equator and the 
place] . 

How tho degrees of lati- 50. Tho number of Yojanas [in • 

sr th ° « ° f °“y 

vice to »&. circle] multiplied by 15(30 and divided 

by [tho number in Yojanas in] tho circumferenco of fcho 
circlo is tho number of degrees [of that are] in tho earth 
or in tho planetary orbit in the heavens. Tho Yojanas are found 
from the degrees by reversing the calculation. 

51. Tho Gods who live in tho 
Parallel sphere. mount Meru observe at their zenith 

[• As the sun or any heavenly body when it reaches tho Primo Vertical 
of any place is called due east or west, so according to the Hindu Astronomical 
language all tho places on the Barth which avo situated on the circle 
corresponding to the Prime Vertical are duo east or west from tho place and not 
those which are situated on the parallel of latitude of the place, that is, the 
places which havo tho angle of position 90° from any place are due east or 
west from that plaoo. And thus all t directions on the Earth arc shown by 
means oftheauglo of position in the Ilindu Astronomical works. U. 1>.J 

1 ) 



122 


Translation of the * [III, 52. 

the north polo, while the Daityas in VadavInala the south 
pole. Bnt whilo the Gods behold the constellations revolving 
from left to right, to tho Da.ityas they appear to revolve from 
right to left. But to both Gods and Daityas tho equatorial 
constellations appear to rovolve on and correspond with tho 
horizon. 

Dimensions of tho Earth’s 52. Tho circumference of the earth 
circumference. has been pronounced to be 49(i7 

Yojanas and tho diameter of the same has been declared to 
bo 1581 Jr Yojanas in length : tho superficial area of the Earth, 
liko tho net enclosing tho hand ball, is 78,53,031 squaro 
Yojanas, and is found by multiplying tho circumference by 
tho diameter.* 


. 53. The superficial area of tho 

The error of Lalla is ox- § 1 

posed in regard to the super- Earth, liko the net enclosing the 
lleial area of tho Earth. . . , , , 

« hand ball, is most erroneously stated 

by Lalla: the true area not amounting to one hundredth 
part of that so idly assumed by him. His dimensions are 
contrary to what is found by actual inspection : my charge of 
error therefore cannot bo pronounced to be rude and uncalled 
‘ for. But if any doubt bo entertained, I beg you, 0 learned 
mathematicians, to examino well and with tho utmost impar- 
tiality whether the amount stated by mo or that stated by 
him is tlio correct ono. [Tho amount stated by Lalla in his 


• [The diameter and the circumference of tho Earth hero mentioned are to 
each other as 1250 : 31)27 and the demonstration of this ratio is shown by 
Bhasbaracuarya in the following manner. 

Take a radius equal to any large number, suoli as more than 10000, and 
through this determine tho sino of a smaller arc than even the 100th part of 
the circumference of tho circle by the aid of the canon of sines (Jyotpatti,) 
and tho aitio thus determined when multiplied by tlmt number which represents 
the part which tho arc just taken is of tho circumference, becomes the length of 
ciroumference because an arc smaller than the 100th part of tho oiroumfercnco 
of a circle is [scarcely different from] a straight line. For this reason, tho cir 
oumfcrence equal to tho numbor 62832 is granted by Aryaduatxa and the others, 
in the diameter equal to the number 20,000. Tlyjugli the length of the circum- 
ference determined by extracting the rquare root of the tenfold sjuare of tho 
diameter is rough, yet it is granted for convenience by ShiduabAciiAbya, Brah- 
Maovpia and the others, and it is not to v be supposed that they were ignorant 
of this roughness.— B. D,] 




’ 123 


. III. 50.] tfiilha h ta-s'i ronutni 

work entitled Dh^iddiiida-tantra is 285,(13,38,557 square 
• Yojanas, which he appears to have found by multiplying tho 
square contents of the circle by the circumference.] . 

Shows the wrongness ( of 51. If a picco of cloth be cut ill 
tho Kule given by Leila. ft c i rcu l ar f orm w ith a* diameter equal 

to half tho circumference of tho sphere, then half of the sphcro 
will be (entirely) covered by tliat circular cloth and there will 
still bo some cloth to aparo. • 

55. As the area of this pieco of cloth is to be found nearly 
2J times tho area of a great circle of tho sphere : and tho 
area of the picco of cloth covering tlio other half of the sphere 
is also tho same ; * f <: r v/tf 

50. Therefore the area of the whole sphere cannot bo more 
than 5 times the area of tho great circle of tho sphere. How 
tlien has lie multiplied [the area of the great circle of tho 
sphere] by tho circumference [to get tho superficial contents 
of tho sphere] ? # 

57. As the area of a great circlo [of the sphere] multiplied 
by tho circumference is without reason, tho rule (thcreforo of 
bALLA for tho supci'ficial contents of tho sphere) is wrong, 
and the superficial area of tho Earth (given by him) is conse- 
quently wrong. 

Otherwise 58, 59. Suppose tho length of tho 

[equatorial] circumference of the globe 
equal to 4 times tho number of sines [visa. 9(3, there being 24 
sines calculated for every 3°|, which number multiplied by 
4 = 90] and such oblong sections equal to the number of tho 
length of the said circumference and marked with tho vertical 
linos [running from polo to polo], as thcro aro aeon formed by 
nature on tho XnwlX fruit marked off by tho linos running 
from the top of it to its bottom. 

* Lot the diameter of a sphere be 7 : the circumference will bo 22 nearly. 
The area of a circle whoso diameter is 7 will be about 881 ; that of a circle 
whose diameter is 11 (\ circumference) will bo about 80? this 89? is little less 
than 24 times 88*. L. W. 


D 



124 


Translation of the c 


[III. 60. . 


CO. Tf wo determine the superficial a^ca of ono of these 
sections by means of its parts, wo have it in this form. Snm of - 
all the sines diminished by half of the radius and divided- by 
the same.* , 

i * 

• The correctness of this form is thus briefly Illustrated by Bha'skaha'cha'eya 
in his commentary. 

Let gah a ,g be the section in which ab, he,c dfi o. 
and Vi* e,tf M Ac. are each equal to 1 cubit and 
also aa x aw equal to 1 cubit i then hh„ co„ dd„ Ac. 
are proportional to tho sines mb,nc,od, Ac. and are 
thus found. 

mb. 

If lea or rad : give, aa 1 (= 1) : : mb : bb x = — — 

Mad. 

M 

If Bad : 1 : : nc : ee, = — 

Mad 
m od 

again Bad :1s : od : dd, — ~ — 

, Mad 
Ac. 

Now an , , bb„ pc„ Ac. being found, the contents of 
each of aa x b,b, Mq e A «*i d,rf, Ac. the part of the 
acction is found by taking half the sum of aa, A bb x 
bb, A re„ cc, A d(l, Ac. and multiplying it by ab 
(which is equal to eaoh of be, cd, Ac.) hore ab is assumed ns 1 and the whole 

Burlhco each of ao,M, ttb^c as a plane, for an are of 3*£ is scarcely different 

from a piano. 

Now to And the sum of aa x b x b , bb x e t e Ac. we havo 

aa x -f- hb x b\ + ce, ee, + dd. 



-X 1 + - 


XI + - 


- X 1 + 


2 2 

adding those and leaving out 1 multiplier, wo have 

i aa, *f bb, + ce x + dd, + Ac. 

Substituting the values of aa x , bb„ Ac. we hare 
mb no id 

i J 1 t» — + Ac. so on for the assumed sines 

M M M 

_ M \M 

hut t ----- — 

By substitution we get 

R mb no 

— + — + — + &c.. 

EMM 
B + »S + »e + od + Ac .. -JJ U 

m ■■■■■■■ s n s ■ i ■ 

M 

It it evident from this that the sum of al) tho sines diminished by tho half 
of the Badiua and divided by the Radius is equal to tiie contents of tlie upper 
half of the section, therefore by dividing by $ Bad we get the whole section 
instead of only the upper half of it. 

s&m of all the sines —JR. 

i. e. contents of the whole section = — A. 

k* 


M 



SiiUuintaJiromani, 


125 


III. 05.] 


Gl. As the superficial area of one section thus determined 
is equal to the diameter of the globe, the product found by 
nmltiplying the diameter by the circumference has therefore 
been asserted to bg the superficial contents of a sphere. 

Tho grand deluges or dis- 02. Tho earth is said to swell to 
solutions. tho extent of one Yojana equally all 

around [from the centre] in a day of Braiima by reason of 
the decay of the natural productions which grow upon it : in 
the Brahma deluge that increase is again lost. In tho grand 
deluge [in which Bhatima himself as well as all nature fades 
away then] tho Barth itself is reduced to a state of nonentity. 

03. That extinction which is daily 
Are four-fold. taking place amongst created beings is 

called tho Datnandixa or daily extinction. Tho Brahma ex- 
tinction or deluge takes place at tho eml of Brahma's day : 
for all created beings fire then absorbed in Brafinui’s body. 

Gl. As on the extinction of .Brahma himself all things are 
dissolved into nature, wiso men therefore call that dissolution 
the PrAKIJITIKA or resolution into nature. Things thus in a 
state of extinction having their destinies severally fixed arc 
again produced in soparato forms when nature is excited (by 


tho Creator). 

Go. Tho devout men, who have destroyed all their virtues 
and sins by a knowledge of tho soul, having abstracted their 
minds from worldly acts, concentrate their thoughts on the 


Here, by substituting the values of tho 21 sines stated in tho Ganita dha xa 
we have 

A =■ 30|$ ris. the diameter of the globe where tho circumference — 96. L. W. 
[llcre, the demonstration of tho rule (multiply tlio superficial area of the 
sphere by the diameter and divide the product by 6) for finding out the solid 
content of the sphere is shown by Bha'skaba'cka'bxa in the following manner. 

Suppose in the sphere the number of pyramids, the height of which is equal 
to the radius and whose bases are squares having sides equal to 1, equal to the 
number of the superficial area of the sphere, then 
The solid contents of every pyramid = & ft. 

ss £ diameter 

and the number of pyramids in the sphere is equal to the numbor of the 
superficial contents of the sphere. 

The folid content of the sphere =* ft diameter x superficial area.— B.D.J 



120 


Translation of Hip * [HI. 60. 

Supremo Being, and after their death, as they attain tho stato 
from which there is no return, tho wise men therefore denomi- 
nate this state tho Atyantika* dissolution. Thus the dissolu- 
tions are four-fold. 

« I 

60. Tho earth and its mountains, 

Thouniverw. ,, ~ „ 

the Gods and Danavas, men and 

others and also the orbits of tho constellations and planets 

and tho Iokas which, it is said, aro arranged ono abovo tho 

other, .aro all included in wliat has been denominated tho 

BkaiimInpa (universe). 

Dimension, of tin, B B ah- ° 7 - So,nc astronomers lmvo assert- 
Vk'viik. cd tlio circumference of tho circle of 

Heaven to bo 18,712,009,200,000,000 Yojanas in length. 
Some say that this is the length # of tho zone which binds tho 
two hemispheres of the Bhahmanoa. Somo PaurXnikas say 
that this is tho* length of the circumference of tho Lokaloka 
Pabvata .* 


* Vide versos G7,G8,69, BnA'sKAKA'cTTAnYA does not answer tho objection which 
these Tones supply to his theory of the Earth being the centre of tho systoin. 
Tho Bun is hero made the principal object of tho system— tho centre of the 
•Bbahma'n?a— tho centre of light whoso boundary is supposed iked : but if tho 
Bun moves then tho Hindoo HiuiiarA'jrnA must be supposed to bo constantly 
changing its Boundaries. Subbuji Bapu had not failed to use this argument in 
favour of the Newtonian system in his S'iromaxi PbakaYs, vido pages 65, 6G. 
Biia'hkara'ciia'iita however denies that ho can father the opinion that Lhis is 
the length of tho circumference limiting the Bbauma'n^a and thus saves him- 
self from a difficulty. L. W. 

[Mr. Wilkinson lias thus shown tho objection which SubbiVje Bapti made to 
tho assumption of the Sun’s motion, but I think that the objection is not a 
judicious one. Because had the length of the circumference of the Bbahma'niia 
been changed on account of the alteration of tho boundary of tho Sun’s light 
’ with him, or had any sort of motion of the stars been assumed, as would have 
been grunted if tho ourth is supposed to bo fixed, then, the inconvenience would 
have occurred ; but this is not the case. In fuct, as we cannot fix any boundary 
of the light which issued from the sun, tho stated length of the circumference 
of the Bbahmanua is an imaginary ono. For this reason, BuIseabachabya 
doos not admit tiiis stated length of the circumference of the Bbahma'xda. 
He stated in his Ganita'uiiyaya' in the commentary on the verse G8t!i of this 
Chapter that “ those ouly, who have a perfect knowledge of the Bbauua'itqa 
as they have of an a'nvala' fruit held in their palm, can say that this length of 
the circumference of the Bkahjca'bqa is the true one that is, as it is not in 
man’s power to fix any limit of the Bbahma'x?a, the said limit is unreasonable. 
Therefore no objection can be possibly made) to the system that tho. Sun moves, 
by assuming such nn imaginary limit of the BaiHXixpA which is little less 
* impossible than tho existence of the heavenly lotus.— B. I).j 



Siilhdnta-s'iivmanL 


127 


IV. 3.] 

68. Those, however, who havo had a most perfect mastery 
of the clear doctrine of tho sphere, havo declared that this is 
the length of that circumference* bounding tho limits, to which 
tho darkness dispelling rays of tho Sun extend. 

G9. But let this be tho length of tho circumfcrcnco of tho 
BkahmXnda or not [of that I haVo no suro knowledge] but it 
is my opinion that each planet traverses a distaneo correspond- 
ing to this number of Yojanas in tho course of A^Kali'A nr 
a day of BkaumA and that it has been called the Kiiakaksua by 
tho ancients. 

End of third Chapter called the Bhuvana-kos'a or cosmo- 
grapliy. 


CHATTER IV. 

Called Madiiya-oati-vasana.* 

On the principles of the links fpr findiiuj the mean places of 
the Fbuiets . 

Places of the several 1. Tlio seven [grand] winds havo 
" md8 * thus been named : viz. — 

.1st. Tho Aval™ or atmosphere. 

2nd. The Pravalia beyond it. 

3rd. Tho Udvaha. 

4th. The Samvaha. 
oth. Tho Siivaha. 

Cth. Tho Parivaha. 

7th. Tho Parfvvalia. 

2. The atmosphere extends to tlio height of 12 Yojanas 
from the Earth : within this limit are tho clouds, lightning, &e. 
Tho Pravalia wind which is above the atmosphere moves con- 
stantly to tho westward with uniform motion. 

3. As this sphere of tho universo includes the fixed stars 
and planets, it therefore being impelled by the Pravalia wind, 
is carried round with the stars and planets in u constant 
revolution. 



128 


Translation of the # [IV, 4 

An illustration of the 4. Tho Planets jnoving eastward 
motion of tho ptaeta. k tho IIoayens ^ gW 

appear os if fixed on account of the rapid motion of tho sphcic 
of .the Heavens to the west, as insocts moving reversely, on a 
whirling potter’s wheel appear to be stationary [by reason 
of their comparatively slow motion] . 

Sidereal and terrestrial G. If a star and tho Sun riso simulta- 

days and thiir lengths. ncously [m my ^ ^ ^ ^ 

riso again (on the following morning) in GO sidereal ohajik^s : 
tho Sun, however, will riso later by tho number of asus (sixths 
of a sidereal minute), found by dividing tho product of the 
Sun’s daily motion [in minutes] and the asus which tho si<ni, 
in which the Sun is, takos in rising, by 1 800 [tho number of 
minutes which each sign of tho ecliptic contains in itself]. 

G. Tho timo thus found added to tho GO sidereal giiatikas 
forms a truo terrestrial day or natural day. The length of 
this day is variable, as it depends on tlio Sun’s daily motion 
and on the timo [which different signs of the ecliptic take] 
in rising, [in different latitudes : both of which are variable 
olemonts] .* 


• [Ilad tlio Sun moving with uniform motion on the equinoctial, the 
each niiuuto of which rises ill ouch asu, the number of asus equal to 
the number of tho minutes of tho Sim’s daily motion, being added to 
.the GO sidereal giiatikas, would have invariably mode the exact length of tlio 
true terrestrial day os Lalla and others Bay. Hut this is not the case, became 
the Sun mores with uncquul motion on tlio ecliptic, tho equal portions of which 
do not riso in equal times on account of its being obhquo to the equinoclionnl. 
Therefore, to find tho exact length of Iho true terrestrial day, it is necessary to 
( determine the time which the minutes of tho Sun’s duily motion take in rising 
' and then add this time to GO sidereal ghatika's. For this reason, the terres- 
trial day determined by Lalla and others is not a true but it is a menu. 

Tho difference between tlio obliquo ascension at tlio beginning of any given 
day, and that at tho end of it pt at tho beginning of the next day, is the time 
which the minutes of tho Sun’s motion at tho day above alluded to take iu 
rising, but as this oannot bo easily determined, tlio ancient Astronomers having 
determined tho periods which the signs of tho adiptio take iu rising at a given 
place, find tho timo which any portion of a given sigii of tlio ecliptic takos in 
rising, by the following proportion. , 

If 30* or 1800' of a sign : take number of tho asus (which any given sign 
of tho ecliptiu takes) in rising at a given place : : whut time will any portion of 
the sign above alluded to take in rising F 
The calculation which is shown in the 5th verso depends on this proportion.— 
1 ). D.] 



[V. 10.] 


Sidhinta-s if omani 


129 


Revolutions of tlu So, In 7 ‘ A sidereal day consists invari- 

y »r are lest than the ably of 60 sidereal GHATfKXs ; a mean 
evolution* of stars by one. , _ . T 

. savana day of the Sun or terrestrial 

lay consists of that time with an addition of the number of 
lsus equal to the number of the Sun's daily mean motion 
in minutes] . Thus the number of terrestrial days in a year 
s less by ono than tlifc number of revolutions made by tho 
fixed stars. « 

8. Tho length of the (solar) year is 

length of •*, year. ggj ^ 

30 TALAS, 224 

f I talas reckoned in Bh<Jmi savana or terrestrial days : The 
T * T th of this is called a sauba (solar) month, viz. 30 days, 26 
uhajikAs, 17 palas, 31 vitalas, 52 4 pravitalas. Thirty 
sXvana or terrestrial days make a savana mouth.* 

Length of lunar month 9. The time in which the Moon 
n lunation. [aftor being in conjunction with tho 

Sun] completing a revolution with tho difference between tho 
daily motion and that of the Sun, again overtakes tlio Sun, 
(which moves at a slower rate) is called a Lunar month. It is 
29 days, 31 GjiatikAs, 50 palas in length.f 
Tho reason of additive 10. An AdhimXsa or additivo month 
months called Adhima'bab. w hi c h i a lunar, occurs in the duration 

of 324 saura (solar) months found by dividing tho lunar month 
by tho difference botwocn this and the saura month. From 


• [Here a solar year consists of 365 days, 16 giiatikXs, 30 palas, 22} 
vipaXiAB, i. e. 365 d. 6 h. 12 m. 9 t. and in Surya-siddha'hta the length of tho 1 
year is 365 A 15 y. 31 p. 31. 4 0 . i. e. 365 d. 6 A 12 tn. 36. 56 «.-B. J).j 
[f That lunar month which ends, when the Sun is in If ksha stellar Arie* is 
cnlled ohaitra and that which terminates when the Sun is in vpiBiUBUA stellar 
Taurus, is oalled Vais 'akiia and so on. Thus, the lunar months corresponding 
to the 12 stellar signs Mbsba (Aries) V^ishabiia (Taurus) Mithuva (Gemini) 
Kaua (Cancer), Sinha (Leo), Kanta' (Virgo), Tula' (Libra), VriB'ciiika 
(Scorpio), Dhahu (Sagittarius), Makaba (Caprioomus), Kumbha (Aquarius) v 
and Mika. (Pisces), are Chaitba, Vais'a'rktia, Jyeshtha, A'ska'diia S'ra'vana, " 
Bha'dbapada, A's'wika, Ka'btika, Ma'boasTusiia, PAUsnA, Ma'giia,' and 
Pha^guha. If two lunar months terminate when tho Sun is onl' in onn stellar 
sign, the second of these is called Aduiva'ba an additive month. The 30th 
part of a lunar month is called Tithi (a lunar day).— JB. D.J 

E 



130 ’ Translation of thn % [IV. 11, 

# 

this, tho number of the additive months i^a kalpa may also 
be found by proportion.* 

11. As a mean lunar month is shorter in length than a 
mean sausa month, the lunar months aro therefore more in 
number than tho* saura in a kalpa. The difference between the 
number of lunar and saura months in ,a kalpa is called by 
astronomers the number of Aduimxsas 'in that period. 

Tho refljon of subtractive 12. An AVAMA or subtractive day 
day called Avama. which is SXVANA occurs in G4 T ! T tithis 

(lnnar days) found by dividing 30 by tho difference between 
the lunar and sXvana month. From this, the number of avamas 
in a yuoa may be found by proportion.*! 

13.J If the Aduimxsas aro found from saura days or months, 
then the result found is in tho lunar months, [as for instance 
in finding the Aharvana. If in tho saura days of a kalpa : aro 


4 [After the commencement of n yuoa, a lunar month terminates at tho 
end of Amava'sya' (new moon) audm sauba month at tho mean yuisiiabha* 
BANKRa'nti (i. o. wlien tho mean Sun enters the second stellar sign) which takes 
place with 64 g. 27 p. 31 «. 621 P> after tho new moon. Afterwards a second lunar 
month ends at the 2nd now moon after which tho MrrnuHA-SANOA'NTi takes 
place with twice tho Gliatis. Ac. above mentioned. Thus tho following San- 
kka'ktis karka Ac. take place with thrice four times Ac. thoso GuAfis, Ac. In 
this manner, when the Sahkra'mti thus going forward, again takes place at new 
moon, tho number of tho passed lunar mouths exceeds that of tho saura by oue. 
This one month is called an additivo month: and the sauua months which an 
additive month requires for its happening cun be found by tho proportion as 
follows. 

As 5ft ghttli*, 27 p. Ae. tho difference between a lunar and a saura 
: One saura mouth 

: : 29, 31, 50 the numbor terrestrial day Ao. in a lunar month 
: 32, 16, 31, &o. tho number of saura mouths, clays, Ac.— R. D.] 

t [At the beginning of a kalpa or a yuoa, the terrestrial and lunar 
days begun simultaneously, but tho lunar day being loss than the terrestrial day, 
terminated before the end of tho terrestrial day, i. e. before the next sun-riso. 
The interval between the end of the lunar day and the next sunrise, is called 
Avama-s'esua tho remainder of the subtractive day. This remainder increases 
every day, therefore, when it is 60 Ghatikas (24 hours), this constitutes a 
Avama day or subtractive day. The lunar days in which a subtractive day 
occurs, ore found by the following proportion. 

If Od. 28 g, 10 p. the difference between the lengths of terrestrial and of a 
luuar month. » 

: 1 lunar month or 30 tithis 

t : a whole terrestrial day : 64-^titlUs nearly.— B. D.] 

I The objects of these two verses seems not to be more than to assert that 
the fourth term of a proportion is of the same denomination as the 2nd*— L. W. 



JSidMuUi-tfiromaiiL 


131 


A question. 


IV. 16.] 

so many AdiiimA^s : : then in given number of solar days j 
how many AdhimXsab ?] If the AdhimXsas are found from 
luqar days or months, then the result is in saura months, and 
thcnMnaindcr is of^ the like denomination. 

14. [In like manner] the avamab or subtractive days if 
found from lunar days, are in sXvana time : if found from 
s .{ van A time they are lunar and the remainder is so likewise. 

1 5. Why, 0 Astronomer, in find- 
ing the Abargana do you add sauiia 
months to the lunar months Cuaitha Ac. [which may have 
(‘lapsed from tlio commencement of the current year] : mid 
tell mo also why the [fractional] remainders of Aim I mamas and 
Avama days arc rejected : for you know that to give a true 
result in using the rule of proportion, remainders should bo 
taken into account ? 

feuon or omitting (o in- 10 ’* As 1,10 ,unartnm ’ th ™ ls nt 
elude tlio Aduimasa s'rsua the change of tlio Moon and tlio 
in finding the Auauqana. § . , . , 

8UAKA month terminates when the Sim 

enters a stellar sign, the accumulating portion of an Aduimasa 
always lies after each new Moon and before the Sun enters tlio 
sign. 


* [The meaning of these 4 verses will bo well understood by a knowledge of the 
rule for finding tlio Auaikuna, we therefore show the rule here. 

In order to find the Ahargasa (elapsed terrestrial days from the commenee- 
ment of the Kalpa to the required time) Astronomers multiply the number of 
bauiia years expired from the beginning of the Kall*a by 12, and thus they get 
the number of 8AVRA months till the last Mebua Sakkrahti (that is, the time . 
when tlio Sun enters the 1st sign of the Zodiac called Aries.) To these months 
they add then the passed lunar months Ciuitba Ac., considering them as bauba. * 
These bauha months become, up to the time when the Sun enters the sign of the 
Zodiac corresponding to the required lunar mouth. They multiply then the num- 
ber of these mouths hy 30 and add to this product the number of the passed 
Tunis (lunar days) of live required month considering them as sac ha days. The 
number of bauba days thus found becomes greater than that of those till the 
end of the required tithi by the aduimasa b'stiia. To make these bauba davs 
lunar, they determine tlio elapsed odditivo mouths by the proportion in the 
following manner 

At the number of bauba days in a Kalpa 
: the number of additive months in that period 
: : the number of liURA days just found 
: the number of additive mouths elapsed 

K 2 



132 


Translation of the * [IV. 17. . 

» 

17. Now the number of tithis (lunar cjpys) elapsed since 
tho chan go of the Moon and supposed as if sauba, is added to • 
the number of sauba days [found in finding the Ahabgan*] : 
but as this nnmber exceeds the proper amount by the qyAitity 
of tho Adhimasa-s'esha thereforo tho Adhimas-sesha is omit- 
ted [to bo added] . 

18. [In tho same manner] there is always a portion of a 
Avama-s*esha between tho time of sun-rise and tho end of tho 


[preceding] tithi. By omitting to subtract it, the Aitahgana 
is found at tho time of sun-riso : if it were not omitted, tho 
Ahaboana would represent tho timo of tho end of .tho tithi 
[which is not required but that of tho sun-rise] . 

Reason of tho correction 19 ’ 20 > 21 and 22 ; ** th ° tn, °* 

called the UdayAjttaba terrestrial day is of variable length, the 
mm&e Aitaroaxa has been found in mean 

terrestrial day&: tho places of the planets found by this 
Ahargaxa when rectified by tho amount of the correction 
called tho Udayantara whether additivo or subtractive will bo 
found to bo at tho time of sun-rise at LankI.* The ancient 


If these additive months with their remainder be added to the sauba days 

• above found, the sum will be the number of lunar days to the end of tho sauba 
days, but we require it to the end of tho required tithi. And as tho remainder 
of tho additive months lies between the end of the Tirm and that of its corre- 
sponding sauba days, they therefore add tho whole number of Adiii-masas just 
found to that of tho sauba days omitting the remainder to And the lunar days 
to the ond of tho required tithi. Moreover, to inako these lunar days terrestrial, 
they determine Avaha subtractive days by the proportion such as follows. 

As the number of lunar days in a Kalpa 
• the number of subtraotive days in that period 

• : : the number of lunar days just found 

, : the number of Ayaxa olapsed with their remainder. 

If these AYAXAS be subtracted with their remainder from the lunar days, the 
difference will be tho number of the Ayaxa days olapsed to the ond of the requir- 
ed tithi; but it is required at the time of sun-rise. And as the remainder of the 
eubtraotivo days lies between the end of the tithi and thesun-rise,they therefore 
subtract the Ayaxas above found from the number of lunar days omitting their 
remainder i. e. Avaxa-b'bsha. Thus the Ahaboana itself becomes at the sun- < 
rise.— B. D.] 

[If the Sun been moving on the equindbtial with an equal motion, tho 
terrestrial day would have been of an invariable length and consequently the 
Sun would hare reached the horizon at LavkI at the end of the Ahaboa&ta 
which is an euumeration of the days of invariable length that is of the mean 
terrestrial days. But the Sun moves on the ecliptio whose equal parts do not 



133 


l\\ 23.] pidhuntas'iromanl . 

Astronomers have got thus rectified the places of the planets 
by this correction, as it is of a variable and small* amount. 

{Die difference between the number of asus of the right 
ascemion of the mean Sun [found at the end of the Ah a ru ana] 
and the number of asus equal to the number of minntes of 
the mean longitude .of tho Sun [found at tho same time] is 
tho difference between the true and mean aiiauganas.* Mul- 
tiply this difference by tho daily motion of tho planet and 
clivido the product by the number of asus in a nycthemeron.f 
The result [thus found] in minutes is to be subtracted from 
tho places of the planets, if the asus [of tho right ascension 
of the mean Suu] fall short of the kal.(s or minutes [of tho 
mean longitude of tho Sun], otherwise tho result is to bo 
added to tho places of tho planets. Instead of tho right as- 
cension, if obliquo ascension bo taken [in this calculation] 
this Udayantara correction which is to be •applied to tho 
places of the planets, includos also tho chara correction or 
tho correction for tho ascensional difference. 

Reason of the correction 23. The places of tho planets 
called tho Dsa a'ntaba. which are found being rectified by 

this Udayantara correction at tho time of sun-riso at Lanka * 
may bo found, being applied with tho Desantaua correction, 
at the timo of suu-riso at a given place. This Dksantara 
correction is two-fold, one is cast and west and the other 

rise in equal periods. For this reason, the Sun does not come to tho horizon at # 
Lanka' at the end of the Ababgana. Therefore the places of tho planets * 
determined by the mean Ahakgasta, will not beat the sun-rise at Lanka'. Hence • 
a correction is necessary to be applied to the places of the planets. This 
correction called TJdayAntaba has been first invented by BaXsiiABAcnluTA who 
consequently abuses them who say tliafc the places of tho planets determined by 
the mean Ahaboana become at the time of the sun-rise at Lanka.— B. D.J 
* The difference between the mean and true AhAbgAnAs is that port of 
the equation of time which is duo to the obliquity by the ecliptic.— L. W. 
t [This calculation ia nothing else than the following simple proporiioiw— 

If the number of Aaus in a nycthemeron 
: daily motion of the planet 

: : tho difference between the true and moan AuaboanaS 
give.— B. 1).] * 



131 * Translation of the # [IV. 21. . 

is north and south. This north and soutj} correction is called 
ciiara. * ' 4 

24. Tho line which passes from Lank£, Ujjayin/, Kj#a- 
kbhetka and other places to Meru (or tho North Pole the 
Earth) lias been denominated tho MadhyaeekhI mid-lino of 
tho Earth, by tho Astronomers. Tho sun rises at any place 
east of this lino before it rises to that line : and after it has risen 
on the line at places to its west. On this account, an amount 
of the correction which is produced from tho difference be- 
tween the time of sun-rise at the mid-line and that at a given 
place, is subtractive or additive to the places of the planets, 
as tho given pluco be cast or west of the mid-lino [in order 
to find the places of tho planets at the time of sun-rise at the 
given place] . 

25. As tho [small] circle which is described around Mem 
or North Pole v>f tho Earth, at the distance in Yojanas reckon- 
ed from Meru to given place and produced from co-latitude 
of tho pluce [as mentioned in tho verso 50th, Chapter II T.] 
is called rectified circumference of tho Earth (parallel of latitude) 
[at that place] therefore [to find this rectified circumference], 

* tho circumference of the Earth is multiplied by the sine of co- 
latitude [of the given place] and divided by the radius. 

End of 4th Chapter called Madhya- Gati Vasana. 


* This amount of correction is determined in tho following manner. 

Tho yojanas between the midlino and the given place, in the parallel of 
latitude at that place, which is denominated Hmim-PABIDUI ara called, 
Dm'a'ntaba yojanas of that place. Then by the proportion. 

As the number of yojaNab in the Spashta-pakidw : 60 ghatikas : : Dksa'n- 
TABA yojanas : the difference between the time of sun-riso at nridliiie and that 
at a given place. This difference called Deb'a'ntaba ghatika'b is the longitude 
iu time east or west from Lanka'. Again 

Aa GO oiiatika's : daily motion of the plnnct : : DbbANTA'ea ghatika'b : 
the amount ot tho correction required. 

Or this amount can be found by using the proportion only once such as follows 
■ -r.'j tho number of yojanab in the Spasuta-Pabidui : daily motion of tho 
planet : : Drs'AntabA yojanas: the same amount of the correction above 
found.— B. D.] 



t Siilhanta-*'imnani. 


135 


V. 


20 


* 


CHAPTER Y. 

OKthc principle^ on which the Buie* fur failing the true 
places of the Planets are groutuleil. 

n lf , . • 1, The pianos of a Sphere arc 

On the canon of Bines. § 1 1 f 

intersected by sines of bhuja and 

koji,* as a picco of cloth by upright and transverse threads, 
lieforc describing the spheric, I shall first explain tho canon 
of sines. 

2. Tako any radius, and suppose it the hypothennso (of 
a right-angled triangle). The sine of biiuja is the base, ami 
the sino of koji is tho square root of tho difference of tho 
squares of the radius and the base. The sines of degrees of 
iniujA and KO'pi subtracted separately from the radius will 
bo tho versed sines of koji and bhuja (respectively). 


[* The rniVJA of any given nrc is that nro, less than 90°, the sine of which 
i« equal to tho sine of that given are, (tho consiclerntiun of tho positiveuess ami 
negation of the sino is hero neglected). For this reason, tin* lMUJA of flint 
lire which terminates in the odd quadrants i. c. tho 1st and 3rd is that part 
of I ho given are which fulls in tho quadrant where it terniinnlea, and tho 
Biiuja of tho arc which ends in tho even quadrants, i. o. in the 2nd and 4th, 
is that arc which is wanted to complete tho quadrant where tho given aro is 
ended. 

The Koti of any aro is tho complement of tho bhuja of that arc.. 


Let Iho 4 quadrants of a circle 
A 11 C D be successively A B, B 
C 1) and 1) A, then the iuiujas 
of the arcs A P„ A B P», A U P al 
A 1) P 4 will b« A P |t C P B , C P„ 
A P 4 and tho complements of 
theso UHL-JAS aro the arcs BP,, 
B P g , U l* a , 1) P 4 respectively.— 
B. l>.] 




136 


Translation of the t 


[V. 3. 


3. The versed sino is like the arrow i|tersocfcmg % tlie bow 
and the string, or the arc and the sine.* 

The square root of half the square of the radius is the sjne 
of an arc of 45\ The co-sino of an arc of 45* is of tho/Samo 
length as the sine of that arc. * * 


* These methods are grounded upon the following principles, written 
Bhi bkaba'oiubya, in the commentary Vasaxa'-bha'bhya. 


by 


(1) Let tho are A B = DO* and A C = 
45® 

A D (- 1 A B) = sin. 46° ; and let 
0 A or 0 11 = the radius (R) then A B* 
= 0 A’ +OB> = 20A* =2R* 

.'.«! = 

and A D = 1 A B = \/& 

2 


or sin. 45° 




(2) It is evident and stated also in tho Lila'vati, that the sidoof a regular 
hexagon is equal to the radius of its circumscribing circle (i. e. ch. 60° = R). 
Hence, sin. 30* = i R. 

(3) Let A B be the half of a given arc A P, whose sino P M and versod sine 
A M are given. Then 

ap= v^pmhTam 1 

and J A P s= A N = sin. A B 

sin. A B = \ V'pM’ + AM*" 

(4) The proof of the last method by Algebra 

cos = R — versed sine 
,\ cos 1 == R« — 2 R . v + e 1 
subtracting both sides froniR 2 , 

R*— cos* - 2R,ti — o” 
or sin.* = 2 R . c — e* 
adding v 4 to both sides 
•in.” + e*ss! 2R. e 
and 1 (sin. 1 + v‘) = J R . v 
extracting the square root, 

i */.!«.■+»• = ynrr 

by the preceding method , 

4 J\ sin,” + t” = the sine of half the given are \ 




Sidhdntas'iromani. 


V. 7.] • Sidhdnta-8'iroma$i. ‘ 137 

4: Half the ipdius is the aine of an are of 30°: The 
co-sine of an arc of 30* is the sine of an arc of 60°. 

^Half the root of the sum of *the squares of the Bine and 
veratd sine of an arc, is the sine of half that arc. 

5. * Or, the sine of half that arc is tho square-root of half 
the product of the radius and the versed sine. 

The sines and co-sides of the halves of the arcs before found 
may thus be found to any extent. » 

6 . Thus a Mathematician may find (in a quadrant of a 
circle) 3, 6 , 12, 2t &c., sines to any required extent.* 

Or, in a circle described with n given radius and divided 
into 360°, tho required sines may be found by measuring their 
lengths in digits. 

Reason of correction which 7.f As tho Centre of tho circle of 

frora? tlio^ mean "place* o'f “a , tll ° constellation of the Zodiac coin- 
P ,unet * cidcs with tho centra of the Earth : 


• [When, 24 sines are to be determined in a quadrant of a circle, tho 3 sines, 
i. e. 12th, 8th and 161 h, can be easily found by tho method here given for finding m 
the sines of 46°, 30°, and tha complement of 30°, i. e. 6C« Then by means 
of these three sines, the rest can bo found by the method for finding the sine 
of half an arc, as follows. Fiom the 8th sine, the 4th Bnd the co-sine of tho 4th 
i. e., the 20th sine, can be determined. Again, from tho 4th, the 2nd and 22nd,* 
and from the 2nd, the 1st and 23rd, can be found. In like manner, the 10th 
14th, 6lh, 19th, 7th, 17th, lltli, and 13th, can also he found from the 8th sine. 
From the 12th again, tho Glh, 18th, 3rd, 21st, 9th ami 15th ran be determined, 
and the radius is the 24th sine. Thus all the 24 sines are found. Several 
other methods for finding the sines will be given in the sequel. — B. J).] 

[t Bda'skara'chabya maintains that the Earth is in tho centre of the 
Universe, and the Sun, Moon and the five minor planets, Mars, Mercury, &c. 
revolve round the Earth in circular orbits, the centres of which do not coincide 
with that of the Earth, with uniform motion. The circle in which a planet* 
revolves ia called Pbatiy^itta, or exoentrio circle, and a circle of the same sixe, 
which ia supposed to have the same centre with that of the Earth, is called 
Kaksha'vritta. or ooncentrio circle. In the circle, the planet appears to revolve 
with unequal motion, though it revolves in the excentrio with equal motion. 
The place where the planet revolving in the excentrio appears in the concentric 
ia its true plaoe and to find this, astronomers apply a correction called mahda- 
ph ala (1st equation of the centre) to the mean place of the planet. A mean 
planet thus corrected ia called makda-spabh^a, the circle in which it revolves 
XAVDA-PKATfvfiTTA (1st excentrio) and ita farthest point from the centre gf the 
concentric, MANDOCHOH (1st higher Apsis). As the mean places of the Sun and 
Moon when corrected by 1st equation become true at the centre of the' Earth, 
this correction alone is sufficient for them. But the five minor planets, Mara, 
Mercury, Ac. when corrected by the lit equation are not true at the centre of 
the Earth but at another place. For this reason, astronomers having assumed a 

V 



138 


Translation of the 


[V. 8. 

# 

and the centre of the circle in which the planot revolves does 
not coincido with the centre of the Earth: the spectator, 
therefore, on the Earth does dot find the planet in its mpfn 
place in the Zodiac. Hence Astronomers apply the correction 
called bhuja phala to the mean place of the planet [to get 
tho true place] . 

Mode of, illuitration of 8. On the northern side of a wall 
the above M. running duo east and west, let the 

teacher draw a diagram illustrative of the fact for the satis- 
faction of his pupils. 


A verse to encourage those 
who may bo disposal to de- 
spond in consequence of Lhe 
difficulties of the science. 


9. But this science is of divine 
origin, revealing facts not cognizable 
by the senses. Springing from the 


tho concentric circle as Bccond excontrro of those fire planets, tale another 
circle of the same size and of the samo centre with tho Earth as concentric, and 
in order to find tjte placo where tho planet revolving in tho 2nd oxceutrio 
appears, in this concentric, they apply a correction called s'foHiiA-PiiALA, or 2nd 
equation of tho centre, to the mean place corrected by tho 1st equation. Tho 
MANJOA-BPASttTA planet, when corrected by the 2nd equation is called s'pasiita, 
or true planot, the 2ndcxoenttic, a'fuURA-PKATiVRrm, and its farthest point from 
1 the centre of the Earth, s'iohbocucii the 2nd higher Apsis. 

If a man wishes to draw a diagram of the arrangement of tho planets accord- 
ing to what we have briefly stated here, he should first describe the excentrio 
.circle, and through this excentrio tho concentric, and tlu>n he may determine 
‘ the placo of the mahda-spasuta planet in the concentric thus described. Again, 
having assumed tho concentric us 2nd excentrio ami described tho concentric 
through this 2nd excentrio, lie may find tiio place of the true planet. This is 
the proper way of druwing the diagram, bub astronomers commonly, having 
first described the concentric, and, through it, the excentrio, find the corrected 
mean place of the planet in the concentric. After this, haring described tho 
2nd excentrio through tho same cunccntrio, they find the true place in tho 
concentric, through the corrected meun place in the same. These two modes 
of constructing the diagram differ from each oilier only in the respect, that iu 
*the former, the concentric is drawn through the excentrio circle, and in the latter, 
, the excentrio it drawn through the concentric, but this can easily be understood 
that both of these modes are equivalent and produce the sumo result. 

In order to find the 1st and 2nd equations through a different thcoiy, astro- 
nomers assume tliat tho centre of a small circle called ZJfcHOCHCHA-vaiTTA or 
epioyole, revolves in the concentric circle with the mean motion of the planet 
and the planet revolvos in the epicycle with a reverse motion equal to the mean 
motion. Bba'skaka'ciia'rya, himself will show in the sequel that the motion of 
the planet is the same in both these theories of exoentricB and epicycles. 

„ jfcjatdbe observed here that, in the ease pi the planets Mars, Jupiter and 
Saturn, the motion in the excentrio is in fact their proper revolution, in their 
orbits, and the revolution of their b'iohrochoha, or auick apogee, corresponds to 
a revolution of the Sun. But in the case of the planets Mercury and Venus, 
the revolution in the excentrio is performed in the same time with the Sun, and 
. the revolutions of their s'iohhochchab are in fact their proper revolutions in 
their orbits.— B. D.] 



Sidhantas'iromani. 


m 


w 12.] 

supreme Brahma Jumself it was brought down to the Earth 
by Vasishtha and other holy Sages in regular succession ; 
though it was deemed of too secret a character to be divulged 
to men or to tho vulgar. Hence, this is not to bo communi- 
cftted*to those who revile its revelatious, nor to ungrateful, 
evil-disposed and bad men : nor to men who take up their 
residence with its professors for but a short time. Those 
professors of this science who transgress theso limitations 
imposed by holy Sages, will incur a loss of religious merit, and 
shorten their days on Earth. 

_ 4 „ 10. In tho first place then, de- 

construction of a dm- r 

gram to illustrate the ox- scribe a circlo with tho compass opened 

centric theory. ^ tho length of tho radius ( 3438 ). 

This is called tho kaksitavjutta, or concentric circlo ; at tho 
centre of tho circle draw a small sphere of the Earth with a 
radius equal to ^th* of the mean daily molioM of tho planet. 

11. In this concentric circle, having marked it with 300°, 
find tho place of tho higher apsis and that of tho planet, 
counting from tho 1st point of stellar Aries; then draw a * 
(perpendicular) diameter passing through tho centre of tho 
Earth and tho higher apsis (which is called uchcha-rekhX; 
tho lino of the apsides) and draw another transverse diamotor 
[perpendicular to the first] also passing through the centre. 

12, On this line which passes to tho highest apsis from tho 
centre of tho Earth, take a point at a distance from the Earth's 
centre equal to tho excentricity or tho sine of the greatest 
equation of the centre, and with that point as centre and the 
radius [equal to the radius of tho concentric], describe the 
prativ&itta or exccntric circle; the ucncuA-REKHA answers 
the like purpose also in this circle, but make tho transverse 
diameter different in it. 

* AU the Hindu Astronomers seem to coincide in thinking that the horizontal 
parallax pabaiul-laxbaiu of all the planets amounts to a quantity equal to 
y^th of their daily motion.—!. W. * 

F 2 1 



140 


Translation of the 


[V. 13. 


< 

13 and 14.* Where the uchcha-bekha .perpendicular dia- 
meter (when produced) cuts the excentrio circle, that is the • 



[ In fig. lit let G be the centre of the coneentrio circle A B C D, T the place 
of the atellar Aries, A that of the higher ape is, and M that of the mean planet 
in it : then ]£ A will be the uchciia-bekita (the line of the apaidea). Again 
let E 0 be the excentricity and HFLG the excentrio which haa 0 for its 
centre j then H, T P, will be the places of the higher apsis, the stellar Aries 
and the planet respectively in it. Henoe H P will be the kxhdha * P X the 
l ine of the nnmu j P I the co-sine of the xikdba. 

""-■'TMFkEUDM which is more than 9 signs and Mess than S is called XBIOADI 
fi. e. that which terminates in the six signs beginning with Caprioornus) and 
that which ia above 3 and less than 9 is oalled xabxxadx (i. a. that which ends 
in the six signs beginning with Cancer), « 

Thus (Fig. 1) that which terminates in G H F is HSXOADX xxxdba, and 
4 that which ends in F L G is Kabxya'di.— B. D.] 


Sidh&nta-Jiromani . 


141 


■ V. 15.] 


place of the higher «psis in it also. From this mark the first 
• stellar Aries, at the distance in degree of the higher apsis 
in antecedentia : the place of the* planet must be tlion fixed 
counting the degree^ from the mark of the l§t Aries in the 
usual oriler. 

The distance between tho higher apsis and tho planet is call 
ed the kendra.* Tho right line lot fall from the planet on the 
rcncHA-REKHA is the sino of bhuja of tho kendra. Tho right 
lino falling from the planet on the transverse diameter is tho 
cosine of the kendra, it is upright and the sine of bhuja is a 
transverse line. 


The principle on which 
the rule for finding the 
amount of equation of centre 
is based. 

of the kendra is above 


15. As tho distance between tho 
diameters of the two circles is equal 
to the excenfcricity and tho co-sino 
and below the excentrieity when the 


kendra is mrioadi and karkyKdi (respectively), t 


* The word Kinds* or centre is evidently derived from tho Greek word 
Ktrrpov and means tbetrue contre of tho planet.— L. W. 


+ fin (Fig. 1) P K is the sphuta koti and P E the xabjia (tho hypothonuse) 
which cute the concent rio at T. Hence tho point X will bo the apparent place 
of the planot and T M the equation of the centre. 

This equation can bo determined as follows. 

Draw M n perpendicular to E T, it will be tho sine of the equation and tho 
triangle P M « will be similar to the triangle P E K. 


.\PE:EK = PM:M»i 


FU.SK 

lienee M n =■ — — sine of the equation ; 

PE 


EO.EK 
" PE 


,forPM = IK=*EO 


Now, let A =* kinds*, a — the distance between the centres of the two 
circles excontrio and concentric, x = sine of the equation, and A = hypothenuse : 
then the sphuta xoyi c= cos. A ± a, according as the Kbvdb* is mbigadi or 

XASXTADI, and A ^ sin . 1 k db (cos. A 1 fl)* 

hence by substitution 

a . sin. k s . sin. k 


t/sin.* A + (oos. k ± a)* 



142 ^ Translation of the • [y, jg 

16 and 17. Therefore the sura or, diffewnce of the co-sine 
and exoentricity (respectively) is here the sphut* ko?i (i. e 
the upright sido of a right-ahglcd triangle from the pW 0 ' 
the planet in tl*e excentric to the transverse diameter to the 
concentric,) the sine of the bhuja [of the kendra/ is the 
bhuja (the base) and the square-root of tho sum of the squares 
of tho fJPliUTA kott and bhuja is called karna, hypothenuso, 
This hypothenuso is the distanco between the Earth’s centre 
and the planet’s place in tho excentric circle. 

Tho planet will bo observed in that point of the concentric 
cut by tho hypothenuse. 


The equation of the centre is tho distanco between tho 
mean and apparent places of tho. planet : when the mean 
placo is moro advanced than the apparent place then the 
equation thus found is subtractive ; when it is behind tho true 
place, the equation is additivo.* 


18. The moan planet moves in its 
MANDA-rfiATivRTTTA (first excentric); 
tho HANDA-sPASHfA planet (i. e. whoso 
mean place is rectified by tho first equation) moves in its 
s 'io HRA- peati vjjitta (second excentric). The manda-spasttta 


The reason for assuming 
tho KAND-sPAsnpA planet aa 
a mean in finding the 2nd 
equation. 


It also follows from this that, when cos. k is equal to a in the kabkyadi 
XitSDitA, then A will bo equal to sin. fr, otherwise A will always be greater than 
sin. k and consequently x will be less than a. Hence, when A is equal to Bin A, 
x will then be greatest and eqnal to <r, i. e. tho equation of tho centre will be 
greatest when tho hypothenuso is equal to the sine of tho BXNDRa, or when the 
• planet reaches tho point in tho excentric cut by the transverse line in the 
concentric. Therefore, tho centre of tho excentric is marked at the distance 
equal to tho cxcentricity from the centre of the concentric (as stated in the 
V 12th.) — B. D.] 

• [Thus, the mean planet, corrected by the 1st equation, becomes 
KANDA-spashi'A and this process is called the if abda ' process. After this, 
the xavda-bfabhta when rectified by the bi'ghha Praia, or 2nd equation, is 
.mJ liajsPASHTA planet, and this 2nd process is termed the s'ighba process. Both 
of these processes, mabda-sfashta and BPArtrjA are reckoned in the timan- 
pALA or the orbit of the planet as fainted at by Bhaskabachabta in the 
commentary called Vababa-bhabhta in the sequel. These places are assumed 
for tho ecliptic also without applying any correction to them, because the 
correction required is very imall.T-B. D.] 



143 


y 23.] •Sidhdnta-sirmani. 

is therefore here asaumed as the mean planet in the second 
process (i. e. in finding the second equation).* ' , 

ft- reason for the inven- 19* The place in thq concentric 
lion oi^he higher apsis. j n which the revolving planet in its 
own exhentric i3 seen by observers is its true place. To find 
tho distance between the true and mean places of tho planet, 
the higher apsis has bc'eu inserted by former Astronomers. 

20. That point of the exccntric which is most distant from 
the Earth has been denominated the higher apsis (or 
cchceta) : that point is not fixed but moves; a motion of tho 
higher apsis has therefore been established by thoso con- 
versant with tho science. 

21. The lower apsis is at a distance of six signs from the 
higher apsis : when tho planet’ is in cither its higher or 
lower apsis, then its true place coincides with its mean 
place, because the line of the liypothenuse fallS on tho mean 
placo of the planet in the concentric. 

22. As the planet when in the higher apsis is at its 
greatest distanco from the Earth, and when in tho lower 

The cause of variation of at toast distance, therefore its 

apparent aize of planet’s dwc. disc appears small and large accord- 
ingly. In like manner, its disc appears small and largo accord- 
ingly as tho planet is near to and remoto from the Sun. 

23. To prevent tho student from becoming confused, 
I have separately explained tho proof of finding the equation 
by the Phativkitta Biianoi of the diagram of the exeunt ric. 
I shall now- proceed to explain the same proof in a different 
munuor by tho diagram of a NfoHOCHCHA-viUTTA (epicycle). 

• [For this reason, having assumed the UAXDA-gPASlTf-A planet for the 
mean, which manda-bpasuta can be determined in the concentric by describing 
the cxcentric circle Ac. through the mean planet and mavdocucua, malu^the 
place of the stellar Aries from the manda-svabhta place in tho reverse order of 
the signs and then determine the place of the s'lOHBOCncnA in the order of 
the signs. Through the places of the stellar Aries and s'rGunocnCKA describe 
the 2nd exccntric circle &c. in the way mentioned before, and then find the 
place of the true planet in the concent nc.—B. 1).] 



144 


1 'mutation of tho t 


[V. 24. 


Construction .of Di.gr.rn 21 Takin & th « mean P 1 *** of the 
to illustrate the theory of planet in the concentric as the centre, 

with a radius equal to the excentroity 
of tho planet,^ draw a circle. This is called NfcHofocm 
veitta or epicycle. Then draw a line from the centre* of the 
Earth passing through tho mean place of the planet [to the 
circumference of the epicycle] . ' 

25. That place in the epicycle most distant from the 
centre of the Earth, cut by the line [joining the centre of the 
Earth and mean place of the planot] is supposed to be the 
place of the higher apsis: and the point in the epycicle 
nearest to the Earth's ccntro, the lower apsis. In the 
opicycle draw a transverse line passing through the centre of 
it [and at right-angles to the above-mentioned lino which is 
called here uchcha-tiekha]. 

26. As th6 mean planet revolves with its kendra-gati 
( tho motion from its higher „apsis) in tho 1st and 2nd epicy- 
cle marked with the- 12 signs and 360 degrees towards tho 
reverse signs, and according to tho order of the signs respec- 
tively from its higher apsis. 

27. Mark off therefore tho places of the first and secoud 
KENDRAS or distances from their respective higher apsides in 
the manner directed in the last verse : tho planet must bo 
fixed at those points. [Here also] Tho (perpendicular) line 
from tho planet to tho uchcha-rekha is tho sine of the bhuja 
of the kendra : and from the planet on the transverse line 
is the cosine [of the kendra] .* (See note nut page.) 

To flud the hypothenuse 28 and 29. The BHUJA PHALA and 
tnd the equation of centre. K0TI PHALA 0 f the KENDRA which are 

found [in tho Ganitadhyaya] arc sine and cosine in the epicycle. 
As tho kotiJphala is above the radius (of the concentric) in 
mrigadi kendra and within the radius in karkyadj-kindra, the 
Bum and difference, therefore, of the koti phala and the radius 
is here the sphuta-koti (upright lino), the bhuja phala is 
the bhuja (the base) and the karna hypothenuse (to co^lete 



145 


• V. 27.] • Sidhdnta-tiromani . 

the right-angled triangle) is the line intercepted between the 

• centre of the Earth and the planet. The equation of the 
centre is here the arc [of the concentric] intercepted between 



[In fig. 2, let A B C D be the concentric, r the place of the stellar Aries, B the 
centre of the Barth, M the mean place of the planet in the concentric, A / i g t 
the Bpicvole, A the place of the higher apsis in it, E A the uokoha»bbkha' 
l the place of the lower Apsis, P that of the planet, ft P the KiXDJtA, F ft the 
■ine of the uitsba and P i the cosine of it. 

The sine and oo-«ine of tiie ksndba in the exoentric, reduced to their 
dimensions in the epicycle in parts of the radius of the concentric, ere named' 
bhuja-fhaIiA and boji-phaia respectively in the GavitXdeyata. That 
As the radius or 860° of the concentric 
t the sine and oosine of the nxsiA in the exoentric 
# : : exoentricity or the periphery of the epicycle 
: bhwa-fhala and koji-phala respectively. 

Pwaribre the bhuja*pkala and xoti-fhala must be equal to the line and 
coahJWtbe xhtbba in the epicycle.— B. 0.] 

G 



146 Translation of the* [V. 30. 

9 * 

the mean place of the planet and the point* cut by the hypo, 
thcnuse-. The equation thus found is to be added or subtracted 
as was before explained.* ‘ , * 

. 30. The plapet appears to move forward from mandochcha, 

Construction of tho mixed OT Ut ‘Wf a P S ^> the « oentr >C 
diagrams of tho excentrio circle with its KENDRA-OAfl (the nio- 

and epicycle. tion from its mandochcha) and in tho 

order of the signs and to the East : From its si'ghrochcha, 
2nd higher apsis, it moves in antecedents or reversely, as 
it is thrown backwards. 

31. When the epicycle however is used, the reverse of this 
takes place, the planet moving in anteccdentia from its 1st 
higher apsis and in the order of the signs from its 2nd higher 
apsis. Now as the actual motion in both cases is the same, 
while the appearances are thus diametrically opposed, it must 
be admitted therefore that theso expedients are tho mere 
inventions of wise astronomers to ascertain the amount of 


equation. 

• In (Fig 2) E k is the spiiuta-koti, P E the hypothenuse, T the apparent 
place of the planet in the concentric and T M the equation of tho centre. This 
equation can also be found by the theory of the epicyole in the following 
manner. 

Draw T » perpendicular to E M, then T n will bo the sine of the equation ; 
let it bo denoted by x, tho kksdra in the excentrio by A, the excentricity by a, 
Mid the liypolhenuau by A ; then 

B : sin le = a : P A the bhfja-puala 
a sin A 

,\ the BUUJA-PHALA = — , 

B 

Mow. the triangles E T a and E P A are similar to each other 
B P ! P i = B 1’ : T s 
or A : P A = B : x 
PA X B 

... x — . j 

A 

that is, the bhujA'PHALA multiplied by the radius and divided by thfe hypothec 
nuse is equal to the sine of the equation. 

a sin A 

But P A * 

B * 

.*. by substitution 

a sin A B a sin A 

* =*= X — «=•— , the sine of the equation as 

BAA 

found before by the theory of the ex centric in the note on the verses 15, 16 and 
17.— B. D.] 



V. 37.] 


• Sidhdnta-s'imnani. 


U7 


32. If the diagrams (of tho exccntrio and epicycle) be 
drown unitedly, and the place of the planet be ifiarked off in 
tiro manner before explained, then the planot will necessarily 
be in the point of tho intersection of the oxcentric by the 
epicyclb. 

33. [In illustration of these opposito motions, examine an 
oil-man’s scrow-press.] ' As in the oil-man’s press, the wooden 
press (moving in tho direction in which tho bullock fastened 
to it goes) moves (also itself) in tho opposito direction to that 
in which the bullock goes, thus tho motion of tho planet, 

1 hough it moves in tho oxcentric circle, appears in antece- 
dents in the epicycle. 

34. As the centre of tho 1st epicycle is in tho concontric, 

Explains why the 5 minor let the planet therefore move in tho 

)!T.nd Td'oqi/to concentric with its moan motion : Iu 
their true places. the concentric [at tlinit point cut by 

tho first hypothenuso] is tho centre of the si'oiira nichochcha, 
vrttta or of tho 2nd epicycle: In the second or r'ic.hra 
epicycle is found tho true place of tho planot. 

35. The first procoss, or process of finding tho 1st cqua- 

tion, is used in tho first place, in order to ascertain tho position 
of tho centre of the sfannA nIchochcha v;iitta or of tho 2nd 
epicycle, and tho 2nd process, or the procoss of tho 2nd 
equation, to ascertain the actual place of the planet. As these 
two processes are mutually dependent, it on this account 
becomos necessary to have recourse to the repetition of tlicso . 
two processes. - 

36 and 37. Some say that tho hypothenuso is not used in 

_ , . e the 1st process, because tho differcnco 

sion of hypothenuso in the (in tho two modes of computation) 
kakdi precise. j 8 inconsiderable, but others maintain 

that since in this process the poriphery of tho first opicyclo 
being multiplied by the hypothenuse and divided by tho radius 
becomes true, and that, if the hypothenuso then be used, the 
result is the same as it was before, therefore the hypothenuse is 
o 2 



148 


Translation of the • [V. 38, 

not employed. No objection is to be made why this is not 
the case in tho 2nd process, because the proofs of finding the 
equation are different here.* ' * 

38. As no observer on the surface of tho Earth sees the 

BwonofNmxinu. P lanet m0Tin g “toe excentric, do. 

fleeted from his zenith, in that place 
of the concontric, where an observer situated* at the centro of 
the Earth observes it in the eastern or western hemisphere, 
and at noon both observers see it in the same place, thoreforo 
the correction called Natakarma is declared (by astronomers). 
The proof of this is tho same as in finding tho parallax.f 


• [The bhvja-phala, determined by means of the sine of the first xendba of 
tho planet (i. e. by multiplying it by the periphery of the 1st epicycle and 
dividing it by 360») has been taken for the sine of tho 1st equation of the centre : 
and what we have shown in the note on the Y. 28 and 29, that the bhuja- 
PHala, when multiplied by the radius and divided by the liypothenuse, become* 
the sine of the equntion may be understood only for finding the 2nd equation of 
the five minor planets and not for determining the 1st equation. 

Some say that the omission of tlA hypothenuso in the 1st process has no 
other ground but the very inconsiderable difference of the result. But bbahxa- 
GUPTA maintains that the periphery of the 1st epicycle, varies according to the 
hypothenuso { that is, their ratio is always the same, and the periphery of the 
1st epicycle, mentioned in the oaxitadhyAya, is found at the instant when the 
, hypothenuso is equal to the radius. For this reason, it is necessary at first to 
find the true periphery through tho liypothenuse and then determine the 1st 
equation. But, he declares that by so doing j also the sine of the equation 
beoomes equal to the bhuja-phala as follows. 

As B : 1st periphery = the liypothenuse : the true periphery 
P X h 

the true periphery , and consequently the bhuja-phaia in 

B 

P X h sin it 

• the true epioyole =* — - .X I 

B 860® 

PX* sin* B 

.% the sine of tho 1st equation s= — x X — and abridging = 

B 860* h 

• P . sin * 

— - which is equal to the bhuja-phala. Hence the hypothenose is not 
8600 

vied in the 1st process. » 

Brahmagupta's opinion is much approved of by Bha'ixaba'oha'bta.— ' B. D.] 
f But this is not the case, beoause the xatakabx which Bha'skaea'cri'eya has 
stated in the GaxitAdhyIya has no oonneotion with the fact stated in this 
i’loxa and tlierefore many say that this c'loxi doss not belong to the 
teat.— B. D.] 



V.’40.] • Sulhdntas'iromani. ' 149 

39. The mean# motion of a planet is also its true motion 

Explains where the mean when the P 1 *^ reaches ‘that^ point in 
«nd true motions of all the the excentric cut by the transverse 

p ^ diameter which passes through the 

centre? of the concentric : and it is when the planet is at that 
point that the amount of equation is at its maximum. [Lalla 
has erroneously asseriod that the mean and true motions 
coincide at the point where the concentric is cut by its ex- 
centric.]* 

40. Having made the oxcentric and other circles of thin 

Manner of observing the pieces of bamboo in the manner ox- 
retrogrruioa Ac. offlmieta. befor0j haying clmnged 

the marks of the places of the planet and its s'Iciikochcha 2nd 
higher apsiB with their daily motions, an astronomer may 
quickly show the retrogressions, &c.f 


* The ancient aatonomers Lalli, 8'bipati &c, any that the true motion of a 
planet equals to its mean motion when it aeadies tlio point of intersootion of the 
concontrio and excentric. But Bha'skaka'chauya denying this, says, that when 
the planet roaches the point when the transverse axis of the concentrio cuts • 
the exoentrio and when tlio amount of equation is a maximum, the true motion 
of a planet becomes equal to its mean motion. For, suppose, p„ p gt p„ & o. t 
are the mean daces of a planet found on successive days at sun-rise when the 
planet proceeded from its higher or lower apsis and e„ ko. are the amounta 
of equation, then p ,± e u p ai ± « a> p t ± e„ &o. will be the true places of Use 
planet, 

P*-Pi ± (/»-*»). P.-Pn ± («•— P*-P, ± (e 4r e,),&o. will bo the 
true motions of the planet on successive days. Now, ai the duFerenco between 
the true and mean motions is callod the oatiphala, by cancelling therefore, 
jp a — !», Pi,— p B , &o. the parts of the true motions which are equal to the mean 
motion, the remaining parts e a — <?„ e a — e, &o. will evidently be the OAtiphalas 
that is the differences betwe en t wo suc c essive amount s of. equation .are the 
OATIPHALAS. T hus. it is plain that the oatiphala entirely depends upon the* 
amount of equation, but as the amount of equation increases, so the oatiphala^ 
is decreased and therefore whon it is a maximum, the oatiphala will indifiutely 
be decreased i. e. will be equal to nothing. Now as the amount of equation 
becomes a maximum in that place where the transverse diameter of the con- 
centrio circle cuts the excentric, (see the note on verses 15, 16 and 17) the 
oatiphala, therefore becomes equal to nothing at the same place, that is, in 
that very plaoe, the true motion and mean motions of a planet are equal to each 
other. Having thus shown a proof of his own assertion, Bkaskaba'cha'kva 
says that what the ancient astronomers stated, that the true and mean options 
of a planet are equal to eaoh ether when the planet comes in the intersection 
point of the ooneentrio and excentric circles, is entirely ungrounded.— B. D.] 

t According to the method above mentioned, if the place of tlie higher apsis 
and that of the planet be changed, and the planet's plaoe be marked, the motion 
of the planet will be in a path like the dotted line as shown in the diagram, 
gee Diagramfacittg Mil pnge* 


150 ' Translation of the , [V. 41. 

41. The word kendra (or m vrpov) means the centre of a 

Th, ot'm in Ten- oircl ° : ifc is 0n Uiat ftccount “PP^d 
fion of the appellation of to the distance between tho planet aifd 

mMm& * higher apsis, for the centre of tho 

NiCHOCiiCHA-vijiTTA or epicycle, is always ‘at the distance of 
the planet from the place of the higher apsis. " 

42. The circumferenco in yojanas' of the planet’s orbit 

Sphcta-xamha or cor- being multiplied by the s'Ighra-karna 

wcted orbit. ( or 2nd hypothenuso), and divided by 

the radius (3438) is si'Huta-kaksh/C (corrected orbit). Tho 
planet is (that moment) being carried [round the earth] by 
the pkavaha wind, and moves at a distance equal to half tho 
diameter of tho sftiuta-kaksha from tho earth’s centre. 

43. When tho sun’s manda-fhala i. o. the equation of the 

Reason of Buja'ktaba centre is subtractive, the apparent or 

correction. r real timo of sun-rise takes placo before 

tho time of mean sun-rise : when the equation of tho centre 
is additive, tho real is after the mean sun-rise, on that account 
tho amount of that correction arising from the son’s manda- 
pttala converted into asus* of time has been properly declared 
to bo subtractive or additive. 

44. Those who have witB as sharp as the sharp point of the 
inmost blado of the dorbha or darbha grass, find the subject 
above explained by diagrams, a matter of no difficulty whatever : 
but men of weak and blunt understanding find this subject 
.as heavy and immovable as tho high mountainf that has been 
*shcfrn of its wings by the thunderbolt of Indra. 

End of Chapter V. on tho principles on which tho rules for 
finding the true places of tho planets are grounded. 

It ii to be obierved here that when the planet comes to the places a, a Ac. 
in the dotted line, it is then at its higher apsis, when it comes to the places c, 
e and c, it is at its lower, and when it comes to b t b Ao. it appears, stationary : 
and Jrhen it is moving in the upper arc b a%, its motion being direct appears 
quicker, aud when in the lower aro b o i, its retrograde motion is seen.— B. J).J 

• [These levs are equivalent to that part of the equation of time, which is 
due to the unequal motion of the sun on the ecliptic.— B. I).] 

t Mountains are said by Hindu theologians to have originally had wings. 



. VI. 3.] 


Sidhdnta-s'iromani. 


lol 


CHAPTER VI. 

Called Golabanjjha, on the construction of an 
Amitiary Sphere. 

* 

1. Let a mathematician, who is as skilful in mecbanics ns 
in liia knowledge of tlio sphere, construct an ancillary sphere 
with circles made of polished pieces of straight burnt) oo ; and 
marked with the number of degrees in the circle. 

2. In the first place, let him mark a straight and cylindri- 
cal dhruva-yashti, or polar axis, of any excellent wood ho 
pleases : then let him place loosely in the middle of it a small 
sphere to represent tho earth [so that tho axis may movo 
freely through it] . Let him then firmly secure tho spheres 
beyond it of the Mooir, Mercury, Venus, tho »Sun, Mars, 
Jupiter, Saturn and tho fixed stars : Beyond them let him 
place two spheres called khagoaa and duiguola unconnected 
with each other, and fastened to the hollow cylinders [in 
which tho axis is to bo inserted] .* 

[Description in detail of tho fact above alluded to.] 

3. Fix vertically the four circles and another circle called 

mi , ..... horizon transversely in the middle of 

Tlio prune vertical, llio * 

meridian and the kosayuit- them, so that one of thoso vertical 

circles called Samamant»ai.a, prime 
vertical, may pass through tho cast and wost points of tho 
horizon, the other called yXm yottaka- v rItta, meridian # tlio' 

* The sphere of the fixed stars which is mentioned here is called the bitaooia 
starry sphere. This bhagola is assumed for all the planets, instead of fixing 
a separate sphere for each planet. This sphere oonsists of the circles ecliptic, 
equinoctial, diurnal circles, &o. which are moveable. For this reason, this 
sphere is to be firmly fixed to tho polar axis, so tliat it may more fredv by 
moving the axis. Beyond this sphere, the bhagola celestial sphere which 
consists of the prime vertical, meridian, horizon, Ac. which remain fixed in a 
given latitude is to be attached to the hollow cylinders. Having thus sojlflhtely 
fixed these two spheres, astronomers attach, beyond these, a third splieFe in 
which the circles forming belli the spheres sbaGola and bhaGola are mixed 
together. For this reason the latter ia called driggula the double sphere. 
And as the spherical fingers are well seen by mixing together the two spheres 
khagola and bhagola, the third sphere which is the mixture of the two 
spheres, is separately attached.— B. D.J 


152 


Translation of the 


[VI. 4. 


north and south points, and the remaining two called kona- 
v$ittas tho N. E. and S. W. and N. W. and S. E. points. 

4. Then .fix a circle passing through the points of tlfo 

The unmasdala ,or six horizon intersected by the prime verti- 

o’eiock line. csa.1^ and passing also through the 

south and north poles at a distance below and above the ■ 
horizon qqual to the latitude of the place. This is called the 
UNMAwp AtA, or s ix o'c lock lino, and is necessary to illustrate the 
increase and decrease in the length of tho days and nights.* 

5. The equinoctial (called NXpf-VALAYA), marked with 
60 ghatis, should bo placed so as 
to pass through the east and west 

points of the horizon, and also to pass over tho meridian at a 
distance south from the zenith equal to tho latitude, and at a 
distance north of the nadir also equal to the latitude of the 
place [for whicti the sphere is constructed] . 

6. Let the azimuth or vertical circle be next attached 

. . , .... within the other circles, fixed by a 

Agtmntn or vortical circle. 

pair of nails at the zenith and nadir, 
so as to revolve freely on them : [It should be smaller than 
tho other circles so as to revolve within them] . It should be 
capable of being placed so as to cover tho planet, wherever it 


The equinoctial. 


7. Only ono azimuth circle may be used for all tho planets ; 
or olse eight azimuth circles may be made, viz. one for each 

• of tho 7 planets and the 8th for the nonagesimal point. Tho 

• azimuth circle for tho nonagesimal point is called the DfliK- 

SKEPA-ypiTTA. 


* The circle of deolination or the hour circle paeeiug through the east and 
Vest points of the horison is called UzrxixgALi in Sanskrit ; but I am not 
•cquaiuted with any corresponding term in English. In the treatise on 
astroMomy iu the Encyclopedia Metropolitan! the prime vertical is named the 
six o'clock line. This term (six o'clock line) should, I think, be applied to the 
UXMAvpAZiA, because it ia always six o'clock whan the sun arrives at this circle, 
the TOXAvpAiA. Tie prime vertical or the saxa-xaxdala of the Sanskrit 
cannot, with propriety, be called the six o'olook line; because it is only twiee a 
year that it is six o'clock when the sun is at this circle, the prime vertical.— 



153 


VI. U.] •Sulh&ntas'iromanL 

8. Lot two hollow cylinders project beyond the two poles 

The D?mgola. north and soutl1 of toe khagola ce- 

% lestial sphere, and on these cylinders 

let the skilful astronomer place the d$iggola double sphere 
as follows. * 

9. When the* system of the khagola, celestial sphere, is 
mixed with the ecliptic,' and all the other circles forming the 
bhagola (which will be presently shown) it is then called 
DtyiGGOLA, double Bphere. As in this the figures formed by 
the circles of the two spheres khagola and bhagola are seen, 
it is therefore called d^liggola double sphere.* 

THE BHAGOLA. 

10. Let two circles be firmly fixed on the axis of the poles 
answering to tho meridian and horizon (of the khagola) ; 
they aro called tho adh£ra-v$ittas, or circles of support: 
Lot the equinoctial circlo also be fixed on therA marked with 
60 ghatis like the prime vertical (of the khagola). 

11. Make tho ecliptic (of tho same size) and mark it with 

The Ecliptic ^ signs; in this tho Sun moves: and 

also in it revolves tho Earth's shadow 
at a distance of 6 signs from tho Sun. Tho kranti-pXta or 
vernal equinox, moves in it contrary to tho order of the signs : 
The spASHTA-piTAS [of the other planets] havo a like motion : 
the places of these should be marked in it.-f* 


* See the note on 2 Verse. 

t [The Sun revolvos in tho ocliptic, but the planeti, Moon, Mara, &o. do not . 
nsvolre in that circle, and tho planes of their orbits are iuolincd to that of the 
eoliptio. Of the two points where the planetary orbit outs the plane of tin • 
ecliptic, that in which the planet in its revolution rises to the north of the 
ecliptio is called its pa't a or. seconding node (it is usually called the mean 
pa'ta) and that which is at the distance of six signs from the former is called 
its babhadbha pa'ta or descending node. The pa'ta of the Moon lies in its 
ooncentric, because the plane of its orbit passes through the centre of the 
conoentrio, i. e. through the centre of the Earth } but the pa'tab of the other 
planets are in their second oxcentric, because the planes of their orbits pass 
through the centres of their 2nd exoentrioa, which oentres lie in the pleno the 
eeliptic. ' When the planet is at any other place than its nodes, the disfanoe 
between it and the plane of the eoliptio is called its north or south latitude ae 
the planet is north or south of the ecliptic. When the planet is at the distance 
of 3 signs forward or baokward from its pa'ta, it is than at the neatest distance 
north or sonth from the ecliptic : This distance is its greatest latitude. Thus, 



154 


Translation of the * [VI. 12. 

12. Let the ecliptic be fixed on the ^equinoctial in the 
point of vernal equinox kbXnti-pIta and in a point (autumnal 
equinox) 6 signs from that : it should be so placed that the 
point of it, distant 3 signs eastward from the vernal equinox, 
shall be 24° north of the equinoctial, and the 3 signs westward 
shall be at tho same distance south from the equinoctial. 

13. Divide a circ le called kshepa-V^itt a representing the 

t orbit of a planet into 12 signs and 

Planet's orbit. 

mark in it the places of the spashta* 
pItas, rectified nodes, as has been before prescribed [for the 
ecliptic]. Then this circle Bhould be so placed in connection 
with the ecliptic as it lias been placed in connection with tho 
equinoctial. 

14. The ecliptic and tho kshepa-v&itta should bo so 
placed that tho latter may intersect tho former at the [rectified] 
ascending and 4 descending nodes, and pass through points 
distant 3 signs from the ascending node east and west at a 
distanco from tho ecliptic north and south equal to tho 
rectified greatest latitude of tho planet [for the time] . 

15. Tho greatest (mean) latitudes of the planets being 
multiplied by the radius and divided by the sighra-karna 


the latitude of tho planet begins from its pa'ta nnd become* extreme nt the 
distanco of 8 sign* irom it, therefore, in order to find the latitude, it is necessary 
to know the distance between the planet and its pa'ta. This distance is equal 
to tho sum of the places of the planet and its pa'ta, because all pa'ta s move 
in antecedents from the stellar aries. This sum is called the tiksiiepa-kiup ba 
or the argument of latitude of the planet. As the pa'ta of the Moon lies in her 
. concentric, and in this circle is her true place, the sum of these two is her 
YlKSHXX'A-KiiCDBA, but the pa'ta of any other planet, Mars, Ac. lies in its 2nd 
* excentrio and its xanda-ipashta place (which is equivalent to ita heliocentric 
place) ia in that circle, therefore its yiksotpA’Xixdha is found by adding the 
plaoe of ita pa'ta to its xaxda-bpabhta place. The sfabhtA'Pa'ta or the 
planet is that which beinp added to the true plaoe of tho planet, equals its 
viKSHiPA-xavDBA for this reason, it is found by reversely applying the 2nd 
equation to its mean pa'ta. As 

*\ bfashta pa'ta 4* true plaoe of the planet, 

» VDNHIFA-KiroRi, 

*== place of the xahda spashta planet 4 "V*»d pa'ta, * 

bs p, of the m. t. p. ± 2nd equation -f- *• P.? 2ud equation, 
ss true place of the planet 4- mean pa'ta ± 2nd equation, 
spashta pa'ta = mean pa'ta f 2nd equation. 

Tlio plaoe rtf this spashta pa'ta is to be reversely marked in the ecliptic from 
the atelier aries.- B. D ] 



Sulhunta-tfiromani. 


155 


VI. 15.] 

second hypothenufb becomes spabhta, rectified. The ksugfa- 
v^itta, or circles representing the orbits of the six planets, 
sfiould be made separately. The Moon and tho rest rovolve in 

their own orbits.* m 

■ • 


• [As the pa'ta of the Moon end her true plaoe lie in her concentric, theeum 
of these two, whioh is called her viKBHBPA-KnnDKA or the araument of latitude, 
must be measured in tho same circle, and her latitude, therefore found through 
her vikbhepa-xmdba, will be as seen from the oentre of her conoentrie i. e. 
from the centre of the Earth. But the pa'ta of any other planet and its makiu- 
bpabiita place (which is its heliocentric plaoe) lie in its 2nd escentrio, therefore 
its latitude, determined by means of its vikbuipa-xshdiu, which is equal to 
the sum of its xanda-spabhta placo and pa'ta and measured in the same circle, 
will bo such as seen from the oentre of its 2nd exoentrio and is called its mcuu 
latitude (whioh is equivalent to the heliocentric latitude of tho planet). 

As in Fig. 1, let N E be tho 
quarter of the ecliptic, N 0 
that of the 2nd exccntric, N 
tho node and F tho planet. 

Suppose 0 E and F p (parts of 
great circles) to be drawn from 
O aud P perpendicularly to 
tho plane of the ecliptic : then 
0 E will be tho greatest lati* 
tude and F p the latitude of tho planet at F, by which a s{>ec tutor at the centre 
of the 2ud cxcentric and not at the centre bt the Earth, will see the planet distaut 
from the ecliptic. This latitude, therefore, is called a mean Lutitude whioh can 
be found aB follows, 

sin N 0 : Bin 0 E : : sin N P : sin F p, 

or R . sin F p = sin 0 E . sin N p, 

consequently, in order to determine P p, it is necessary to know previously 0 E, 
tho greatest latitude and N F, the distance of the place of the planet from tho 
node, which distance is evidently equal to tho viKSUKPA-KKiroEA that is, to 
the sum of the makua-spasiita place of tho planet and the mean place of tho 
node. Now tho lutitude of the planet as seen from the centre of tlie Earth is 
called ita true latitude. This true latitude can bo found in the following 



manner, 

Lgt E ho the centre of the earth, 0 that of tho 
2nd exeentrio, F the wanda spAbiiTA place of the 
planet in it : then E F will be the 2nd hypothenuso 
which is supposed to cut the concentric at A : 
then A will be the true place of the planet in the 
concentric. Again let F q be a circle with the 
centre 0, whoso plane is perpendicular to the eclip- 
tic plane and A b another circle with the centre 
E whose place is also perpendicular to the same 
plane, then F q will be the mean latitude of the 
planet and A b will be the true. Let F p and A <* 
lines be perpendicularly drawn to tiie plane of the 
ecliptic, these lines will also be at right angles to 
the line E p : then F p will be the sine of the 
mean latitude F a and A a that of the true lati- 
tude A b. Now by the similar triangles E P p 
and E A a, 





156 


Translation of the 9 [VI. 16. 


16. The declination is an arc of a gre&t meridian circle: 

' . catting the equinoctial at right angles. 

Declination and latitude. ° 1 oat 

• and continued till it touch the ecliptic. 

• It X sine of the mqpn latitude 

or the sine of tlie true latitude — — — — ' 


but, the sine of tho mean latitude 


ft 

sin 0 E . sin N P 


R 


by substitution 

R sin 0 E . sin N P 

the sine of the true latitude X ■ ■ — 

A R 

sin 0 E . sin N P 


h 

As the latitude of the planet is of a smaller amount, the arc of a latitude it, 
therefore taken in the Siddhantas instead of the sine of the latitude. 

0 1 . sin N P 

Hence, the true latitude = , 

that is, the sine of the argument of latitude multiplied by the greatest 
latitude and divided by the 2nd hypothenuse is equal to the true latitude of the 
planet. 

Now in the Bhaooia, a circle should be so fixed to the ecliptic, that the 
former may intersect the latter at the bpabiita-p£ta and the point six signs 
from it, and whose oxtreme north and south distance from the ecliptic may 
he such that the distance between the circle and the ecliptic at the place of the 
true planet may be equal to the truo latitude of the planet. This circle ia called 
the yihavpala or Y1K8UIPA-YRITTA and its oxtreme north and south distance 
from the ecliptic is called the true or rectified extreme latitude of the planet 
which can be found as follbws. 


let N be the spashta- 
PATA,N P the YIKSnXPA-XBK- 
DUA, P p the true latitude, 
E 0 the true extreme latitude i 
then 

sin N o : sin E 0 : : sin 
, N Piiin Tp 



L.sinNP 


* ain NO. sin Pp 

ain E 0 3 ■ ■ — ■ ; 

•in N P 

R.P p 

or E 0 i 

ain N P ■ 

but be taken for the mean extreme latitude the P p = 

R L.ainNP R.L 
" B0? *7inNP * A ' 

This is the mean oxtreme latitude stated in the OayitadhtXya multiplied 
by the radius and divided by the 2nd hypothenuse equals the true or notified 
extreme latitude, — B. D.] 



Sidhdntas'ir omani. 


157 


• VI. 21.] 


celestial latitude if in like manner an arc of a. great circle 
' (which passes through the ecliptic poles) intercepted between 
tho ecliptic and the kshepa-v&itta. 

The corrected declination [of any of the small planets and 
Moon] is the distance of tho planet from tho oquinoctial in a 
circlo of declination. ' 

a 

17. The point of intersection of tho equinoctial and ecliptic 

- . circles is tho kranti-pXta or inter- 

Prcceuion of the equinox. 

secting point for declination. Tho 
retrograde* revolutions of that point in a Kalpa amount 
to 30,000 according to tho author of tho S rya-sidd itXnta. 

18. Tho motion of tho solstitial points spoken of by Mun- 
jala and others is tho saiuo with this motion of tho equinox : 
according to these authors its revolutions arc 199,609 in a 
Kalpa. 


19. Tho ploco of tho krAnti-pXta, or the amount of tho 
precession of the equinox determined through tho revolutions 
of tho krXnti-pXta must bo added to the place of a planet ; 
and tho declination then ascertained. Tho ascensional differ- 
ence and periods of rising of the signs depend on tho 
declination : henco the procession must be added to ascertain 
tho ascensional difference and horoscope. 

20. Thus the points of intersection of tho ecliptic and tho 
orbits of tho Moon and other planets are tho kshkpa-patas, or 
intersecting points for the kshepa celestial latitude. Tho 
revolutions of the kshepa-pXtas are also contrary to tho order ' 
of tho signs, hence to find their latitudes, the places of tho 
kshepa-pXtas must be added to tho places of the planets 
(before found). 

21. As the xanda-spashta planet (or the mean planet cor- 
rected by the 1st equation) and its ascending node rqgplve 
in the s'iohra-pbativ$itta or 2nd excentric, hence the amo&nt 
of the latitude is to be ascertained from (the place of) the* 
xanda-spashta planet added to the node found by calculation. 

• The motion of the KrIkti^ta is in a contrary direction to that of the 
order of the iigne.—L. W... ■* . 



158 


Translation of the m [VI. 22. 

22. Or the amount of the latitude may %e found from the 

spashta -planet added to the node which the sighra-phala 2nd 
equation is added to or subtracted from accordingly as it was 
subtractive or additive.* , 

r 9 

As the Moon's node revolves in tho concentric circle, the 
amount of the latitude, thereforo, is # to 1 bo found from the 
true plaeo of the Moon added to the mean node. 

23. Tho exact revolutions of tho nodes of Mercury aud 
Venus will be found by adding the revolutions of their s'Iohba- 
kendras to the revolutions of their nodos which havo been 
stated [in tho GanitXdhyXya] : if it bo asked why these smaller 
amounts havo been stated, I answer, it is for greater facility of 
calculation. Hence their nodes which are found from their 
stated revolutions are to bo added to tho places of their 
s'fomiA-KENDRAS [to got tho exact places of tho nodes] .f 

24. To find the kgndra [of any of tho planets] tho place 
of tho planet is subtracted from tho sTgiirociiciia : then take 


* [See the nodes on V. 11, and V. 13, 14, 15. — B. D.] 
t [In all the original astronomical works, the sum of the pa'ta and s'loirnocH- 
CHi of Mercury and Venus, is assumed for tlieir vikshepa-kbndba, and 
through this, their latitudo is determined. But the latitude thus found would 
bo at tho place of their a'fflHBOCHCiTA and not nt their own place, because their 
places are different from those of their s'foHBOCHCHAS. To remove this 
difficulty, Biu'bkha'cha'hya writes. <( Iho exact revolutions Ac." But the 
difficulty arises in tho supposition that, the eBrtli is stationary in the centre of 
the universo and all the planets revolve round her, because we are then bound 
to grant that tho mean puces of Mercury and Venus are equal to that of the Sun, 
and hence their places will be different from those of their bTouuoouchab. 
But no inconvenience occurs in the supposition that, the Sun is in tho centre 
of the universe and all the planets together with the earth revolve round him. 
For, in this cuso the {daces of tho B'faHBOCHCnAS of Mercury and Venus arc 
their own heliocentric places, and consequently the sum of the places of thoir 
B'feHBOCHCBAB and pa'tas will be equal to the sura of their own places and 
those of their pa'tas, that is to their vikbhspakekdua. For this reason, 
their latitude found through this, will be at their own places. Bow, it is a 
curious foot that, the revolutions of the, pitas of Meroury and Venus, stated 
in the original works, are such as ought to be mentioned when it is supposed 
that the Sun is in the middle of the universe and the planets revolve round 
him,"Snd not when the Earth is -supposed tb be stationary in the centre of the 
uhiverse. From this fact, we can infer that the original Authors of the As- 
tronoAioal works knew that all tlie planets together with the Earth revolve 
round the Sun, and consequently they stated the smaller amounts of the 
revolutions of the Pa'tas of the Mercury anA Venus. ’When this is the cssc, 
why is it supposed that ill the planets rpvoFve round the Earth, because the 
Spherics oan more easily be understood by this supposition than by the other.— 
B. D.] i 



VI. 27 .] .Sidhanta-siromani. * 159 

the kgndra with tljB pata added [to get the exact amount of 
the f£ta or node] and let the placo of the planet be added 
thereto, [we thus get tho vikshepa-kendra or the argument 
of tho latitude of Mercury or Venus] . Thcreforo from tho 
s'fGiiROtHCHAS of these two planets with tho fxtab added, 
their latitudes ar<5 directed by the ancient astronomers to be 
found.* * 

25 and 26. The tatas or nodes of these two pluncts added 
to the s'ighrochchras from which the true places of tho 
planets have been subtracted, becomo sfashta or rectified. 
It is the s'pashta-pxta which is found in tho miagola (above 
described). 

In tho sphere of a planet, tako the ecliptic above described 
as the concentric circle, to this circlo tho second cxccntric 
circle should bo attached, as was explained before, and a circlo 
representing the orbit of a planet (and which consequently 
would represent the real second exceutric) should be also 
attached to tho latter circlo with tho amount of latitude 
detailed for it. In this latter circlo mark off tho mean places 
of tho nodes of the (superior) planets, and also mark in it tho 
mean placo of the nodus of Mercury and Venus added to their 
respective s'faHRA-KENDKAs.t 

27. Next th o ahobatka-vbittas or d iurnal circles, must be 
Diurnal circles cdled made oil both sides of tho equinoctial 
a iio&'axh a-y Hitt as. [ al id parallel to it] at every or any 

degree of declination tliat may be required : — and they must 
all bo marked with 60 ghatis : The radius of tho diur nal 
circle fon wh ich the Sun m ay move on any day] is called 
dyujyI. 


• [Let, A = s'foHEOCHCHA or the placo of 2d higher apsis. 
k ss the ^fOHtU’KIKD&A. 
p = the place of the plenet. 

• = pa'ta or the place of the ascending node, 
and N. = the exact pa'ta. 
then k = h—p ; and k — k + « = A. — p + *1 
V TliSHBPA. kbxdra or alvamciit 4 of latitude of Mercury or Venus = 
ff. + p5BS jt.-p + A + U.-B.D.J 

t [See the note on rerses 13, 14 and 15 :-B. D.] 





160 


Translation of the , [VII. 1. , 

28. From the vernal equinox mark the ^2 signs in direct 

order., and then let diurnal circles be attached at the extremity , 
of each sign. * 

29. On either side of the equinoctial, three diurnal circles 

should be attached in the order of the sigrfs : these agftin will 
answer for the three following signs. * 

The bhaoola has thus been described. This is to be known 
also as tjie khjschara-gola, the sphere of a planet. 

30. Or in the plane of the ocliptic bind the orbits of Saturn 
and of the other planets with cross diameters to support them, 
bat these must be bound below (within) the ecliptic in succes- 
sive circles one within tho other, like the circles woven one 
within the other by the spider. 

31. Having thus secured the bhaoola on tho axis or 
yashti, aftor placing it within the hollow cylinders on which 
the khagola is 'to be fastened, make tho bhaoola revolvo 

it will do so freely without reference to the khagola as its mo- 
tion is on the solid axis. Tho khagola and d^iggola remain 
stationary whilst the bhaoola revolves. 

End of Chapter VI. on the construction of an armillary 
sphere. 


CHAPTER VII. 

Called TripWna-vIsahX on the Principles of the Rules for 
resold n<j the questions on time , space , and directions . 

The woeniionel difference 1 . T he time called CHABA-KHPND A 

•nd its pliec. or ascensional difference is found by 

that arc of a diurnal circle intercepted between tho horizon 
aud the six o'clock line. The sine of that arc is called the 
kojya* in the diurnal circle: but, when reduced to relative 



VII. 5.] Siddhdnia-firomaQi, ' ]fii 

value in a great ^circle, it is called chabajya or sine of as- 
censional difference.* 

• 2. The horizon, as s een at the_ eqiyttor^or in a right 
sphere, is denominated in other places [to the north, or Boutli 
of the* equator] jbhfe unmanp ala six o'c lock line : but as the 
Sun appears at' any place to rise on its own horizon, the 
difference between the 'times of the Sun's rising [at a given 
place and the equatorial region uuder the same meridian] is 

| the ascensional difference. 

3. When the sun is in the nor- 

Determination of tho .. - . , 

question when tt» cimu them hemisphere, it nsea at any 

* Bd p kca ("° rth ° fth9 •*■*«) 

it does to that on the equator: but 
it sets after it sets to that on the equator. Therefore the 
correction depending on .the ascensional difference is to be 
subtracted at sunrise of a given place from the place of the 
planet [at sunrise at tho equator] and to be added at sunset 
to the place of the planet [as found for the sunset at the 
equator] . 

4. When the Sun is in the southern hemisphere the reverse 
of this takos place, as the part of tho unmandala in that 
hemisphere lies below the horizon. Tho halves of the sphere 
north and south of tho equinoctial are called tho northern and 
southern hemispheres. 

Cause of increase and decrease O d ifc » in consequence of 

in length of days and nights. this ascensional difference that] the . 

days are longer and the nights shorter (than they are on the . 

• [The times found by the arcs intercepted between the horizon and the 
six o’clook line, of the three diurnal circles attached at tho end of tho first 9 
signs i. e. Aries, Taurus and Gemini ar e called the CHARA-ka'lab or the ascensional 
difforanoM of these signs, and t he differences of these 0Haba<Xa£a8 are called the 
oiuba-kh ANQA8 of tuose three signs. 

A«, where the falabha is 5 digits or the latitude is nearly 221* north, the is- 
censional differences of the 8 first signs are 207, 541 and 642 asub, and^he dif- 
ferences of those i. e. 297, 244 and 101 are the ohaba»khaB9AS of those signs. 

These are again the ohaba*xhav?as of the following three signs inversely i. s. # 
101, 244 and 297 asus. 

Thus the OHABA-XHAVpil of the first six signs answer for the follqping six 
signs,— ‘B. D.1 



162 


Translation of the 


[vn. r,, 


.equator) when the Sun is in the northern^ hemisphere: and 
that the days are shorter and the nights longer when the Sun • 
is in the southern hemisphere. For, the length of the night 
is represented by that arc of the diurnal circle below fcho 
horizon, and the length of the day by {hat arc abota the 
horizon. 

6. But at the equator the days and nights are always of the 
same length, as there is no unman^ala there except tho 
horizon [on the distanco between which, the variation in tho 
length of days and nights depends]. 

A circumstance of peculiar curiosity, howover, occurs in 
those places having a latitude greater than 66* N. viz. than tho 
complement of the Sun’s groatest declination. 

Determination of plao. and 7 ‘ Wltenover thenorthem declina- 
time of perpetual day and tion of tho Sun exceeds the comple- 
•’ nient of the latitude, then there will be 

porpetual day for such time as, that oxccss continued ; and when 
the southern declination of the Sun shall exceed tho comple- 
ment of the latitude, then there will be perpetual night during 
the continuance of that excess. On mkru, therefore, day and 
night are each of half a year’s length. 

8. To the Celestial Beings [on 
heru at the north polo] the equinoc- 
tial is horizon : so also is to the daityas [at tho south pole] . 
For, the northern and southern poles are situated respectively 
.in their zeniths. 

. 9. The Celestial Beings on meru behold the Sun whilst ho 

is in tho northern hemisphere, always revolving above the 
horizon from left to right : but daityas the inhabitants of the 
southern polar regions behold him whilst he is in the southern 
hemisphere revolving above their horizon from the right to 
the l&t. 

10. Thus it is day whilst the Sun 

M night and the day is visible, and night whilst he is m- 
t of the m*n. visible. As the determination of 


Place of ynu. 



VII. 15.] Siddhdntas'it'omani. * 165 


night and day is jnode in regard to men residing* on the but- , 
faco of the Earth, so also is that of the pit&is or deceased 
ancestors who dwell on the npper part of the Moon. 

11. As for the doctrine of astro- 
,J!a TS lo S ers > *at it was day with tho Gods 

professors or ba'nhitkas.. at merit whilst the Sun was in the ut- 

• taeIyana (or moving from the winter 
to the summer solstice) and night whilst tho Sun was in the 
dakshinAyana (or moving from tho summer to the winter sol- 
stice), it can only be said in defence of such an assertion, that it 
is day when the Sun is turned towards the day, and it is night 
when turned towards the night. Their doctrine lias reference 
merely to judicial astrology and the fruits it foretells. 

12. By tho degrees by which tho Sun proceeds in his nor- 
thern course to tho end of Gemini, he moves back from that 
sign : entering also the same diurnal circles in Ilia descent as ho 
did in his ascent. Is it not therefore that tho Sun is visible in 
his descent to the Gods in the place where he was first seen by 
them in his ascent ? 


• 13. The pitris reside on tho upper 

riTHiaf h0fth0 ^ ° f tb<> P arfc °f the Moon and fancy the foun- 
tain of nectar to be beneath themselves. 
They behold the Sun on tho day of our amavXsyX or new Moon 
in their zenith. That therefore is the time of their midday. 

14. They (i. e. the pitris) cannot see tho Sun when he is 
opposite tho lower part of the Moon : it is therefore, midnight, 
with tho pitris on the day of the puknim! or full Moon. Tho , 
Sun rises to them in tho middle of the Krishna paksha or dark 
half of the Moon, and sets in the middle of the s ukla paksiia 
or light half of the Moon. This is clearly established from 
tho context. 


15. As BrahmA being at 8h .im- 

The explanation of a day mense distance from the Earth, always 
of Brahma • ^ . 

sees the Sun till the timo of the pra- 

laya or general deluge, and sleeps for the same time, therefore 



164 TrantlaHan of th*. [VII, 16. 

the day and* night of Brahma are together 2000 mahxyugar 
in length. * 

16. As the portion of the ecliptic 

& * Wch k more oblit i oe the otw > 

riton. rises and sets in "a shorter tine and 

that which is more upright takes a longer time in rising and 
setting, hence the times of rising of the several signs are 
various [even at the equatorial regions]. 

17. The (six) signs from Capricorn to Gemini or ascending 
signs which are inclined towards the south with their respec- 
tive declinations whilst they riso even at the equator are still 
more inclined towards the south in the northern latitudes (on 
account of the obliquity of the starry sphere towards the south) ; 
hence they arise in still shorter times than they do at the 
equator. 

18. At the 'equator, the [six] signs from Cancor or de- 
scending signs incline whilst they rise to the northerly direc- 
tion, but they will have upright direction in consequence of tho 
northern latitude, hence they rise in longer times [than they 
do at the equator.] The difference between the period of the 
rising of a sign in a given latitude, and at the equator under 
the same meridian, is equivalent to the charakiianpa of that 
sign. 

19. Each quarter of the ecliptic rises in 16 ghatis or 
6 hours to those on the equator: and the 6 signs of the 

■ northern as well the 6 of the southern hemisphere appear to 

■ rise each in 12 hours or 30 ghatis in every or any latitude. 

20. The three signs from the commencement of Aries to 
the end of Gemini, i. e. the first quarter of the ecliptic, pass 
the unkanqala in 15 ghatis; but the horizon [of a place in 
north latitude] is below the unman?ala, they therefore pre- 
viously pass it in time less than 15 gratis by the chabakhahpas. 

^ 21. The three signs from the end of Virgo to the end of 

Sagittarius, i. e. the 3rd quarter of' the ecliptic, pass the unman- 
in 15 ghatis; but they pass the horizon of a place 



105 


VII. 23.] Siddhdnta-i'iromani. 

afterwards which in above the unmanjjala [in north latitude] 
in 15 ohatts added to tho charakhanjjas. * 

•22. The three signs from tho end of Gemini to tho end of 
Virgo, i. e. the 2nd quarter of the ecliptic or those from the 
end of {Sagittarius td the end of Pisces i. e. the 4th quarter of the 
ecliptic, pass the horizon in tho time equal to the remainder 
of 30 GHAfis diminished »by the tiino which the first or third 
quarter takes to pass tho horizon respectively. For this reason, 
the times which the signs contained in the 1st and 4th quar- 
ters of the ecliptic, or ascending signs, and those contained 
in the 2nd and 3rd quarters, or descending signs tako to pass 
the horizon at a given place are found by subtracting the 
ciiarakhandas of the signs from and adding them to the times 
which those signs take in rising on the equator respectively.* 
23. Having placed tho. 1st Aries in tho horizon and set tho 
sphere in motion, the tutor should show tho abave facts to the 


* The time* taken by the several signs of the ecliptic in rising at tho equator 
ami in northern latitudes will be soou from the following memo, according to 
the SlDDH AKTA. 



Times of rising at 
equator in side* 
real time. 

| Ascensional rlitfer- 
| ences in 22» 30' 

1 north latitude si- 
| dereal time. 
Times of ruing in 
22° 30 # north la- 
titude sidereal 

time. 


Aries, 

ASUS. 

1670 

48178. 

— 297 

ABU8. 

1373 

■\ These 3 and the last 
( 3 signs take less time to 
(rise in north latitude 

Taurus, 

1793 

— 214 

1549 

Gemini, 

1937 

— 101 

1836 

Cancer, 

1937 

+ 101 

2038 

s than at the equator. 


1793 

+ 244 
+ 297 

2037 

These 6 signs take a 



VlPffO. as HSU. 

1670 

1967 

longer time to rise t in 
‘north latitude than at 

Libra, 

1670 

+ 297 

1967 

Scorpio, 

1793 

+ 244 

2037 

the equator. m 

Sagittarius, 

1937 

+ 101 

2038 

Capri nom, 

1937 

-101 

1836 


Amiwiiii. ....... ........ 

1793 

-244 

1549 


Puoes,.... 

1670 

-297 

1373 

LfW. 






16G 


Translation of the [VII. 24. 

pupils, thaiftkoy may understand as well yhat has been ex- 
plained as any other facts which have not been now mentioned. 

. 24. In whatever^ time any sign rises above the horizen 
[in any latitude] the sign which is the 7th from it, will take 
oiactly the samb timo in setting : as one &alf of the aecliptic 
is always above the horizon [in every latitude], 

25. When the complement of latitude is less than 24° (i. e. 

than tlitf extreme amount of tlio Sun's declination taken to bo 

* 

24° by Hindu astronomers) then neither the rising periods of 
the signs, nor the ascensional differences and other particulars 
will correspond with what has boon hero explained. The facts 
of those countries (having latitudes greater than 06") which 
are different from what lias been explained on account of their 
totally different circumstances, are not hero mentioned, as 
those countries arc not inhabited by men. 

20. That paint of the ecliptic which is (at any timo) on 
Etymology of the word the eastern horizon is called the laona 
IA0NA * or horoscope. This is expressed in 

* signs, degrees, &c. reckoned from the first point of stellar 
Aries. That point which is on the western horizon is called 
the asta-laona or setting horoscope. Tlio point of the 
ecliptic on the meridian is called tlio madhya-laona or middle 
horoscope (culminating point of the ecliptic).* 

• [When the place of tho horoscope is to bo determined nt. a given time it is 
necessary at first to ascertain the height and longitude of tho nonagesimal point 
from tho right ascension of mid-heaven, and then by adding 3 signs to the 
longitude of the nonagesimal point, the place of the horoscopo is found : but as 
‘ this way for finding the place of the horoscope is very tedious, it has been 
. determined otherwise in tho Sidihta'htas. 

As, from tho periods of risings of the 12 signs of the ecliptic which are 
determined in the Siddhantas, it is very easy to find tho time of rising of any 
portion of the ecliptic and vice versa, we can find a portion of the ecliptic 
corresponding to the giren time from sun-rise through the longitude of the 
Sun then determined and the given timo. The portion of the ecliptic which 
can be thus found is evidently that portion of the ecliptic intercepted between 
the place of the Sun and the horizon. Therefore by adding* this portion to the 

! >lace the Sun, the place of the horoscope ii found. Upon this principle, the 
bllowing common rule which is given in the Siddbaxtas for finding the place 
of the horoscope is grounded. 

~~ Find first the true place of the Sun, and add to it the amount of the procession 
of the equinox for the longitude of the Sun. Then, from the longitude of the 
' Sun, tlie sign of the ecliptic in which the Sun lies and the degrees of that sign 



VII. 27.] Sidilhdntas’iromani. 107 

it 

27. If when yotf want to find the laqna, the given oii ajis 
oro savana-ghatis, then they will bo- 
JS , pbM ,l of > 'th? d Su g n *t come sidereal by finding the Sun’s 
the time of question in order instantaneous place i. ft. the place of 

to find lagna. i r 1 

* . the Sun for tho hour given. Tho times 

which he has passed, and those which ho haB to pass, are known. Thua tho 
degrees which the Sun hus passed, and tlioso which ho has to pass, are palled tho 
bbuktIns'as and bhogyaks'as respectively. Now the time which tho Sun 
requires to pass the bhogyans'as is called the buogya time, and is 'found by 
the following proportion. 

If 30« 

: the period of rising of tho sign in which tho Sun is 

: : BHOGYita^AB 

: buogya time. 

In the same manner, the bhukta time can also bo found through tho 
biiuktXsbab. 

Now from tho time at the end of which tho horoscope is to be found, and 
which is c&llod tho isuta or given time, subtract the buogya tune just found, 
and front tho remainder subtract the periods of risings of tho next successive 
signs to that in which the Sun is a; long as you can. Then at Inst you will find 
the sign, the rising period of which being greater than tho remainder you will 
not bo able to subtract, nnd which is consequently called tho as'uddha sign, 
or tho sign incapable of being subtracted, and its rising period, as'uddha 
rising. From this it is evident that 'tho as'uddha sign is of course on tho 
horizon at tho given time. Tho degrees of tho as'uddha sign which arc abovo 
the horizon aud therefore called the bhuxta or passed degrees, are found as 
follows. 

If tho rising period of tho as'uddha sign 

: 30o 

s : tho remainder of tho given time 

: the passed degrees of tho as'uddha sign. 

Add to these pnssed degrees thus found, the preceding signs reckoned from 
the 1st point of Aries, and from the Sum, subtract the amount of the procession 
of the equinox. Tho remainder thus found will be tho placo of the horoscope 
from tho stellar Aries. . 

If the time at the end of which tho horoscopo is to be found, bo given before 
sun-rise, then find tho bhukta, or passed time of the sign in which tho Bun 
is, in the wtfy above shown, and subtract it and the rising periods of the pre* 
ceding signs from the given time. After this find the degrees ot the as'uddha 
sign correspon din g to the remainder of tho given time which will evidently be the 
buogya degrees of the horoscope by proportion aB shown above, and subtract 
tho sum of the BlIOGYA degrees of the horoscopo, the signs the rising periods of 
which are subtracted and tho bhukta degrees of tho sign in which the Sun is 
from the Sun'a place and the remainder thus found will be the place of the 
horoscope. 

Thus we get two processes ; one when the given time at the end of which the 
horoscopo is to be found, is after Bun-rise, and the other when that time is given 
before sun-riso, and which aro consequently oalled xbaha, or diivcfj' apd 
yyutkbaha or undirect processes respectively. , 

It is plain from this that if tho place of the Sun and that of the horoscope be 
known, the given lime from sun-rise at the end of which the horoscopo is found 
can be lpiown by making the sum of the buogya time of the sign in wli(ph the 
Sun and the buukta time of the horoscope and by adding to this s*in the 
rising periods of intermediate signs.— B. ]).] 



168 


Translation of the 


[VII. 27. 


of rising of the signs which are sidereal qiust be subtracted 
from tkpso ghatw (of the question) reduced to a like deno- 
mination. When the hours of the question aro already sider&l, 
there is no necessity for iinding the sun's real place for that 
time.* * * * 


* [If it be asked whether the time at the end qf which the horoscope is to be 
found is terrestrial or sidereal time ; if it be terrestrial, how it is that you 
subtract from that the rising periods which are of different denomination on 
account of their being sidereal, and why the sun's instantaneous place i. e. the 
place determined for the hour given is used to ascertain the bhogya time, the 
given time is reckoned from sun-rise and the bhooya degrees of the sign m 
which the sun is, rise gradually above the horison after sun-rise. Hence the 
bhogya degrees of the sign of the Sun’s longitude, determined at the time of 
sun-rise, should be taken to find the place of the Horoscope, otherwise the place 
of the Horoscope will be greater than the real one. As for example, take the 
time from sun-rise, at the end of which the Horoscope is to be found, equal to 
60 sidereal ghatis and 44 asttb when the Sun is in the vernal equinox at a place 
where the Palabha is 6 digits or the latitude is 22*} nearly, and ascertain the 
place of the Horoscope through the instantaneous place of the sun. Then, the 
place of the Horoscope thus found will bo greuter than tho plaoo of the Sun 
round at the time *f next sun-rise, but this ought to be equal to it, and you will 
not be able to make this equal to the place of the Sun determined at the time of 
next sun-riae, unless you determine this through the place of tho nun ascertained 
at sun-rise, and not through the Suh's instantaneous place. Hence it appears 
wrong to ascertain the place of the Horoscope through tho Sun’s instantaneous 
place. But the answer to this is as follows. 

The ghatis contained in the aro of the diurnal circle intercepted between that 
point of it where the Sun is, at a given time and the Horizon are the bAvana or 
terrestrial gkatib, but the ghatis contained in tho aro of the diurnal circle in- 
tercepted between that, point of it where the Sun was at the time of sun-rise and 
the Horizon are the sidereal, ghatis. Thus it is plain from this that if the 
Sun’s place determined at the time of sun-rise bo given, the time between their 
place and the Horizon reckoned in the diurnal circle v ill evidently bo the side- 
real time and consequently the place of the Horoscope determined through this 
will be right. But if tho instantaneous place of the Sun be given, the time given 
must bo the ba yana time, because let the instantaneous place of the Sun be 
assumed for the Sun’s place determined at the timo of sun-rise, then the time 
between this assumed instantaneous place of tho Sun and tho Horizon, wliich is 
sAyana, will evidently be the sidereal time. Hence the fact as stated in the 
verse 27th is right. 

Therefore if the Sun's instantaneous place and tho place of the Horoscope be 
given, the time found through these will be the bAtana time, but if the place of 
the Horoscope and tlint of the Sun determined at the time of sun-rise be given, 
the time ascertained through theso will be the sidereal time. And if you wish to 
fiud the sAyaka time through the place of the Horoscope and that of the Sun 
determined at the time of the sun-rise assumed tho sidereal time just found as a 
rough sAvana time and determined through this the instantaneous plaee of the 
Bun bj the following proportion. ♦ 

If 60 GHATIS 

: Sun’s daily motion 

: : these rough bAtana GHATIS , 

„ • the Sun’e motion relating to this time i and add then this result to 
the place of the Sun found at tha time of sun-rise. The sum thus found will be 
lira instantaneous plao^ of the Sun nearly. Find the timo again through this 
r 



VII. 32.] Siddhdnto-tftromani. M 59 

9 

28. In those Gentries having a north latitude of 69* 20* tho 

signs Sagittarius and fcapricornus are 
^termination of latitudes .... _ . 

in which different signs are never visible : and tho signs gemim 

hoMon ftb<>Te * Ud bdow [ he an ^ cance r remain always above the 
horizon. • 

29. In those places having a northern latitude of 78° 1 S', the 
four signs scorpio, Sagittarius, capricomus, and aqugrius are 
never seen, and the four signs taurus, gemini, cancor, *ond leo, 

, always appear revolving above tho horizon. 

30. On that far-famed hill of gold Merit which has a lati- 
tude of 90° N. the six signs of tho southern hemisphere never 
appear above the horizon and tho six northern signs are 
always above tho horizon. 

. 31. Lalla has doclorod that when the asus of chara- 

KtfANpA [in any latitude] are oqual to 
An error of Lalla exposed. , . , 1 

the time which any sign takes to rise 

on the equator, then that sign will always remain visible above 

the horizon : but this assertion is without reason. Wero it so, 

then in places having a latitude of 66*, tho whole twelve signs 

of the ecliptic would always be visible, and would all appear 

at once on all occasions, as tho times of their rising on tho 

equator are equal to the asus of their CHAKA-giiANpAS : but 

this is not the fact. 

32. Lalla has also stated in his work on tho sphere that 
where the north latitude is 66° 30', 
L “ gro “ error ot Sagittarius and capricomus are not 1 
visible, and also that in north latitude 
75°, scorpio and aquarius are never there visible : but this also 
is an idle assertion. How, my learned friend, has he managed 
to mako so gross and palpable an error of three degrees?* 

instantaneous plaoe of the 8un, and through this time ascertain the instantaneous 
place of the Sun. Thus you will get at last the exact sAyawa time fiom sun-rise 
to tho hour given by the repetition of this prooeas. As the 8un is taken here 
for an example, you ean find the bXta^A time of any planet or any planetary • 
time from the planet’s rising to the hour given by the repetition of the aforesaid 
proeesa.— B. D.] * 

• [BbAixabAosauta means here that Laxla mentioning the degrees of lati- 
tudes, lias committed » grand mistake in omitting 3 degrees, because he has 
* x 



170 


Tramhtion of the [VII. 33. 

33. The altitude of the polar star and fts zenith distance 
as found by observation, give respectively the latitude and 
t he lambaksa or compl ement of the latitude. Or the zenith 
distance and altitude of the Sun at mid-day when on the equi- 
noctial give the latitude and its complement. 

84. The unnata the time found in that arc of the diurnal 
circle which is i ntercepted between fhe eastern or weatem 
h orizon 'and the planet abo ve it. This is used in 

finding the shadow of the planet. The sine of the unkata 
which is oblique, like the aksha-karna, by reason of the lati- 
tude, is called chhedaka and not s'anku because it is upright.* 

35. In order to find the shadow of the Moon, the udita 
(the time elapsed from the rising of a planet) which has been 
found by some astronomers by means of repeated calculation 
is erroneous, for the udita, (found by repeated calculation) is 
not savana. the labour of the astronomer that does not 
thoroughly understand mathematics as well the doctrine of the 


* 

stated in his work that Sagittarius and oapricornus are always visible in a plaoe 
bearing a latitude 66® 30', and scorpio and aquarius at 73° N., whereas this is 
not the earn, those signs are always visible in the places bearing the latitudes 
69* 80' and 78 • lb' respectively as shown in the verses 28 and 29.— B. D.] 

* [When the Sun is above the Horuon, the shadow caused by a gnomon 12 
digits^ high, is called the Sun’s shadow according to the s'iddhanta languages, 
and having at first determined the sine of tho Sun's altitude and that of it 
complement through hit UDITA time, astronomers ascertained this by the follow- 
ing proportion. 

Aa the sine of the Sun’s altitude 
: the sire of its complement 
1 1 gnomon of 12 digits 
i the shadow caused by the gnomon. 

Thus they determine the shadow of all planets, Moon, Ac., and that of the fixed 
atari. Though the light of the five small planets, Mars, Ac., and the fixed stars 
ia not so brilliant, like that of the Sun and Moon, as to make their shadow 
visible, yet it is neoeasary to determine the shadow of any heavenly body in order 
to know the direction in which the body may be. Because, if the length and 
direction of the shadow of the body be known, the direction in which it is can , 
he ascertained by spreading a thread from the end of ito shadow through that of 
the gromon. For, if yon will fix a pipe in the direction of the thread thus 
spread, you will ace through that pipe the body whose shadow is used here. 

The time given for determination of any planet’s shadow must be the liVAVl 
time, because it ia necessary to determine the degrees of altitude of a planet 
to know ita shadow, and the denees 'can be determined through the time 
contained in that arc of the diurnal oirole intercepted between the planet and 
horiaon. Bat the time contained in this are cannot be other than tne iXtaKa 
time.— B. D.] 



Skhlhdntastromani. 


VII. 30.] 


171 


sphere, in writing a book of instruction on the science is utter- 
ly futile and useless.* 

* 36. The degrees of altitude are found in the d^inmanpala 
or vertical circle, being the degrees of 
tndDfiwrA.* 011 °* elevation in it above 1 the horizon; the 
• degrees of zenith distance are (as their 
name imports) the de'greos in the same circle by which the 
object is distant from the zenith or mid-heaven of the observer : 

pbigjy! is the sine of the zenith distance. 

37. When the Sun in his ascent arrives at the prime verti- 

cal, the s'anku found at the moment is 
the a,AMA-aJ^KUj the s'ahcts found at 
the moments of his passing the kona- 
v$itta and the meridian' are respectively termed tho kona- 
s'anku and madhya-s'anku. * 

38. One-half of the vertical circle in which a planet is 

_ . • observed should be visible, but only 

Season of the correction , * • 

of parallax to the sine of alti- one-half less the portion opposite tho 

radius of the Earth is visible to observ- 
ers on the surface of the Earth. Therefore x ‘j part of tho daily 
motion of the planet observed is to bo subtracted from tho sine 
of altitude or from the s'anku to find the shadow : [inasmuch 
as that amount is concealed by, or opposite to, the Earth] . 

39. The aqbX (the sine of amplitude) is t he sine of the arc 

of the horizon intercepted between tho 
“ d Pn“e vertical and the planet's diurnal 
circle in the east or west i. o. between 

* [In order to determine the Moon’s shadow at a given time at full moon, 
some astronomers find her udita time i. e. the time elapsed from her rising to the 
hoar given bj the repeated calculation, through her instantaneous place and the 
plaoa of the horoscope determined at the given hour. But the/ greatly err in 
this, because the time thus found will not be the s'avava time and consequently 
they eannot use this in finding the Moon's shadow. Tbeir way for finding the 
udita time by the repeated eauulation would be right, then only if the given 
place of the Moon would be suoh aa found at the time of her rising and not )m 
instantaneous place. Because her vdita time found through her instantaneous 
place beeomes i'atava at once without having a recourse to the repelted caleu- • 
lotion, aa it is shown in the note on the verse 27 of this Chapter.— B 0.] 

>K 2 





J72 


Translation of the [VII. 40 . 

the east or west point • of the horizon, and the point of the 
horizon at which the planet rises or sets. The line connecting 
the points of the extremities of the east and west agb! Is 
called the udayasta-sutra, the line of rising and setting. 

' 40. The b'anku-tala or base of the s'anku stretclfts dur- 
ing the day to the south of the udyXsta-suI'RA; because the 
diurnal circle have during the day a southern inclination (in nor- 
thern latitude) above the horizon. But, below the horizon 
at night, tho base lies to tho north of the udayXsta-sutra as 
then the diurnal circles incline to the north. Tho sankc- 
tala’s place has thus been rightly defined. 

41. Tho s'anku-tala lies to the south of tho extreme point 
of aura when that aqrX is north and when the ag&A is south, 
the s'anku-tala lies still to the south of it. Tho differenco 
and sum of the sine of amplitude and s' anku-tala has been 
deno minated thfe bAhu or bhuja ; it is the sine of tho degrees 
lying between the prime vertical and the planet on tho plane of 
tho horizon. 

42. [Taking this bAhu as ono side of a right-angled 
trianglo.] The sino of tho zenith distance being the hypothe- 
nuse then the third side or tho KOfi being the square root of 
the difference of their squares will bo found : it is an east and 
west portion of tho diameter of the prime vertical.* 

I now propose to explain tho triangles which aro created by 
reason of the Sun's varying declination : and shall then proceed 
.to explain briefly also the latitudinal triangles or those created 
.by different latitudes. [The former ore called krAnti-kshetbas 
and the lattor aksha-kshetbas.] 

* Vida accompanying dia- 
gram. 

a being plaoe of the Sun : d its 
place of airing in the horixon : 

A A the UDAYA'axA-atfm d f 
the Aefti' : a b the b'asxu-taza : 

" then eg ii the ba'htj end the 
triangle « * g ie the ono here 
» represented to.— L. W. 





fiiddhdnla-airmani. 


173 


1st. The sine of declination 
* the radius of diurnal circlo corO 


= hypothonuse. 
== hypotbonuso : 


VII. 45.] 

In the 1st triangle of declination. 

= bhuja or base, 

. , , , ,1° ^° r “] = Koti or per- 

responding with tho declination >■ , 

_ r . , pen&cular, 

above given J ♦ 

and radius of largecircle 

2nd. Or in a right sphere. 

The Bine of I, 2 or 3 signs 

Tho declination of 1, 2 or 3 signs in six’! L 

o’clock line J 

44. Sines of arcs of diurnal circles cor- ^ 

responding with the declination L = kojis. 
above given J 

Those sines being converted into terms of a large circle : 
and their arcs taken, they will then express the times in asus 
which each sign of the* ecliptic takos in rising*at tho equator 
i. c. the right ascensions of .those signs or the lankodayas, 
that is tho 2nd will bo found when tho 1st is subtracted from 
two found conjointly, and the 3rd will bo found when tho 
Bum of tho 1st and 2nd is subtracted from three found con- 
jointly. 

45. In tho right-angled triangle formed by tho s'anku 

T riangl e* arise from lati- or gnomon when the Sun is on tho 

equinoctial.* 

1st. The s'anku of 12 digits = the koti. 

Tho paladhX or tho shadow of s'anku ^ 

= tho BHUJA 

f = tho KARNA Or 
\ hypothenuso 
= BHUJA. 

= KOTI 

= hypothentise. 


tude. 


or gnomon 

and the aksha-kar^a 

or 2nd. The sine of latitude 
The sine of co-latitude 
and radius 


This triangle is found in the plane of the meridian. 


[• The right angle triangle* stated in the fire Terse* from 45 to 49, are clearly 
. Men by fastening aome diainetrial thread* within the artmiiary* sphere. A* 



[VII. 46.. 


174 


Translation of the 


46. Or the sine of declination reckoned;* 
on theu^MANpALA from the east and west line J 
KujyjC, tjie sine of ascensional difference') 
in the diurnal circle of the given day J 


KOfl. 

BHUJA. 



Let Z G N H be the meridian of the given place, G A H the diameter of the 
horizon, Z the Zenith, P and Q the north and south poles, £ A F the diameter 
of the equinoctial, P A Q that of the six o’clock line, O/O that of one of the 
diurnal circles, aud E B, / h the perpendiculars to G E. Then it is dear from 
this that 

Z E or H P :* the latitude, 

A B s* the sine of it, 

E B s= the co-sine of it, 

A /a* the declination of a planet revolving in the diurnal 
circle whose diameter is 0 D, 
and A g = the atm or the sine of amplitude, 

/ g = the x ojya', 

A e the sama-ba'kxu or the sine of the planet's altitude* 
when it reaches the prime vertical. 
rg = the TADDnaiTi/ 
e/= the tajjdhjih — 

fk = the WMAirpAJLA sfarxu or the aine of the planet’s 
altitude when it readies the six o’clock line, 

A A = the AGtRA'Di-XHASpA or the 1st portion of the sine 
of amplitude, 

and k g == the aoba'sra-xbahpa or the 2nd portion of the sine 
f of amplitude; • 


■VII. 48.] 


Siddhdnta-tfirmanl 


17 5 


The sine of amplitude in the horizon = hypothenuse 
• This is a well known triangle. * 

47. Or the sama s'anku in the prime ver- ■) 
tical being 

* The sine of amplitude =*= bhuja 

The taddh^iti in the diurnal circle = hypothenuse 
Or 




Taking the sine of declination 
and the sama-s'anku 
Taddh^iti minuH kujya 
48. The unhanpala s'anku being 
The sine of declination will then be 
And AGRADi khanda or 1st portion of the 
sine of amplitude will bo 


} 


= BHUJA. 

= hypothenuse 

= KOfl, 

= BHUJA 

= hypothenuse 
= koti 


Therefore, with the exocptiotl of the first and lost the other six triangles 
stated in the verses are these in succession. A E B, A g /, A eg, A ef, Afh 
and 0 fh and the first triangle you vfill gU by dividing the three sides of the 
E B 

triangle A E B by — and for the last see the note on the verse 49. 

12 

It is clear from the above dosoribed diagram that all of these triangles are 
similar to eaoli other and consequently they can be known by menus of propor- 
tion if any of them he known. 

The siddiiAntis, having thus produced several triangles similar to theso 
original by fastening the threads within the armillary sphere, find answer* of 
the several questions of the spherical trigonometry. Borne problems of the 
spherical trigonometry can be solved with greater facility by this Sidshamta 
way than the trigonometrical way. As 

Problem. The aenith distances of a star when it has reaohed the prime 
vertical and the meridian at a day in any plaoe are known, find the latitude in 
the plaoe. 

The way for finding the answer of this problem according to the siddhahta is 
as follows. 

Draw C c J. A Z, (See the proceeding diagram) then Coe will be a lati- 
tudinal triangle. 

Now, let a b C o, the sine of aenith distance, 
b = A c, the co-sine of Z e, 
o = A e, the SAXA-i’AKU, 
and a « the latitude. 

ThenCo=\/«* + (&-«)% 
and Co:Oe : : AE : AB, 

or : « j s »d : tin * ; 

a X Bad 

V. Sin* — -r: -T-rrr . 


(J-e)*.-B. D.] 



176 


[TO. 49. . 


Or • 

I 

Making the unmanjjala sanku = KOfi 

the aobagra-khanpa or 2nd portion of tho^ wrriTr 

sine of amplitude is 

the KujYA'then becomes 
40.* Tho s'anku being 
and the sanku-tala 
Then, the chhkdaka or h^tti 
Thoso who have a clear knowledge of tho spherics having 
thus immediately formed thousands of triangles should explain 
the doctrine of the sphere to their pupils. 

End of Chapter VII. on tho principles of the rules for 
resolving the questions on time, Bpaco and directions. 




= KOTl 
= BHUJA 
= hypothonu.se 


Chapter VIII. 

Called Grahana Vasana. 

hi explanation of the cause of eclipses of the Sun and Moon. 

1. Tho Moon, moving like a cloud in a lower sphere, 

Th. rf dir*. OVertak0S Sn “ ^ rea80n 0f itS 
, tiona of tho beginning and quicker motion and obscures its shin- 

. r ing disk by its own dark body :] hence 

it ariscB that the western side of the Sun’s disk is first ob scured, 
and that the eastern side is tho last part relieved from the 
Moon's dark body ; and to some places the Sun is eclipsed and 
to others is not eclipsed (although he is above the horizon) 
on account of their different orbits. * 1 

* * This triangle differs from the 1st of the 47th Terse only in this respect that 
the base of the triangle in the 47th Terse iS equal to the sine of the *>holB ampli- 
i t*de while the base round when the Sun is not in the prime vertical, will always 
be more or less than the sine of amplitude and is therefore generally called 
bahxutAla. — L. W. 



VIII. 6.] 


Suldlidnta-s'iromoni. 


2. At the chan|p of the Moon it often so happotis that an 

* # T!»o chum of the parallax obaerver P laced afc M of 

in longitude and that iu Earth, would find the Sun whon far 

* s from the zenith, ot^cured by the 

intervening body o^ the Moon, whilst another observer on 
the surface of the Earth will not at tho samo time find him 
to bo so obscnred, as tHe Moon will appear to him .[on tho 
higher elevation] to bo depressed from tho line Of vision 
, extending from his eye to tho Sun. Henco arises tho necessity 
for t he co rrection of par allax in celestial lon gitude and parallax 
in latitu de in solar cclipsos in consoquence of tho difference of 
the distances of the Sun and Moon. 

3. When tho Sun and Moon are in opposition, the Earth's 

Tho raison of the com*. slladoW 0nTclo P M tho Moon in **■ 
tioii of parallax not being noss. As the Moon is actually onve- 
neeesaart in lunar eclipses. . , . , . . .. 

loped in darkness, its echpso is equally 
seen by every ono on tho Earth's surface [above whoso horizon 
it may be at the time]- : and as tho Earth's shadow and tho • 
Moon which enters it, are at tho same distance from tho Earth, 
tlioro is thereforo no call for tho correction of the parallax in a 
lunar eclipse. 

4. As tho Moon moving eastward enters tho dark sha- 

. .. v dow of tho Earth : th erefore its eastern 

The cause of the uircc- 

tions of the beginning and s ido is firat of all involved in obscurity, 
end of tho lunar eclipse. , .. , „ 

and its wostom is the last portion of 

its disc which omorgos from darkness as it advances in its * 


course. 

5. As the Sun is a body of vast size, and the Earth insigni- 
ficantly Bmall in comparison : tho shadow mado by tbo Sun 
from the Earth is therefore of a conical form terminating in a 
sharp point. It extends to a distance considerably beyoud 
that of the Moon’s orbit. 

6. Tho length of the Earth's shadow, and its breadth at the* 
port traversed by the Moon, may he easily found by propor- 
tion. 



178 


Translation of tb [VIII. 7. 

In tho lunar eclipse tho Earth’s shaded, is northwards or 
southwards of *tho Moon when its latitude is south or north. • 
Hence the latitude of the Moon is here to be supposed inverse 
(b e. it is to be parked revorsly in the projection to find the 
centre of tho Eafth’s shadow from the Moon.) * 

7. As the horns of tho Moon, when it is half obscured form 

The determination of tho ° btnS ° the duration 

coverer in* the eclipse of the of a lunar eclipse is also very great, 
San and Moon. henco the coverer of tho Moon is 9 

much larger than it. 

8. The horns of tho Sun on tho contrary when half of its 
disc is obscured form very acuto angles : and tho duration of a 
solar eclipse is short : henco it may bo safely inferred that tho 
dimensions of the body causing the obscuration in a solar 
eclipse are smaller than and different from the body causing 
on eclipso of the Moon.* 

9. Tlioso learned astronomers, who, being too exclusively 
, devoted to the doctrine of tho sphere, believe and maintain 

that RXhu cannot bo tho cause of the obscuration of tho Sun 
and Moon, founding their assertions on the above mentioned 
contrarieties, and differences in the parts of the body first 
obscured, in tho place, time, causes of obscuration &c. must 
be admitted to assert what is at variance with the Saniiita, 
the Vedas and FurInas. 

10. All discrepancy, however, botween tho assertions abovo 
‘ referred to and the sacred scriptures may be reconciled by 
1 understanding that it is the dark Bahu which entering the 

Earth’s shadow obscures the Moon, and which again entering 
the Moon (in a solar eclipse) obscures the Sun by tho power 
conferred upon it by the favour of Brahma. 

• [3*d the Sun’s corerer been the same iritli that of the Moon, his horns, 
when he is half eclipsed, would hare formed, like those of the Moon obtuse 
angles. For the apparent diameters of the Sun and Moon are nearly equal to 
• each other. Or the Moon when it is half eclipsed would hare represented its 
horns, like those of the Sun, forming aonfe angles, if its ooverer had been tho 
same wkh that of the San. But as this is not tlie case, tho corerer of the Moon 
is, of course, different and much larger than that of the Sun.— B, D.] 



S'uldh&nta-sironioni. 


179 


VIII. 15.] 

a 

11. As the spectator ia elevated above the centre of tlie 

What {■ the oatue of paral- ^li by half its diaittctor, there- 
lax, and why it ia calculated fore sees the Moon depressed from its 
from the radius of the Earth. , r „ , . , , . . , 

place [as found by a calculation made 

for the'centro of thd Earth] . Honco the parallax in longitude 
is calculated from the radius of the Earth, as is also the parallax 
in latitude. ' 

12. Draw upon a smooth wall, the sphere of the earth 

Construction of diagram reducod to "V convenient scale, and 
to illuatrate the cause of the orbits of the Moon and Sun at 
parallax. proportionate distances : next draw a 

transvorso diameter and also a perpendicular diameter to both 
orbits.* • 


13, 14 and 15. Thoso points of tlio orbits cut by this 
diameter are on the (ratioual) horizon, And tho point above 

• Fig. 1. 

• In Fig. 1, let E bo tho 
oentro of the earth; A a 
spectator on her surface ; C 
1>, F G tho vertical circles 
passing through the Moon M, 
and the Sun S; D, G tho 
points of the horizon out by 
tho vertical circles C D, F 
G ; and 0, tho zenith in tho 
Moon’a sphere, and F in that 
of tho Sun. Now, lot E M S 
bo a line drawn from tho 
centre of the Earth to the Sun 
in which the Moon lies always 
at the time of conjunction, and 
A S the vision line drawn 
from the spectator A to the 
San. The distance at which 
the Moon appears depressed 
from the vision line in the 
vertical circle is her parallax 
from the Sun. 

When the Sun reaches the zenith F, it is evident that the Moon also will then 
be at C and the vision line, and the line drawn from the centre of the Barth will 
be coincident. Hence there is no parallax In the zenith. 

Thus the parallax of the Moon from the Sun in the vertical circle^ is fyere 
Shown by means of a diagram which becomes equal to the difference between the 
parallaxes of the Sun and Moon separately found in fhe vertical circle as stated by 
B&A'nAVA'oiiA'BYi. in the chapter on eclipses in the commentary va'smu'bha* 
bhya. and the theories and methods afro also given by him on the parallaxes of 
the Sun and Moon. This parallax in the vertical circle which arises «f rom the 
zenith distance of the planet is called the common parallax or the parallax in. 
■H itudc. 

L 2 




ITptiee, it is evident from this that when the sin (d + p) *= R or d+p = 
PO®, then the parallax will be greatest and if it be denoted by P, * 

tinPX rintf+ri 

imPss* and sinp = - 


Now, the parallax is generally so small that no sensible error is introduced by 
making sin p 5= » and sin P a P x 

P X sin (d+p) 

i 


A gain, •‘for the reason jpst mentioned sin d f« assumed for sin (d + p) in the 
UddhXxtai, 

P. sin d 


that is, the common parallax of a planet is found by multiplying the greatest 
psnllax by the sine of the senith distance and dividing the product by the 






Siddhdnta-s' iromoni. 


181 


VIII. 20.] 


the parallax arises.* [For this purpose] let him draw one lino 
passing tho centre of tho earth to the Sun's disc : and another 
Which is called the d^iksIjtra or lino of vision, lot him draw 
from the observer on the Earth’s surface to tho Sun’s disc. Tho 
minuteS contained ih the arc, intercepted between theso two 
lines give tho Moon’s parallax from the Sun. 

16. (At tho new Moon) tho Sun and Moon wilj always 
appear by a lino drawn from the centre of tho earth to bo 
in exactly the same place and to have tho same longitude: 
hut when tho Moon is observed from the surface of tho Earth 
in tho D^ixstJTRA or lino of vision, it appears to bo depressed, 
and hence tho name lambana, or depression, for parallax. 

17. (Whon the now Moon happens in tho zenith) thon tho 
lino drawn from tho Earth’s centre will coincide with that 
drawn from its surface, hence a planet has no parallax when 
in tho zenith. 

Now on a wall ru nnin g duo notfh and south draw a diagram 
as above proscribed ; [i. o. draw tho Earth, and also tho orbits 
of the Sun and Moon at proportionate distances from the Earth, 
and also the diameter transverse and perpendicular, &c.] 

18. The orbits now drawn, must bo considered as driksi ik- 
pa-v$ittas or the azimuth circles for tho no n age simal. Tho 
suae of the penith distanco of the nonagesimal or of tho latitude 
of tho zenith is tho d&ikshepa of both the Sun and Moon. 

19. Mark tho nonagesimal points on tho d$iksuepa-v$ittas 
at the distance from the zenith oqual to tho latitude of tho 
points. From these two points (supposing them ns tho Sun 
and Moon) find as before tho minutes of parallax in altitude. 
These minutes are h ere Nati- kalIs, i. e . tho minutes of _tho 
p arallax in latitude of tho M oon from, the Sun. . 

20. The difference north and south between the two orbits 
i. e. the measure of their mutual inclination, is tho same, in 
every part of tho orbit as it is in the nonagesimal point, hence, 
this difference called nati is ascertained through tho d&tkshe- 
pa or the sine of tho zenith distanco of tho nonagesimal.* 

[• VHjpn (he p lane t ia 'depressed in the vertical* circle, Us north and south 



182 


Translation of the 


[VIII. 21. 


21. Tho amount by which the Moon is Oppressed below the 
Sun deflected from the zenith [at tho conjunction] wherever it 
bo, is the east and West difference between the Sun and Moon 

in a vertical cirple.* 

• ' • 
distance from its orbit caused by this depression is callpd nati or the parallax 


in latitude. 

As, in Fig. 3, let Z be the zenith ; N the non a-, 
gosimal j Z N P its vertical circle j N s r tho 
ecliptic ; £ its polo j Z i i the vertical circle 
passing through the true place 8 and the depress* 
ed or apparent place t of the Sun j P t r a 
secondary to tho ecliptic passing through the 
apparent place t of the Sun j then if is tho 
spashta iamb an a or tho parallax in longitude 
and t r the NATI or the parallax in latitude which 
oan be found in the following manner according 
to the ainnuANTAS. 

Let Z N be the zenith distanco of tho nona- 
gesimal and Z S that of tho Sun ; then by the 
triangles Z N S, t s r 

•iu Z S : siu Z N = sin s 1 i sin r t, 

* siu 1 1 % sin Z N 

sin r t as • ; 

sin Z S 

Now, 9 1 is takon for Bin 1 1, aud rV for sin r t, 
on account of their being very small 
9t% sin Z N 

S-rtss { 

sin Z S 

but according to the slddhantas 
F . sin Z S 

»< = (see the preceding noto). . (1) 


Nig. 3. 



r < —= - 


R 

P.sinZN 


( 2 ) 


that if, the »ati is found by multiplying the sine of the latitude of the nona- 
gesimal by the greatest parallax and dividing the product by the radius. 

It is clear from this thut the north and south distance from the Sun depressed 
in the vertical circle to the ecliptic wherever he may be in it, becomes equal to 
tho common parallax at the nonagesimal, and hence the XT ATI is to be determined 
from the zenith distance of the nonagesimal. 

For this reason, by subtracting the sati of the Sun from that of the Moon, 
whioh are aeparately found in the way nbove mentioned, the parallax in latitude 
of the Moon from the Sun ia found : and this becomes equal to the difference 
between the mean parallaxes of the Sun and Moon at the nonagesimal. The 
a fret ia ihown by BbIbsabIohIbxa through the diagrams stated in the 
-s 12th Ac. 6 ^ 


At the time of the eclipse as the latitude of the Moon revolving in its orbits is 
very small, the Moon, tlierefore, is not far from the ecliptic j and hence the 
parallax in longitude and that in latitude of the Moon is here determined from her 
* comspondingplaoe in the ecliptic, on, account of the difference being very 

' • [Xccordiug to the technicality of the Siddhantos, the distance taken in any 
eirde from any point in it, is called tho east and west distance of the point, anil 



. VIII. 27.] SidJhanta-ifiromoni. 183 

22. For this r$son, the differenco is two-fold, lieing partly 
• cast and west, and partly north and south. And tliq ecliptic 

is here east and west, and tho circle secondary to. it is north 
and south. (It follows from this, that the east and west 
differenco lies in tho ecliptic, and tho north and south differ- 
ence in the secondary to it.) 

23. The differenco feast and west has been denominated 
UMBAKA or para llax in longitude, whilst that running north 
and south is parallax in latitudo. 

2 A. The parallax in minutes as observed in a vertical circlo, 
forms tho hypothenuso of a right angle triangle, of which tho 
nati-kalX or the minutes of tho parallax in latitude form one of 
tho sides adjoining tho right angle then tho third sido found by 
taking the square-root of the differenco of the squares of tho 
two preceding sides will bo hpkuta- iam d an a- liptX or tho 
minutes of tho parallax m long itude.* * 

25. Tho amounts in minutes of parallax in a vertical circlo 
may bo found by multiplying tho sino of tho Sun’s zenith a 
distance of tho minutes of tho extreme or horizontal parallax 
and dividing tho product by tho radius. Thus tho nati will 
be found from tho D^ixsnErA or tho sine of tho nonagcsimal 
zenith distance.! 

26. The extreme or horizontal parallax of tho Moon from 
the Sun amounts to T y part of the differenco of tho Sun’s and 
Moon’s daily motion. For T l j part of the yojahas, tho distanco 
of which any planet traverses per diem (according to tho bid- 
dhantas) is equal to tho Karth’s radius. 

27. The minutes of tho parallax in longitude of the Moon 
from the Sun divided by the difference in degrees of tho daily 

the distance taken in the secondary to that circle from tho same point, is called 
the north and south distance of that point.— -B. D.] • 

* [See Fig. 8, in which by assuming the triangle rat as a plane right-angled 
triangle , rt — base, at — hypothenuso and ir = perpendicular, and therefore 
er=V i B.D.] * 

f [This is clear from the equations (1) and (2) shown in the preceding largo 
note.— B. D.] 



184 


Translation of the [VIII. 28. « 

motions of tlio Sun and Moon will bo concerted into anAfis 
[i. e. thg timo between the true and apparent conjunction] .* • 

If tho Mo, on be to tho cast [of thenonagesimal], it is thrown 
forward from tho Sun, if to the west it is thrown backward (by 
tho parallax). * 

28. And if the Moon be advanced from* tho Sun, then it 
must bo, inferred that tho conjunction has already taken placo 
by reason of tho Moon's quicker motion; if depressed behind 
tho Sun, then it may be inferred that the conjunction is to , 
come by tho same reason. 

Hence tho parallax in time, if tho Moon bo to the oast [of 
the nonagosimal] is to be subtracted from tho end of tho titht 
or tho hour of ecliptic conjunction, and to bo added when tho 
Moon is to tho west [of the nonagcsimal] . 

29. The latitudo of tho Moon is north and south distance 
botweon tho Sun and Moon, and tho nati also is north and 
south, lienee tho sxra or. latitudo npplied with the nati or 
tho parallax in latitudo, becomes tho apparent latitudo (of tho 
Moon from tho Sun). 

Valana or variation (of the ecliptic). 

[Tho deviation of the ecliptic from tho eastern point (in 
reference to the observer's place) of a planot's disc, situated 
in tho ecliptic is called tho Valana or variation (of the ecliptic). 

It is evident from this, that tho variation is equivalent to the 
arc which is tho measuro of tho angle formed by tho ecliptic 
• and tho secondary to tho circlo of position at tho planet's 
' placo in the ecliptic. It is equal to that arc also, which is tli^ 

* It is dear from the following proportion. 

If difference in minutes of daily motions of Sun and Moon. 

: 60 ghatis — whet will 

: : given Laubana— ka'ias or minntes of the parallax give j f 

60 X given minntes of the parallax 

or * ■■■■■ 

diff. in minutes of Sun's end Moon's motions 
given minutes of the parallax 

sa - ss acceleration or delay of con* 

dlflj in degrees of Sun's and Moon's motions 

junction arising from parallax.— L. W. 



Siihlhdnta-8 f irMiioni, 


185 


. VIII. 29.] 

measure of the an gib at the place of the planet in tlio elliptic 
• formed by the circle of position and the circle df latitude. It 
is Very difficult to find it at once. For this reason, it is 
divided into two parts called the a'ksua-valana (latitudinal 
variatiod) and the Iyana-valaxa (solstitial variation). The 
aksha-valana is tho arc which is the measure of the angle 
formed by the circle of position, and tlio circle of declination at 
the place of the planet in the ecliptic, and the Iyana-valana 
is the arc which is the measure of the angle formed by tlio 
circle of declination and the cirelo of latitude. This angle is 
equivalent to tho angle of position. From the sum or differ- 
ence of those two arcs, tho arc which is tho measure of tho 
angle formed by tho circle of position and tlio circle of latitude 
is ascertained, and hence it is sometimes called tho sVashta- 
YAiiANA or rectified variation. 

Now, according to the phraseology of the SiiuihXntas, tho 
point at a distance of 90* forward from any place in auy 
circle is the east point of that place, and tho point at an equal 
distance backwards from it is the west point. And, tho right 
hand point, 90* distant from that place, in tho secondary to 
the former circle, is the south point, and tho left hand point, 
is the north point. According to this language, tho deviation 
of the east point of the place of the planet in tho ecliptic, from 
the east point in tho secondary to the circle of position at the 
planet's place, is tho vaj,ana. J3ut tho secondary to tho circlo 
of position will intersect tho prime vertical at a distanco of 
V'f forward from tho place of tho planet, and henco tho 
deviation of the east point in tho eclipt io from tho east point 
in the prime vertical is the valana or variation, and this results 
equally in all directions. When the east point in the ecliptic 
is to tho north of the cast point in the prime vertical, the 
variation is north, if it be to the south, tho variation is south. 

The use of tho valana is this tliat, in drawing tlie projoc- • 
tions of the eclipses, after the disc of the body which is £o be 
eclipsed is drawn, and the north and south and the cast and 
‘ M • 



186 


Translation of the [VIII. 29. ■ 

west lines are also marked in it, which liqps will, of course 
represent the circle of position and its secondary, the direction « 
of the line uepresenting the ecliptic in the disc of the body can 
easily be found through the valana. This direction being 
known, the exadt directions of the beginning, middle and the 
end of the eclipse can be determined: fiut as the Moon 
revolves in its orbit, the direction of its orbit, therefore, is to 
be found. But the method for finding this is very difficult, 
and consequently instead of doing this, Astronomers deter- 
mined the direction of the ecliptic, by means of tho Moon’s 
corresponding place in it and then ascertain tho direction of 
the Moon’s orbit. 

The valana will exactly be understood by seeing the follow, 
ing diagram 



* Let E PC be the ecliptic, P r the place of the planet in it, 
A h B the equinoctial, V the vernal equinox, D h F the prime 
vertical, h the point of intersection of the prime vertical and 




187 


•VIII. 29.] SiMli&nta-xiromoqL 

the equinoctial, hen^e h the east or west point of the horizon 
• and D h equivalent to the nata which is found* in the V. 36. 
Again, let e P c, a P b and d P / bo the circles of latitude, 
declination and position respectively passing through the place 
of the planet in the ecliptic. s 

* Then, ' * 

the arc f h which is the measure of /. h Pjf = tho^KSHA- 
valana : • 

the arc b e £ r P b = the Xyana- 

VALANA : 

and the arc / r £ r P/= the SPAsnfA- 

VALANA. 

Or according to the phraseology of the SinnuiNTAS 
B the east point of P in tho ecliptic ; 


A the equinoctial; # 

D ... the prime vortical ; 


hence, * • 

the distance from O’ to A or arc D A or / h a the Xksha- . 
valana : 

A to B or arc A E or h c = tho Iyana-valana : 

and Dto B or arc D E or/r = the spashja-' valana 

or rectified variation. 

These arcs can be found as follows 
Let, l = longitude of the planet, 
ft = obliquity of tho ecliptic, 
d = declination of the planet, 

L = latitude of the place, 
n = NATA, 

JS = Xyana-valana, 

y = AKSHA- VALANA, 

and Z = rectified valana. 

Then, in the spherical triangle AYE, 

Bin E A V : sin A V E = sin E V : sin A E, 
cos d : sin e = cos l : sin x t * 

x 2 

t 


or 



188 


Translation of the 


[VIII. 30. 


sin>a; or sine of tho Xyana-valana = « 


ruin e . cos I , 


cos d 


(A) 


. ScoV. 32,33,34. 

This valana is called north or south as 'tho point E be north 
or south to tho point A. • 

And, # in tho triangle A h D, . 

• sin D A h : sin A k D = sin D h : sin DA; 
hore, sin D A h = sin E A V = cos d, 
sin A h D = sin L, 
and sin D h = sin n, 

cos d : sin L = sin n : sin y, 

sin L. sin a 

sin y or sine of the xksha- valana = (B) 

cos d 

SeoV. 37., 

The Xksha-valana is called north or south as tho point A 
be north or south to the poiht D. 

And the rectified vat, ana D E = DA + AE, when tho 
point A lies betwcori the points D and E, but if the point A 
be beyond them, tho rectified valana will be equal to the 
difference between the Xkbha and Xyana-valana. This also 
is called north or Bouth as tho point E be north or south to 
the point D. • 

The ancient astronomers Lalla, S'afpATi &c. used tho 
co-versed sin l instead of cos l and the radius for the cos d 
in (A) and the versed sin n in the place of Bin n and radius 
for the cos d in (B) and hence, the valanas, found by then* 
are wrong. Bit£skaraciixkya therefore, in order to convinco 
the people of tho said mistake made by Lalla, S'afpATi, &c. 
in finding the valanas refuted them in several ways in thq 
subsequent parts of this chapter .-y-B. D.] 

30. In either the 1st Libra or the 1st Aries in tho oqui- 
Atava-vaxana* n °ctial point of intersection of the 

* equinoctial and ecliptic, the north and 

south lines of the two circles i. o. their secondaries are different 



VIII. 36.] Shhlhdnta-H'iromom. ISO 

and aro at a distance* of tho extreme declination (of the Sim 
i. o. 24°) from each other. 0 , 

31. Hence, the Ayana-valana will then bo. equal to the 
sine of 24° : — The north and south lines of Jthese two circles 
however aro coincident at the solstitial poinfs. 

32, 33 and 3t. *And the north and south lines being there 
coincident, it follows as a matter of courso that the east of 
those two circles will bo the same. Hence at the solstitial 
points there is no (Ayana) valana. 

When the planet is in any point of the ecliptic bet ween the 
equinoctial and solstitial points, Ayana-valana is then found 
by proportion, or by multiplying the co-sine of the longitude 
of the planet by the sine of 2 1*, and dividing the product by 
the dyujya or the co-sine of the declination of the planet. 
This A yana -valana is called north or south as tlio planet be 
in the ascending or descending signs respectively. 

Thus in like manner at tlio point of intersection of tho prime 

, vertical and equinoctial, the six o’clock 

iUll-TlUlTA. * 

lino is the north and south lino of tho 
equinoctial, whilst the horizon (of the given place) is tho north 
and south lino of the prime vertical. The distance of theso 
north and south lines is equal to the latitudo (of the placo). 

35, Hence at (the east or west point of) the horizon, tho 
AK8UA-valana is equal to tho sino of the latitudo. At midday 
the north and south line of tho equinoctial and prime vertical 
is the same. Henco at midday there is no aksiia-valana. 

36. For any intervening spot, tho Aksiia-vauna is to be 
found from the sino of tho nata| by proportion. 

First, the degrees of nata aro (nearly) to bo found by 
multiplying the timo from noon by 90 and dividing tlio 
product by the half length of day. 

* [By the distance of any two great circles is here meant an *ro intercepted 
between them, of a great circle through tho polos of which they pass.— It. 

t [Here the kata is the aro of tlio prime vortical intercepted between tlio 
zenith and the secondary circle to it passing through tho place of th« planet.—, 
B.D.] 



190 


Translation of the [VIII. 37. 

37. Then tlio sino of tho nata degrees Multiplied by the 

nine of latitude, find divided by the co-sine of the declination 
of the planet .will be the a'ksha-valana. If the nata be to* 
the, cast, the /ks^a-valana is called north. If west, then it 
is called south (in*tho north terrestrial latitude). P 

The sum and difference of the ayana and aksha-vaeanab 
. must bo taken f6r the spasiita-valana, 

CPASnyi-VAKASA. ... . _ 

» viz. their sum when the ayana and 

Xksita-valanas arc both of the sarao denomination, and their 
difference when of different denominations i. e. one north and 
the other south. 

38. When tlio planet is at either the points of the inter- 
section of the ecliptic and prime vertical, the rpashta-yalana 
found by adding or subtracting the ayana and aksiia-valanas 
(as they happen to bo of the same or different denominations) 
is for that time al its maximum. 

39. But at a point of the ecliptic distant from the point of 
intersection throe signs either forward op backward, there is 
no SPASHTA-VATiANA : for, at those points the north and tlio south 
lines of the two circles are coincident. 

40. However, were you to attempt to shew by the uso of 
tho versed sino, that there was thon no spashta-valana at 
thoso points, you could not succeed. Tho calculation must bo 
worked by the right sine. I repeat this to impress tho rulo 
more strongly on your mind. 

•41. A 5 all the circles of declination meet at tho poles ; it 
’ Another way of refutation is therefore evident that the north i 
of uaing the Tewed sine. and south line perpendicular to the 

cast and west line in tho plane of the equinoctial, will fall in 
the poles. 

42. But all the circlos of celestial latitude meet in the 
pole of the ecliptic-called the kadahba, 24* distant from the 
equinoctial polo. And it is this ecliptic pole which causes and 
makes manifest the valana. 

43. In the ecliptic poles always lies the north and south 



VIII. 49.] SuWianta-ffiromoni. 191 

$ 

line which is perpendicular to the oast and west line in the 
plane of the ecliptic. 1 

To illustrate this, a circle should bo attached to tho sphere, 
taking the equinoctial pole for a centre, and 2 1° for radius. 
This circle is callcll the kadahba-bu hama- v$tta or tho circle 
in which the kadamba revolves (round tho pole). 

The sines in this cfrclo correspond with tho sines of the 
declination. * 

All the secondary circles to tho prime vertical meet in tho 
point of intersection of the meridian and horizon, and this 
point of intersection is called sama i. c. north or south point 
of horizon. 

Now from the planet draw circles on tho sphere so as to 
meet in tlio sama, in the equinoctial pole and also in tho 
ecliptic pole. 

Tho three different kinds of valana will now clearly appear 
between these circles : viz.4ho jtesifA valana is tho distance 
between tho two circles just described passing through the 
sama and equinoctial pole. 

2. Tho ayana- valana is the distance between tho circles 
passing through tho ecliptic and equinoctial poles. 

3. Tho spasiita- valana is tho distance between tho circles 
passing through the sama and kadamba. 

These three valanas aro at tho distance of a quadrant from 
the planet and are tho same in all directions. 

48 and 49. Or (to illustrate tho subject further) making 

Second mode of illastrat- the P lanet afi tho P olci of a sphere; 
ing the Spasiita-yaiaxa. draw a circle at 90* from it : then in 

that circle you will observe the aksha valana — which, in it, 
is the distance of the point intersected by the equinoctial from 
the point cut by the prime vertical. 

The distance of the point cut by tho equiuoctial from* that 
cut by the ecliptic is the Iyana — and the distance between tjiu 
points cut by the ecliptic and prime vertical the jjrAsirf a- 
valana. 



192 


Translation of the [VIII. 50. 

50. In this case the piano of the ecliptic ft always east and 

west — celestial latitude forming its north and south lino. Those 
therefore who (liko s'RfrATi or Lalla) would add the s'aua 
celestial latitude $o hud the valana, labour under a grievous 
delusion. 4 * • 

51. The 1st of Capricorn and tho ecliptic polo reach tliu 
meridian jut the same time (in any latitude) : so also with 
regard to *tho 1st Cancer. Hence at the solstitial points tliero 
is no AYANA-VALANA. 

52. As tho 1st Capricorn revolves in the sphere, so tho 
ecliptic polo revolves in its own small circle (called tho ka- 
DA M RA-ii 1 1 RAMA- V E1TTA rOUIld tllO pole). 

53 and 54. When tho 1st of Aquarius or tho 1st of Pisces 
comes to tho meridian, the distance in tho form of a sine in 
tho KADAikiBA-D h rama- VMTTA, between tlio ecliptic polo and 
tho meridian is the Xyana-valana. This valana corresponds 
with the kbAntijy! or tho sine of declination found from tho 
degrees corresponding to tho time elapsed from the 1st Capri- 
cornus leaving the meridian. 

55. As the vorsed sine is like tho sagitta and the sine is 
tho half chord (therefore the versed sine of the distance of 
the ecliptic pole from the meridian will not express tho proper 
quantity of valana as has been asserted by Lalla. &c. : but 
the right sine of that distance does so precisely). Tho ayana- 
valana will be found from the declination of tho longitude of 
tho Sup. added with three signs or 90°. 

* 56. Those poople who have directed that the vorsed shut* 
of the declination of that point throe signs in advance of tho 
Sun should bo used, have thereby vitiatod tho whole calcula- 
tion. Xksha-valana may be in like manner ascertained and 
illustrated : but it is found by tho ri^lit sine, (and not by the 
versed sine). 

57. He who prescribes rules at variance with former texts 
and does not show the error of their authors is much to be 
blamed. Ucnco 1 am acquitted of blame having thus Hourly 
exposed the errors of my predecessors. 



Sithlhdntas'iromoni. 


193 


V1U. 64.] 

58. The inapplicability of the versed sine may bo further 
Another way of refutation, illustrated as Mows. 'Make Jho eclip* 

of using the Tersed sine. tie pole the centre and draw the circle 

called the jina-v&itta with a radius equal to 24*. 

59. *Then make 'a movoablo secondary circle to the ecliptic 
to revolve on the'two ecliptic poles. This circle will pass over 
tho equinoctial poles, when it comes to the end of the sign of 
Gemini. 

00. By whatever number of dogroes tliis secondary circlo 
is advanced beyond tho end of Gemini, by precisely the same 
number of degrees, it is advanced beyond tho equinoctial polo, 
in this small jina-vjiitta. Tho sine of tlioso dogroes will bo 
there found to correspond exactly with and incroaso as does 
tho sine of tho declination. 

Cl. And this sine is tho Xyan a- valana : This valana is the 
valana at tho end of the dynjya. For the distance between 
the equinoctial pole and planet is, always equal to the arc of 
which the dynjya is tho siuo i. o. the cosine of the declination. 

62. But as tho value of tho result found is required in 
terms of tho radius, it is consequently to be converted into 
those terms. 

As tho jina-v^itta was drawn from tho ecliptic pole as 
centre, with a radius equal to tho greatost declination, so now, 
making tho sama centre draw a circle round it with a radius 
equal to the degrees of the place’s latitude. (This circle is 

Called AKSHA-VfllTTA.) 

C3 and 64. To the two samas or north and south points of 
the horizon os poles, attach a moveable secondary circle to the 
prime vertical. Now, if this moveable circlo be brought over 
the planet, then its distance counted in tho aksha-v$itta or 
small circle from the equinoctial pole will bo exactly oqual to 
that of the planet from tho zenith in the prime vertical! The 

sine of the planet’s zenith distance in tho prime vortical, will v 

when reduced to the value of the radius of aksha-v^itta 
represent tho Iksha-valana. 

• n.. 



194 


Translation of the [VIII. 65. 

65. As in tho Xyana-valana so also in this Xksha-valana, 

the result at thtf end of the dynjyX is found ; this therefore must 
be converted into terms of tho radius. From this illustration 
it is evident tha^ it may bo accurately ascertained from the 
zenith distance ill tho prime vertical. ’ • 

66. I will show now how tho Xksha-valXna may be also 
ascertained from the time from tho • planets being on the 
meridian* in its diurnal circle. [Tho rule is as follows.] Add 
or subtract the s'ankutala [of a given time] to and from tho 

sino of amplitude according as they 

See verse 41, Chap. VII. „ , r „ 6 „ J 

are of tho same or of different deno- 
minations (for tho bahu or bhuja). 

67. The sine of the latitudo of the given place multiplied 
by the sine of the asus of the time from the planet's being on 
the meridian, and divided by the square-root of the differonco 
between the squares of tho bhuja (above found) and of the 
radius, will be exactly the Xk^iia-valana.* 

r * This rule and tho means by which it has been established by BHASXiBJtynA- 
* ya require elucidation. 

BhAsxaba'oiia'bta first directs that the ba'hu or btiuja be found for the 
time of the middle of the eclipse and that a circle parallel to the prime vertical, 
bo drawn haring for its centre a point on the axis of the prime vertical distant 
from the eentre of the prime vertical, by the amount of the ba'hu. From this 

as oentrs and the Hop equal to =^/ rad"— ba'hd* as radios draw a circle paral- 
lel to the prime vertical. This circle called an upaymtta will cut the diurnal 
oirole for tne time on 2 points equally distant from tho meridian. Connect those 
points by a chord. The half of this chord is the Bataohati'ji'A as well in the 
diurnal oirole as in the upavbitta, but as these 2 circles differ in the magnitude, 
, these sines will he the sines of a different number of dogrees in each circle. Now 
the bataohati'jyA is known, but it is in terms of a large circle. Reduce them to 
* their value in the diurnal circle. 

1. If trijyA : natajyA : : dybjya' : sine of diurnal circle. 

Thia aine in diurnal circle is also sine iu rpivarrrA. 

2. If UPA-YfLXTTA-TBlJYA : this sine : : tbjjya equal to akshajya. 

3. pram' ; this result t : THUYA i sine of aKBHA-vaiaba 

now can eel 

and there will remain the rule above stated 
iriTim # akshajya' 

, = sine of amHA-yalaba. 

upaybitta-tkuya' 

' Here onr author makes use of the diurnal circle and upavbitta in tenn of 
the equator and prime vertical, whose portions determine tho yaxaba. The 
smaller birelea being parallel to the larger, the objeet sought is equally attained. 
-L.W. 


« 



VIII. 74.] Sbldhanla-tftromoni. 195 

68. Or the Xk^ha-valana may be thus roughly found. 

Multiply the time from the planet's being bn the. meridian 

and divide the product by the half length of day, tho result 
are the nata degrees. The sine of tliesp nata degreos 
multiplied by thd 'sine of the latitude and* dividod by the 
pynjyA or tho cosine of the declination, will give the rough 

XKSHA-VAIANA. * 

69. Place tho disc of tho Hun at the point at which tho 

„ . ... diurnal cirelo intersects the ecliptic. 

Further illustration. . r 

u he arc of the disc intercepted bo- 

tween these two circles represents tho Ayana-valana in terms 

of radius of tho disc. 

70. This valana is equal to the difference between the 
sino of declination of the centre of the Sun and of the point 
of intersection of the disc and ecliptic ; and it is thus found ; 
multiply the radius of disc by the buogya-khanpa of tho 
bhuja of the Sun's longitude* ancLdivido by 225. 

71. Then multiply this result by sine of 24° and divide by # 
the radius : tho quotiont is the difference of the two Bino of 
declination. This again multiplied by tho radius and divided 
by the radius of Sun's disc will givo the value in terms of tho 
radius (of a great circle). 

72. Now in these proportions the radius of tho Sun's disc 
and also radius are in one case multipliers (being in third 
places), and in the other divisors (being the first terms of the 
proportion) therefore cancel both. There will thon remain* 

»* ulc, multiply the Sun's bhogya khanpa by sino of 24* and * 
divide by 225. 

73. And this quantity is equal to the declination of a point 
of ecliptic 90° in advance of Sun's place. Thus you obsorve 
that the valana is found by the sino of declination as above 
alleged, (and not by tho versed sino). Abandon therefore, 

0 foolish men, your erroneous rules on this subject. # 

74. Tho disc appears declined from the zenith ljke an 
umbrella ; but the declination is direct to the equinoctial pole : 

• »-2 



196 


Translation of the [IX. 1. . 

the proportion of the dynjyX or complement, of declination is 
therefore;required to reduco tlie valana found to its proper ' 
value in tonus of tho radius. 

End of Chapter VIII. In explanation of the cause of 
eclipses of tho Stin and Moon. 


‘ CHAPTER IX. 

Called dqikkar/ma-vasanX on the principles of the Rules fur 
fniling the times of tho rising and setting 
of tho heavenly bodies . 

.1. A planet is not found on the horizon at tho time at 
Object of the correction which its corresponding point in th o 
ecliptic (or that point of tho eclipt ic 

plied to the piece of the having the sAme longitude) reachos 

planet, for finding the point J 

of the ecliptic on the horizon the horizon, inasmuch as it is elevated 

' wllen tho planet roaohw it§ above or depressed below the horizon, 
by the operation of its latitude. A correction called drik- 
kahAma to find the exact time of rising and sotting of a planet, 
is therefore nocessary. 

2. When the planet's corresponding point in the ecliptic 
reaches the horizon, tho latitude then does not coincide with 
the horizon, but with the circle of latitude. Tho elevation of 
the latitude abovo and depression of it below the horizon, is of 

• two sorts, [ono of which is caused by the obliquity of the 

* ecliptic and the other by tho latitude of the place.] HencS* 
the d$ikkab^ma is two-fold, i. e. the Ayana and tho akshaja or 
Xksha. The detail and mode of performing these two sorts 
of the correction are now clearly unfoldod. 

3. When the two valanas arp north and the planet's 

D jgmuA. corresponding point in the ecliptic is 

. in the eastern horizon, the planet is 

thereby depressed below the horizon by sonth latitude, and 
elevated when the planet's latitude is north. 



• IX. 9.] Siddhdtitas'iromoni. ■ 197 

» 

4. When the tVo kinds of yalana are south, then the 
' reverse of this takes plane ; the reverse of this also takes placo 

when the planet’s corro sponding point is in tho western hori- 
zon. 

[And the difference in the times of rising of the planet and 
its corresponding point is called the resultant time of tho 
d?ikkarma and is foundry the following proportions.] . 

Tf radius : Xyana-valana : : what will celestial latitude 
give? 

5. And ^ 

if cosine of tho latitude of tho given > : Xksha-valana . 
placo ) 

: : what will spashja s'ara givo ? 

Multiply tho two results thus found by theso two propor- 
tions, by the radius and divide the products by tho dycjyX or 
cosino of doclination. 

6 and 7. Take tho arcs § of theso two results (which are 
sines) and by tho asus- found from the sum of or tho difference 
between theso two arcs, tho planet is depressed below or 
elevated above tho horizon. The lagna or horoscope found 
by the direct process (as shown in tho note on tho vorso 
20, Chapter VII.) when tho planet is depressed and by tho 
indirect process (as shown in tho same note)* when it is 
elevated, by means of the asus above found, is its udaya 
lagna rising horoscope or tho point of tho ecliptic which 
comes to tho eastern horizon at the same time with the 
v planet. 

When tho planet’s corresponding point is in tho western 
horizon, tho lagna horoscope found then by the rule converse 
of that abovo given, by means of the place of the planet added 
with 6 signs, is its asta lagna setting horoscope or the point 
of tho ecliptic which is on the eastern horizon when the pla&et 
comes to the western horizon: • 

8 and 9. For the fixed stars whose latitudes are very 
considerable the resulted time of the d&ikkabma is found in a 



198 Translation of the [IX. 9. . 

« 

difforont way. Find the ascensional difference from the mean 
declination of the star, i. e. from the declination of its corre- ' 
sponding paint in the ecliptic, and also from that applied with 
the latitude, i. g. from the true declination. The asus found 
from the sum of or tho difference between tho ascensional 
differences just found, as the mean and true declination? are of 
tho different or of tho samo denominations respectively, are tlio 
asus of* depression or elevation depending on tho aksha 
D$ iKKARMA. (Find also tho time depending on tlio Ayana- 
djukkakma) : and from the sum of or the difference between 
them, as they may be of tho samo or different denominations, 
the udaya lagna or asta lagna may bo ascertained as above 
found (in the Gth and 7th versos).* 


* Lot A D B C be tho meridian ; CED the horizon, A tho zenith ; E tlio 
oaet point of the ly>rizon ; F E G the equinoctial ; K the north polo ; L tho 
south ; F tho planet ; p its corresponding point in tho ecliptic ; IT P p J tho 
secondary to tho ecliptic passing through the p’anet P, and lienee p P tho 
latitude. Lot f P g the diurnal circle passing through tho planet P and hence 
p R the notified latitude. 

* Nov, when the corresponding place of tho planet is in tho horizon, it is then 
evident from the accompanying figure, that tho planet is elevated above or 
depnased below the horizon by its latitude p P and as it is very difficult to find 
the elevation or depression at once, it is therefore ascertained by means of its 
two parts, the one of which is from the horizon to tho circle of declination, i. e. 
Q to R. This partial elevation or depression takes place by the planet’s rectified 
latitude pR. And the other part of the elevation or depression is from the 
oirole of declination to the circle of latitude ; i. e. from R to P and this occurs 
by the planet's mean latitude p P. From the sum or differonco«of these two 
parts, the exact elevation of tho planet above the horizon or the depression 
below it, can be determined. When the terrestrial latitude, of tho given place 
ia north and the planet’s corresponding place in the ecliptic is in the eastern 
horizon, the a'ksha-vaiana is then north and tlio circle of declination is 

’ elevated above the horizon to the north. For this reason, when the a'ksha- 
Yalana is north, tho planet will be elevated above the eastern horizon if its 
latitude be north, and if it be south, tho planet will bo depressed below fnar 
horizon. But tho reverse of this takes place whon tho a'ksha-yalana is south 
whiah occurs on account of the south latitude of the given place, i. e. when the 
a'kshA»YaIi49a is south, the circle of declination is depressed below (lie horizon 
to the north and henee the planet is depressed below it, if its latitude be north, 
and if it be south, the planot is elevfted above the horizon. 

Again, when tho planet's longitude terminates in the six ascending signs, it is 
evident that the Atana-valana becomes then north, and the north pole of the 
ecliptic is elevated above the circle of deolination passing through the planot. 
Henoe, when the a'yaka-yalaka is north, the planet is elevated above or 
depressed below the circle of declination by its mean latitude, as it is north or 
south. But the reverse of this takes place, when the a'yawa-yaiaka is south, 
i. e. the planet is depressed below or elevated above the circle of deolination, 
as its latitude is north or south. Because when the a'yava-yaiava is south 



Siddhaiitas'iromoni. 


109 


IX. 9.] 


A/ 



the north polo of the ecliptic lies below the circle of declination and the aouth 
above it. 

Again, when the planet is in the weatem horizon, the circle of declination 
passing through the place of the planet in the ecliptic lies to the north above 
tho horizon, but the ajcbiia-valana, becomes south and hence the reverse takes 
}. M oe of what is said about the elevation or depression when the planet is in the 
eastern horizon. But as to the Xtaka-talaya, it becomes north when the 
longitude of the planet terminates in the ascending six signs and the north pole 
of the ecliptic lies below the circle of deolination. Hence the depression of the 
planet takes*place when its latitude is north and the elevation when the latitude 
is south. But when the longitude of the planet terminates in the discending 
six signs, the Ayawa>7alana becomes then south and the north pole of the 
ecliptic lies above the circle of declination. For this reason, the elevation of the 
planet takes place when its latitude is north, and the depression when it is 
south. Thus in the western horizon the elevations and depressions of the planet 
are opposite to those when the planet is in the eastern horizon. • 

Now, the time elapsed from the plafaet's rising wlien it is elevated above the 
horizon and the time whieh the planet will take to rise when it is depressed 
below the horizon, are found in the following manner. 



200 Translation of tho [IX. 10. , 

« 

10. Tho [Aspash^a] S'aba or true latitude [of the planet] 

To flnd*tl>o nlue of «• mnlti P K ed by the 'Dtotta or cosiao . 
lathi htitode^in term. o( s 0 f declination of the point of the eclfn- 

circle of declination, to ren- . , r r 

dfr it fit to be added to or tic, three signs in advance of the 
•abtraoted from doclipation. ^^g ^ tad & 


See the figure above described in which the apgle Q E B or the equinoctial 
are Q' jf denotes the time of elevation of the planet from Q to R, and tho time 
of elevation of the planet from E to P is denoted either by the angle P X H 
or by the equinoctial arc P / p'. Out of these two times Q' p* and P' jp , we show 
at first how to find F' p\ 

In the triangle P p R, P p = the latitude of the planet, £?p R = the 
i'yava-VALAVA and < P R p sa -J, and 
,\ R : ain Vp R *= sin 2p : sin R P ; 
or if radius 

t sin of a'xasta-yalaita 
= the sine of latitude 
: ain R P. 

Again, by the similar triangles E P R and E P' »' 
sin E P : sin R P «= sin E P* : sin P' p\ 
here, sin K P = cosine of declination and KF = B, 

* R X sin R P 

sin P # p' = 

cos of declination 

Row, the time p* Q' is found as follows. 

* In the triangle p R Q, p R = the spabhta-b'aba which can be found by tho 
rale given in the V. 10 of this* chapter, ^ B pQ- akbba-vaiava and 

R Q p — co-latitude of place nearly 

and sin p Q R : sin R p. Q s s sin p R i sin R Q 

or, if cosine of latitude, 

: sine of AXIHA-YALAKA, 

= BPABHf A-B*AKA 
: sin R Q | 

again, by the triangles E Q R, E Q' p\ 
sin E Q : sin Q R ■= sin K Q' : sin p* Q' j 
here, sin K Q => cosine of deolination and sine E Q' = R, 

R X sin Q R 

• sin p' Q' = 

cos of declination. 

If both of these times thus found, be of the elevation or both of the deprex'TL 
sion, the planet will be elevated above or depressed below the horixon in the 
time equal to their sum, and if one of these be that which the planet takes for 
its elevation and the other for it* depression, the planet will be elevated above 
or depressed below the horixon in the time equal to their diffttenoe as the 
re matnder ia of the time of elevation or of that of the depression. The sum or 
diflbrenoo of the two times just found is called the resulted time of the D*nc- * 
kasha in the S’iddb aviab. t 

That point of the eeliptio whieh ia on the eastern horiion when the planet 
reaches it, ia called the udaya laqka rising horoscope of the planet. As it is 
•neoesaary to know this toaya laoka for finding the time of the planet’s 
rising, we are now going to show how* to find the rising horoscope. If the 
planet Is depressed by the resulted time above mentioned, it is evident that 
when the planet will come to the eastern horiion, its corresponding place in the 



SidilhAnta-siromoni. 


201 


IX. n.] 

vided by the radiul becomes [nearly] the spashta or rectified 
latitude, [i. e. the arc of the circle of declination intercepted 
between the planot’s corresponding point in the, ecliptic and 
the diurnal circle passing through the planet] . This rectified 
latitude is used wh£n it is to be applied to tho*mcan declination 
and also in the Iksha d$ikkarma.* 

11. The celestial latitude is not reduced by Brahmagupta 


ecliptic will be elovated above it by the resnlted time. For this reason, having 
assumed the corresponding place of the planet for the Sun, find the horoscope 
by the direot process through the resulted time and this will be the rising 
horoscope. But if the plnnet ho elevated above the horizon by the resulted time, 
its corresponding place will then be depressed below it by the same time when 
the planet will come to it. Therefore, tho horoscope found by the indirect 
process through the resulted time ; will be the rising horoscope of the planet. 

That point of the ecliptic which is on the eastern horizon when the planet 
cornea to the western horizon, is called tho asta lagna or setting horoscope of 
the planet. As it » requisite to know the selling horoscope for finding the 
time of setting of the planet, we therefore now show the way for finding the 
setting horoscope. It the planot be depressed below the western horizon by the 
resulted time, it is plane that when the planet will reaches it, its corresponding 
place will be elevated above it by the resulted time and consequently the 
corresponding place of tho planet added wfth six signs will be depressod below 
the eastern horizon by tho .same time. Therefore, assume tho corresponding 
place of the planet added with six signs fqr the Sun and find tho horoscope by * 
the indirect process, through tho resulted time and this will bo the asta lamia 
setting horoscope. But if the planet be depressed below the western horizon, its 
corresponding place added with six signs will then bo elevated above the eastern 
horizon by the resulted time and hence the horoscope found by the direct process 
will then be the asta lagna setting horoscope. 

Now the time p' Q' which is determined above through the triangle p R Q, is 
not the exact one, because, in that triangle the angle p Q R is assumed equal to 
the co-latitude of the given place, but it cannot be exactly equal to that, and 
consequently the time p l Q' thus determined cannot be the exact time. But no 
considerable error is caused in the time p f Q' thus found, if the latitude be of a 
planet, aa it is" always small. As to the star whose latitude is considerable, tho 
time »' Q' thus found cannot be the exact time. The exact time can be found . 
aa follows. 

* Seo the preceding figure and in that take R for a star and p the intersecting » 
point of the ecliptic, and the circle of declination passing through the star R then 
pp* is called the mean declination of the star, R p, the rectified latitude and Up 
the rectified declination. 

Now, find the ascensional difference E p' through the mean declination p p* 
and the ascensional difference E Q' through the rectified declination R p' or 
Q Q'. Find the difference between these two ascensional differences and this 
difference will be equal to p ’ Q' i. e. E Q'— E p' = p' Q\ But it occurs then 
when; and R are in the same side of the equinoctial F G and when /is in,o»o 
side and R in the other of the equinoctial, it is evident that p' Q' tu this case 
will be equal to the sum of the two ascensional differences.— B. D.] • 

* This rule is admitted by BrJ^kakAouAkta to be incorreot; but the 
error being small, ia neglected. Instead of using the sir Jf A, the Yaaupi should 
liave been adopted. 


* 9 . 



Translation of the 


[K. 12. 


Ominion of u. l„t “Mother gastronomers to itsvalne 
mentioned correction or re- in declination : and the reason of this • 

duct ion of Celestial latitude „ . , , 

to it. value in- declination, 0mi8S,0n > aeems **> naVO been its 

o&iers BlUI,IIAOUPI * 8nJ 8ma ^ ness of amount. And also it is 
• tli <3 unconnected ‘latitude which is 

used in finding the half duration of the eclipses and in their 
projections &c. * 

12. As the constellations are fixed^ their latitudes as given 
in the books of these curly astronomers are the bpasuat- 
b'aras, i. o. tho reduced values of the latitudes so as to render 
them fit to be added to or subtracted from the declination ; 
and tho dhkuvas or longitude of these constellations are given, 
after being correctedYy tho ayana drtkkakma so as to suit 
those corrected latitudes that is, the star will appear to rise at 
the equator at the same time with longitude found by the 
correction. * 


Let a d be equinoctial aud P the equinoctial pole, 
m d b — Ecliptic 

b s — Celestial latitude, < 

i 0s Celestial latitude reduced to it. value in 
declination is KOfi, 

SC — bhuja being arc of diurnal circle e s g 
s c — Jcb portion of diurnal circle of tlio planet’, 
longitude at h. 

The triangle s c b or * it & is assumed to bo a dig- 

VALANAJA TliYASHA. 

The angle she — ir ava-valava or the angle of 
the inclination of s b which 
goes to ecliptic pole with b c 
which goes to equinoctial 
• pole. 

Hence this triangle she is called big-yalawaja 
TBT 8BA, the angle she varying with tho Aiana-vaiaita. If b were at the t 
Cancer, tlien the north line abo which goo. to the polo would go also to tho 
eoliptio polo. 

Ilence tlu» ABPASiif A ba'ua, and bpa.hta b'aba of a star of 90° of latitude 
being both represented by b e would be the same. To the longitude of a star 
being 270*, its aspasiita ami spasiita sAka would be the same.— L. W. 

[The rule stated in this verse is founded upon the following principle. « 

Assuming the triangle tic kb a plane rieht-angled triangle and the angle 
s b e, a. the dcoliuation of the point of the ecliptic three signs in advance of the 
planet’s corresponding place, because this declination is nearly equal to the 
Jyaxa-vaiaya, we have, 

siu • e b ; cot $ b c sss b s e j 

pr B : wain or nearly the cosine of the declination of the planet's place 9*0 4* 
ss Celestial latitude : rectified latitude. —11. D.] 




IX. 18.] Suhlhdnta-s'iromoni. 203 

13. Those astronomers, whoihavo mentioned that celostial 

_ . . . latitude is an arc of *a circlo of de- 

• Bea'bkaba'chabta ex- * Ul utJ 

pones the incorrect theory clination, are stupid. \Yero the ce- 

lestinl latitude nothing more than an 

tice whi^h is irrecoucillble arc 0 f a circle of declination, then why 
with their own theory. ’ J 

should they or others bavo evor had 

recourse to the ayana* d:&ikkaiima at all? (The planets or 
stars would appear on tho six o’clock line at tho time, that the 
corresponding degree of the ecliptic appeared there.) 

14. How moreover liavo these samo astronomers in deline- 
ating an eclipso marked off the Moon’s latitude in the tuiddlo 
of the eclipso on s pa s u ja- v a r, an a- ku tra or on tho lino denoting 
the secondary circlo to the ecliptic ? and how also have they 
drawn perpendicularly on the valana-sutka or tho lino repre- 
senting tho ecliptic, tlier latitudes of the Moon at tho com- 
mencement and termination of tlio eclipse. 

15. How moreover, have they made the latitiido koji, i. o. 
perpendicular to the ecliptic and thus found tho half duration 
3 f tho eclipse ? If tho latitude Were of this nature, it wouhf 
never t)e ascertained by tho proportion (which is usod in 
Finding it). 

16. A certain astronomer has (first) erroneously stated tho 
Censure of the iwtrono- D&IKKAKMA and VAT, ANA by the versed 

rjSETSfJSS This coarse lias been followed 

eabua and valana. by others who followed him like blind 

men following each other in succession: [without seeing 
dieir way] . 

1 7. Bhahmaoupta’s rule, however, is wholly unexceptionable, 

but it has boon misinterpreted by his 
Prtiie of Bbahxagupta. . 

followers. My observations cannot bo 

said to be presumptuous, but if they are alleged to bo so, 

[ have only to request able mathematicians to weigh them 

jrith candour. # 

18 . The d$ikkarva and Valana found by tho former astro- 



204 Translation of the [IX, 19. • 

nomers through the versed sine are erroneous : And I shall now 
give an ipstancft in proof of their error. • 

19 and 20, In any place having latitude less than 24° |}. 

An instance in proof of multiply the sine of the latitude of the 
the error. • pl ac0 by the radiks and divide the 

product by the sine of 24° or the sino of the obliquity of the 
ecliptic qnd tako the arc in degrees of the result found. And 
find the* point of the ecliptic, the degrees just found in 
advance of the 1st Aries. Now, if from this point the planet’s 
corresponding point oil the ecliptic three signs backwards or 
forwards, bo on the western or eastern horizon respectively, 
then the ecliptic will coincido with the vertical circle, and the 
horizon will consequently be secondary to the ecliptic. Hence 
the planet will not quit the horizon, though it be at a distance, 
of extreme latitude from its corresponding point in the ecliptic 
[which is on the horizon], as tho celestial latitude is perpen- 
dicular to the ecliptic.* . 

21. In this case the resulted times of the d^lik karma being 
of exactly the samo amount but one being plus and the other 
minus, neutralize each other [and hence there is no correc- 
tion] . Now this result would not be obtained by nsing the 
versed sine— honce let the right sine (as proscribed) be always 
used for tho d^ikkaema. 


* [It is evident that the longitude of this point is equal to the ara through 
which it is found, and as the point, of the ecliptic 8 signs backwards or forwards 
'from this point is assumed on the horizon, this point therefore will at that tins 
, be the nonagesimal, and as the longitude of that point or nonageaimal i» lee* « 
than 9C* the declination of this point will be north. This declination equals to . 
the latitude in question. For 

R X sin latitude 

v The sine of the latitude of the point = — ■ ■ - ■ (by the gs* 

■in 24® 

sumption) 

sin 34° X sin longitude of the point 

sin latitude a ■ ■ , but this » sin de- 

Radius 

. dilution. 

V. The deoliuation of that point or nonagesimal equal to the latitude of the 
plaoe. Agd henee, if the latitude be north the nonagesimal will be in the 
' senith. For this reason the ecliptic will ooincido with the vertical circle.— B, D.] 

c 



Siddh&n ta^iromoni. 


205 


IX. 26.J 

22. Again herq* in like manner, it is from the two valanas 
having different denominations, but equal values, that they 
mutually destroy each.othor. By using the versed sine, they 
would not have equal amounts, hence the valanas must bo 
found iJy the right line. 

[In 'illustration of the fact that the valana does not corre- 
spond with the versed sfno, but the right sine Bha'skabXchXrya 
gives as an example.] 

23. When the Sun comes to the zenith [of the place where 
the latitude is less than 24°] , and consequently the ecliptic 
coincides with tho vertical circle, the spashta valana then 
evidently appears to be equal to the sine of tho amplitude of 
the ecliptic point 90° in advance of tho Sun's place in tho 
horizon. If you, my friend, expert in spherics, can make tho 
spashta valana equal to the sine of amplitude by means of tho 
vorsed sine, then I will Hold tho valana found in tho DnfvjjtiD- 
dhida tantra by Lalla and in thoi other works to bo correct. 

[To this BhIskaraciiXrya adds a further most important 
and curious illustration :] 

24. In tho place where the latitude is 66° N. when tho 
Sun at tho time of his rising is in 1st Aries, 1st Taurus, 1st 
Pisces, or in 1st Aquarius, he will then be eclipsed in his 
southern limb, because tho ecliptic then coincides with tho 
horizon. Therefore, tell mo how the spashta valana will be 
equal to the radius by means of tho versed sine ! 

[In the same manner the d^ikkakma calculation as it 
depends on the valana, must be made by tho right sine and 
not by the versed sine and for the same reasons.] 

25. Even clever men are frequently led astray by conceit 

o t error in in their own quick intelligence, by 

and others, stated. their too hasty zeal and anxiety for 

distinction, by their confidence in others and by their oWn 
negligence or inadvertence, when it is thus with the wise,* 
what need I say of fool ? others*, however, have said : — . 

26. Those given to the service of courtezans and bad poets. 



20C 


Translation of the [X. 1, « 

are both distinguished by thoir disregard of*the criticisms and 
reflectiorfs of the world, by their broach of the rules of time * 
and metres, ‘and their destruction of their substance and of 
thoir subject, being beguiled by the vain delight they feel 
towards the object of their taste. 

End of Chapter IX. called Drikkakma-vasan*. 


CHAPTER X. 

Called S'htngonnati-vXsana in explanation of the can sc of the 
Phases of the Moon. 

• 

1. This ball of nectar the Moon being in contact with rays 
of the Sun, is always illuminated’ by her shinings on that sido 
turned towards tho Sun. The side opposite to the Sun dark 
as the raven black locks of a young damsel, is obscured by 
being in its own shadow, just as that half of a water-pot which 
is turned from the Sun, is obscured by its own shadow. 

2. At the conjunction, the Moon is between us and tho 
Sun : and its lower half which is then visible to the inhabit- 
ants of the earth, being turned from the Sun is obscured in 
darkness. 

That half again of the Moon when it has moved to tho 
distance of six signs from the Sun, appears to us at the pentad 
of full Moon brilliant with light. 

3. Draw a line from the earth to the Sun’s orbit at a 
distance of 90° from the Moon, and find also a point in the # 
Sun's prbit (in tho direction wheye the Moon is) at a dis- 
tance equal to that of the Moon from the earth. When the 
•Sun reaches jbhe point just found, he comes in the line per- 
pendicular at the Moon to that drawn from the earth to the 
Moon. Then the Sun illumines half of the visible side of the 

t * 



X. 5.] Sitldhdnta-siromoni. 2( 7 

9 

Moon. That is When the Moon is 85° . . 45* from the Sun 
cast or west, it wi\l appear half full to us.* * . 

* 4. The illuminated portion of the Moon gradually increases 
as it recedes from the Sun : and the dark portjon increases /is 
it apprtmches.tho Sim. As this sea-born globe of water (tho 
Moon) is a- sphere,, its horns assume a pointed or cusped 
appearance (varying in acuteness according to its .distanco 
from the Sun). • 

5. (To illustrate tho subject, a diagram should bo drawn 
Diagram for illustrating as follows). Let the distance north aud 
the subject. south between tho Sun and Moon re- 

present the BHiTJA, the upright distanco between them tho 
koti and the line joining their centres the hypothenuso. Tho 
Sun is in the origin of the biiuja which stretches in the direc- 
tion where tho Moon is, tin) line perpendicular at tho end of 
the bhuja is Kofi at the extremity of which is tho Moon and 
tho lino stretching (from tho.Moim) hi tlio direction of tho Sun 
is tho hypothenuse. The Sun gives light (to tho Moou) through ^ 
the direction of tho hypothenuso. ’ 



* This is Hum illustrated. Let a repre- 
sent tho IGarth, 4 c d tho orbit of the Sun, 
e /= do. of the Moon. Then it ia ob- 
vious that half of the aide of the Moon 
visible to us will be illuminated when 
the Sun ia at o and not at d, when the ’ 
Sun ia at d it will illumine more than 
half of tho Moon'a dice ; 4 e is leas than n 
quadrant by the arre d, the sine of which 
a e or « g in terms of the radius of the 
Sun's orbit, equals to the Moon's dis- 
tance from the earth. L W. 

[The arc 4 e can he found ns follows 
In the triangle ate right angled at e, 
at — 51560 yojanah, ar =j 689377 
It yojasas ancording to the Hiddha'jsuas. 


a e 

Then, coa a a c = = 

a e 

arc b c = 85’ 


51566 

: . 0748 

689377 
,. 45' nearly. 


s con 85® . . 43' 


B. D.] 




Tribulation of the 


PL 6. . 


[For instance 



Let S be the Sun and m the Moon, then nS = bhijja, 
a m = KOji, m S = hypothenuse. Then f.g a line drawn at 
right angles to extremity of hypotenuse will represent lino of 
t direction of the enlightened horns and tbo angle h m d opposito 
to bhuja will bo equal to g m c = the amount of angle by 
which the northern cusp is elevated and southern depressed, — 
wore the Moon at k, there would be no elevation of either cusp 
either way. For the hypothenuse will also bisect the white part 
of the Moon. If the Sun is north of the Moon, thq north cusp 
of the Moon is elevated : if south the southern cusp. L. W.] 

[Mr. Wilkinson has extracted the following two verses from 
1 the GinitadhyAya. 

I. When the latitudo is 66° N. and the Sun is rising in l*b 
Aries, thon the ecliptic will coincide with the horizon ; now 
suppose the Moon to bo in 1st Capricorn, then it will appear 
to be bisected by the meridian and the eastern half will be 
enlightened. 

But according to Brahmagupta (his would not occur, for he 
Jias declared that the koji will be equal to radius in this case 
whereas it is obviously “nil,”* and it is the bhuja which is 
equal to radius when there is no north and south difference 


• XI. 2.] Siddh&nta-8ir<ynoni. 209 

0 

between the Sun qnd Moon then tho koji would be equal to 

* the hypothenuse or radius and the bhuja woulJ bo “ nil.” 

II.* And the Moon's horns are of equal altitude when there 
is no bhuja, whilst they become perpendicular when there .is 
no koju That tho kofi and bhuja shall at one and tho same 
timol/e equal to radius is an obvious incompatibility. But 
what business have I with dwelling on the exposure of these 
errors? Brahmagupta has hero shown wisdom indeed, and 
I offer him my reverent submission !] 

6. I have thus only briefly treated of the principles of tho 
subjects mentioned in the Chapters on Madhyagati &c. fearing 
to lengthen my work; but tho talented astronomer should 
understand tho principles of all the subjects in completion, 
because tins is the result to be obtaiued by a complete know- 
ledge of tho spheric. 

End of Chapter X. called S'uingonnati-vasanX. 


CHAPTER XI. 

Called Yantradhyaya, on the use of astronomical instruments. 

i 

1. As minute portions of time elapsed from sun-rise cannot 

bo ascertained without instruments, t 
Object of the Chapter. ^all therefore briefly detail a few. 

instruments which aro of established use for this purpose. 

2. The Armillary sphere, nadI-valaya (the equinoctial), 
the YASHfi or staff, the gnomon, the oiiati or clepsydra, tho 
circle, the semi-circle, the quadrant, and tho piialaka : but of 
all instruments, it is “ ingenuity” which is the best. 

• 

* BHlSKAtzXcHJtBYA is here very severe on Bbabhagupta who of allhis'pro* 
dccessore is evidently hie favorite, but truth seemed to require this condemn- 
ation. He at the same here docs justice to Auya-buatta and tho author of tffe 
8 6 kya-81 ddh a'm ta. They both justly concur in saying there is no koti in this 
case.— L. W. 


r 



210 


Translation of the 


[XI. 3. 


Un of Armillary Sphoro. 


3 and 4. (Tliia instrument is to be mode os before de- 
scribed, placing tie Buaoola starry • 
sphere, which consists of tjhc ecliptic, 
diurnal circles, $ie Moon’s path, and the circles of declination 
&c. within the kUagola celestial sphere, which consist^ of tho 
horizon, meridian, prime vortical, six o’clock lino, and* other • 
circles ykich remain fixed in a given latitude). Bring tho 
place of the Sun on the ecliptic to the eastern horizon : and 
mark tho point of the equinoctial (in the uiiaoola) intersected # 
by tho horizon, viz. cast point. Having mado tho horizon as 
level os water, turn the uiiaoola westward till the Sun throws 
its shadow on the centre of tho Barth, Tho distance between 
the mark made oh tho equinoctial and tho now eastern point 
of the horizon will represent tho time from sun-rise. 

5 and G. The laona or horoscopo will then bo found in 
that point of the ecliptic which is cut by the horizon. 

Take a wooden circle and divide its outer rim into (50 hha- 
TIkXs : Then place tho twelve signs 
of tho ecliptic on both sides, but 
instead of making each sign of equal extent, they must bo 
mado each with such variablo arcs as shall correspond with 
their periods of rising in tho placo of observation (tho twelve 
periods are to be thus marked on cithor side, which are to bo 
again each subdivided into two horas (or hours), three PRKSir- 
ka^as, into xavATJs'ab or ninths of 3* . . 20' each, twelfths 
■of2 8 . . 10' and into trtns'ans'Xb or thirtieths. These are 
called tho shapvaroa or six classes). These signs, howovar, 
must bo inscribed in tho inverse order of the signs, that is 
1st Aries, then Taurus to the west or right of Aries and so on. 
Then placo this circlo on tho polar axis of the khagola at tho 
centre of tho Earth (tho polar axis should be elevated to tho 
hoight of tho polo). 

, Now find tho Sun’s longitude in signs, degrees, &c. for tho 
sun-ri^o of tho given day (by calculation) and find the same 
degreo in the circlo. Mark there tho Sun’s place, turn tho 


Tho Na'di-valaya. 



Siiblhanta-sit'ornoni 


211 


XT. 8.] 

circle round the aria, so that the shadow of the axis will fall on 
the mark of tho Sftn’s place at sun-riso and then fix tho circle. 
Now as tho Sun rises, tho shadow of the axis will advance from 
the mark made for tho point of sun-riso to tho nadir and will 
indicate tho hour fsom sun-rise, and also tlic laona (horos- 
cope) ; the number of hours will be seen between tho point 
of sun-riso and the sliadpw: anil the laona will be found on tho 
shadow itself. [While the Sun goes from east to west tho 



shadow travels from west to east and hence tho signs witli 
their periods of rising must bo reversed in order — tho aro 
from W to Laona represents tho hour arc : and the Laona is 
at tho word Laona in tho accompanying figure. — L. W 

7. Or, if this circle marked as above, be placed on any axis 
elevated to tho altitude of the pole, then tho distance from the 
shadow of tho axis to the lowest part of tho circle will repre- 
sent tho time to or from midday. 

8. A QHAfi made of copper like the lower half of a water- 

pot, should have a largo hole bored, in 

The OHiT< or dcpiydra. ^ See how ofton it is filled 

and falls to the bottom of the pail of water on which it is 
placed. Divide 60 ohatis of day and night by tho Quotient 



212 


Translation of the [XI. 9 . 

and it will give the measure of the clepsydra. (If it is filled 
60 times, then the ohati will ho of one ghaiika ; if 24 times 
it will be' of one hour or 2 £ ghattkas.) • 

0 . For a' gnomon take a cylindrical piece of ivory, and let 

\ it bo turned on a t lathe, taking care 

Gnomon* . 0 

that the circumference bo equalabovo 

and below. From its shadow may be r ascertained the points 

of the compass, the place of observer, including latitudo &c. 

and times (as has been elsewhere explained). 

10. The circle should be marked with 360° on its outer 

circumference, and should be sus- 

The chaxba ur circle. _ . , 

pended by a string or chain moveablo 

on the circumference. The horizon or Earth is supposed to 

be at tho distance of three signs or 00 ° from the point at which 

it is suspended : the point opposite to that point being the 

zenith. • 

11. Through its centre put a thin axis: and placing tho 
circle in a vertical plane, so as to catch the shadow of tho Sun : 
'the degrees passed over by the axis from tlio place deno- 
minated the Earth, will bo altitude : 

12. And the arc to tho point denominated the zenith, will 
be that of the zenith distance. 

Some former astronomers have given the following rule for 
making a rough calculation, of the time, viz. multiply the half 
length of day by the obtained altitude and divide the product 
.by the meridian altitude, the quotient will be tho time sought. 

13. First let the circle be so held or fixed that any t^o 

To find the longitudes of of the following fixed stars appear to 

plineti by the circle. touch the circumference, viz. Ma- 

ohA (a Loonis, Regulus), Pdshya (8 Cancri), RevatI (£ Piscium) 
and SatatIrakI (or A Aquarii). [These stare are on the * 
ecliptic und having no latitude, are to be preferred.] Or, that 
any star (out of the ChitrA or a Virginia Spica &c.) having 
very inconsiderable latitude, and* the planet whose longitude is 
required and which is at a considerable distance from the star, 
appear to touch the circumference* 



XL 17.] Siddhanta-siromoni . ' 213 

14 and 15. Then look from tho bottom of tho circle along 
( its plane, so that the planet appear opposite the axis ; and still 
holding it on tho piano of the ecliptic, observe also any of tho 
above mentioned stars. The observod distance between tho 
planet and the star, Jf added to tho star's longitude, when tho 
star is. west, and subtracted when cast of tho plnuefcf will givo 
tho planet's longitude. 

The half of a circle is called a chapa 

Semi-circlo and quadrant. ... m , . * 

or semicircle. The half of a semi- 
circle is called TUitfYA or a quadrant. 

1(5. As others have not ascertained happily tho apparent 
time by observations of altitudes in 

Pn ALAK A- YAITTBA. , 

a vertical circle, I have therefore 
laboured myself in devising an instrument called phalaka 
yantra, the uses of which 1 now proceed to explain perspi- 
cuously. It contains in itself the essence of all •our calculations 
which are founded on tho true principles of tho Doctrine of 
the Sphere. 

17. I Bhaskara now proceed, to describe this excellent* 
instrument, which is calculated to 

Addresses to the Sun. 

remove always tho darkness of igno- 
rance, which is moreover tho delight of clever astronomers 
and is founded on tho shadow of its axis : it is also eminently 
serviceable in ascertaining tho time, and in illustrating trntlis 
of astronomy, and therefore valued by tho professors of that 
science. It is distinguished by having a circle in its centre. , 
1 proceed to describe this instrument after invoking that , 
bright God of day, the Sun, which is distinguished by the 
epithets 1 have above given to the instrument viz. ho is 
eternal and removes obscurity and cold : he makes tho lotus 
to flower and is ever shining : ho easily points out tho time 
of the day and season and year, and makes tho planets and 
stars to shine. He is worthy of worship from tho virtuous and 
resides in the centre of his orb* 

• This Terse is another instance of the double entendre, in which eren the 



214 Trnnxlation of tb [XI. 18. 

18. Lot" a clover astronomer mako a mAlAka or board of a 
piano rectangular and quadrilateral form, tlfo height being 00 , 
digits, and tho breadth 180 digits. Let him halve its breadfh 
and at tho point thus found, attach a moveable chain by which 
to hold it : from, that point of suspension lot him draw a per- 
pendicular which is called tho lamba-rekpjX. • 

19. Let him divide this perpendicular into 90 equal parts 
which will bo also digits, and through them draw lines parallel 
to tho top and bottom to tho edges : theso are called sines. 

20. At that point of tho perpendicular intersected by the 
30th sine at tho 30th digit, a small hole is to be bored, and 
in it is to bo placed a pin of any length which is to be con- 
sidered as tho axis. 

21. From this liolo ns centre draw a circle (with a radius 
of 30 digits : tho circle will then cut tho 00th sine), 00 digits 
forming tho dialncter. Now mark the circumference of this 
circlo with 00 ghatis and 300 degrees, each degree being 
subdivided into 10 palah. 

, 22. Lot a thin pattika or index arm with a hole at one 
end ho mado of tho length of GO digits and let it bo so marked. 
[The breadth of tho ond whore tho hole is bored should be of 
ono digit whilst the breadth of the whole patxtkX be of half 
digit. Lot tho patjikA bo so suspended by tho pin above 
mentioned, that ono sido may coincide with 
tho lamba-rkkhL Tho accompanying ligure -<*■-— — — 

. will represent tho form of the vajjikL 

, Tho rough ascensional difference in pa las determined by flio 
khanpakab or parts, being divided by 19, will hero become the 
sino of the ascensional difference (adapted to this instru- 
ment.*) 

• 

best anthers occasionally indulge. All tbe tpithets giren to the instrument 
apply in tlio original also to tho Sun. This kind of double meaning of course 
does not admit of translation.— L. W. 

' • The sines or ascensional difference for each sign of the ecliptic were found 
by the following proportions. 



Sidtlhunta-ffiruinoni. 


215 


XL 26.] 


23. The numbers 4, 11, 17, 18, 13, 5 multiplied severally 
by tlie aksiia-karna and divided by 12, will bo* the kiian^akas 
or portions at the given place ; each of those being for each 
15 degrees (of bhuja of the Sun's longitude) respectively. . 

2 1. Now find the Sun's truo longitude tty applying tho 
precession of the* equinoxes to tho Sun's place, and adding 
together as many portions as correspond to tho nnuJ4 of tho 
Sun's longitude above found, divide by GO and add tho quo- 
tient to aksiia-karna. Now multiply tho result by 10 and 
divide by 4 (or multiply by 2^). The quotient is here called 
tho YASUfi in digits and the number of digits thus found is to 
be marked off on tho arm of the pattika counting from its 
hole penetrated by the axis. 

25. Now hold tho instrument so that tho rays of tho Sun 
shall illuminate both of its sides (to secure its being in a 
vertical circle) : tlm place in the circumference marked out by 
the shadow of the axis is assumed to be the Sun's place. 

2G. Now placo the. index arm on tho axis and putting it, 
over the Sun’s place, from tho poiflt at the end of tho YAsirp 
set off carefully above or below (parallel to tho lamba-ukkiia) 
on the instrument, tho sine of tho ascensional difference aboYO 
found, setting it oft’ above if the Sun bo in tho northern 


If eoaino of latitude : sine of lat. 
or na 12 : Palamu' 


| * ®i| 

; : XI 


what will Bine of declination of 1 
sign or 2 or 3 signs, give. 

XUJYA' of 1, 2 or 3 signs. 


2. If coaine of declination : this reault : : wlmt will radius : sine of ascen- 


sional difference in xalAs. .. , , , 

The arc of thia will give ascensional difference. Tins is the plain rule s but 
Biia'skaua'ciia'mya had recourse to another short rule by which the ascensional 
differences for 1, 2 and 3 signs, for tlirf place in whioh the palabiu' was 1 digit, 
were 10 8, 3i palaS. These three multiplied by palabua' would giro the 
ascensional differences with tolerable accuracy for a place of any latitude not 
haring a greator palabiia' than 8 digits. Now take these three paiAthakaS 
. 10. 8. 3J and multiplied by six, then tho paias of time will be reduced to asub. 
These are found with a radius of 3438 : to reduue them to the value of a radius of 


60X30 

As 3438 : 10 X 6 = GO* : : 30 digits quantity of CIZAXA for 1 

3438 • 

sign In this instrument, but instead of Multiplying the 10 by 6 X 30 or 180 and 
dtfidiug by 3438, the author taking 180 = fa part of 3438» divided at once 
by 10.— L. W. 



216 


Translation of the [XI. 26. • 

hemisphere, and bolow if it be in the southern hemisphere. 
The distance from the point where the sino which meeting the • 
end of tho sino of the ascensional difference thus set off, cuts 
tho circle, to thp lowest part of the circle will represent tho 
ohajis to or after midday.* 



instrument is held in the vortical circle passing through the Sun, S b will then 
be the aenith distance of the Sun. From this the time to or after midday can be 
found in the following manner. 

Let a zz altitude of the Sun, 
d =3 declination, 

A = asoous'onal difference, 
l = north latitude of the place, 
p = degrees in time to or after midday. 

Thou, we have the equation which is common in the astronomical works, 9 
R*. sin a :p R . sin l . sin d 


008 p - 


-s 


cos l . cos d 
R i. sin o tan l . tan d 

cos l . cos d R 

her?, when tho latitude is north, the second term becomes minus or plus as t 
declination is north or south respectively. * 
tan l. tan d 

But — ■ — — sin A or sine of ascensional difference. 

R 

R* . sin a 


cos p* 


• ¥ sin. A. 


cos l . CO# d 



SiJJhdnta-tfi roman!. 


217 


XI. 27.] 

27 . Set off tlie time from midday on the instrument 

To find the place of the counting from tho lamIia-uekhX ; from 
shadow of axis from time. tho cnd of fcho sill0 0 f t \^ time, sct 

off the sine of ascensional, difference in a line parallel to the 


Now. COB l : B =3 le : A i. e. akshakabna (See Chapter VII. t. 45.) 

B ’ A ' 

cos! 12 * * 

A B . sin a ■ 

cos p = — — . — — ■■■■ — :p sin A 
12 cos d 

sin a h B* 

— y . — zf sin A, when y = — . — , which is called 
B 12 cos d 

yastiti and can be found as follows. 

A B* B A 12 R 

12 cos d 12 12 cos d 

R A / 12 versed 

. ( 12 .+ 1 

12 12 V cos d J . # 

When tho buttja of 'tho Sun's longitude is 15, 30, 45, GO, 75, 90, tho valuo of 
12 verted d , 

is 4, 15, 32, GO, 63, G8 sixtieths respectively. Tho differences of 

cos d 

these values are 4, 11, 17, 18, 13, 5 which /iro written in tho text. Multiply 
these differences by A or the akewakakva, divide the products by 12 and tho 
quotients thus found nrc called the kit ani) as for the given place. By assuming 
the uiiuja of the Sun’s longitudo as au argument, And tho result through tho 
KUAN has aud tako r for this result. 


r A / r \ 

Then = (A + 1 

, 60 12 v GO /, 

B / r X 

and hence, y = — I A + 1 

12 V 60 /. 


But in this instrument B = 30 


10 / r \ 

f A + lwhu 

4 V 60 / 


which exactly coincide with tho rule given iu the 


text for determining the YASim. 

Tho value of the yabhtt will certainly be more than 30, because the valuo of 
tho aksuakabna or A is more than 12. 

Now, (see the diagram) suppose m is the end of the yasiiti in the patt ik A or 
index o n which touches the circle in S, then, iu the triauglc o in n 
B i o m — sin m o » : tn h j 
or B s y = sin a : m n ; 
y X sin a 
m n = - ■■■— ; 

s 

and hence, co sp =* m » sin A, 

. Q 



218 


Tran shi turn of fho [XI. 28. 

lamba-REKH^ but below and above according as it was to bo 
sot off above or below in finding the time 'from the shadow, 
(this operation being the reverse of the former). The sirto 
met by tho sine of ascensional difference, thus set off, is the 
new sino across which tho pattika or index is now to bo 
placed till tho yashti-chinha or point of Yashti falls .on it. 
This position will assuredly exhibit tho place of tho shadow 
of tho a?is. 

28, 29 and 80. Having drawn a circle (as tho horizon) with 

„ „ „ a radius equal to radius of a great 

The Yashti or Staff. . 1 . ° 

• circle, mark cast and west points (and 

the lino joining tlieso points is called tho frAcityapakX or oast 
and west line) and mark off (from them) tho amplitude at the 
east and west. Draw a circle from tho same centre with a 
radius equal to cosine of declination i. e. with a radius of 
diurnnl circle, ahd mark this circle with 00 giiatis. Now tako 
tho YASHTI, equal to tho radius (of the great circle) and hold it 
with its point to tho Sun, so that no shadow be reflected from 
it ) tho other point should rest in the centre. Now measure 
the distanco from the end of tho amplitude to the point of tho 
yashti when thus held opposite to tho Sun. This distanco 
applied as a chord within tho interior circlo will cut off, if it bo 
beforo midday, an arc of the number of uhatikas from sun- 
rise, and if after midday an arc of tho timo to sun-set.* 


that ip, tho sino of the Ascensional diffeqpice is subtracted from or added to m n 
tho distance between the end of tho yabtiti and the middle sine, as the Sun 0 be 
ill the north or the south to tho equinoctial. 

Again, by taking m r equal to sin A we have, 
cos p = f» » ¥ sin A = m » $ m r, 

= n r or tt\ 

— cos c i, 

.*. p-~-zci- B.D.J 

**[It i*,plain from this, that the distanee from the point of tho staff to the 
end of the amplitude is the chord of the aro of the diurnal circle passing through 
the Sun, intercepted between the horizon and the Sun. For this reason, tho 
an subtended by tho distance in question in this interior circle described with 
a radius of tho diurnal circle which is' equal to the cosine of the dcclinutiun, 
will denote the time after sun-rise or to sun-sot.— B. 1).] 



XI. 33.] SUhlhdiita-n'iromtnii. 210 

31. The perpendicular let fall from the point of the YASirp 

To find the paiabha with is tlio s'anku or ftino of altitude: the 
tlfe tasuti. place betweon the s'anku and centro is 

equivalent to driaya or sine of zeuith distance. The sino of 
amplitude is the lind between the point of hbrizon at which 
the Sun rises or sets', on which the point of the YAsirp will 
rest at sun-rise and sum-sot, and the cast and west lino the 

PUXCHYAPARA. 

32 and 33. The distance between the s'anku and the 
[TDAYASTA-stfutA, multiplied by 12 and divided by the s'anku, 
will be the PALABIlA. 

Take two altitudes of tlio Sun with the yashti : observe 
the s'ankus of the two times and the buujas. 

Add the two buujas, if one bo north and the other south, 
or subtvact if they bo both- of the same denomination: multi- 
ply tho above quantity (whether sum or difference) by 12 and 
divido by the difference of the twy s'ankus, tlio result will bo 
the palabha.* Tlio difference between the cast and west line, 
and tho root of s'anku is called miftJA. 

* [Lot O be tho oust or west point of tho horizon 0 a , Z the zenith, in S tho 



diurnal circle on which S and * are the Sun’s two places at different times aiftl 
8 m and * » the s'ankus or the sine* of altitudes of the Sun, then Os,#» will he 
the buujas, m m or tp the difference between the buujas and S y the difference 
between tho s'avkus. 

* Q-2 


220 


Translation of thr [XI. ;H. 

If the a ' an K a be observed three different times by the 

To find falabha^ doclin- yash^i, then tho time, declination &c. 

ation, time. & c. from three t i /i. • i i . • 

observations by the Yasuti, m *Y be found (ty simply observing 
of three b'amkus. 4 the Sun). . 

34. First of Ml find three s'ankus : drfiw a line frfim tho 
top of the first to tho top of the last; from the top of tho second 
s'anku, flraw a line to tho oastorn pointund a lino to the western 
point of the horizon, so as to touch the first lino drawn. 

35. A line drawn so us to connect those two points in tho 
horizontal circumfercnco will bo tho udayAsta bIjtka. Tho 
distance between it and tho centre will give the sine of ampli- 
tude. The line drawn tlirough tho centre parallel to the i:i>y- 
asta-hutra at the distanco of tho sine of amplitude is the east 
and west line.* 

30. Find the talaouX as before (and also the aksha- 
karna). Now *the sine of amplitude multiplied by 12 and 
divided by aksh a- kauna will ,be the sino of declination. This 

# again multiplied by tho radius and divided by tho sino of 2 1° 
or tho sino of the Sun's greatest declination, will give the sino 
of tho UHUJA of the Sun’s longitude. 

37 and 38. Which converted into degrees is Sun’s longi- 
tude, if tho observation shall have been made in tho 1st 
quarter of tho year. If in tho second quarter, tlio longitude 
will bo found by subtracting the degrees found from (i signs : if 

Now as tho triangles tan and S a m aro the latitudinal triangles, the 

* tii mglp 8 1 p is also tho latitudinal - 

8 p : * p = 12 : palahua' “ 

12 sp •* 

Pakabua' = — . 

Sp 

It ia when S, * two places of tho Sun are both north or both south to tho 
prime vertical, but when one place is north and other is south, the sum of tho 
BUCJAS is taken.— B. D.] 

* [As it is plain that the tops of the three s'aitktjs aro in the plane of the 
diurnal cjrcle, tho line therefore drawn from tho top of the first i'ankit to that 
of tho last, will also be in the same piano and hence the two lines touching this 
line, drawn from tho top of the middle s'abku one to eastern and the other to 
TOetern point of the horizon, lie in this plane. Therefore, the line joining these 
two points of the horizon is the intersecting line of the plane of the diurnal 
circle and that of the horizon, and consequently it is the udaya'iia buiba. 

!>•] 



221 


XT. 41.] Svhlhdnta-iiromani. 

in the 3rd quarter, 6 signs must bo added : if in the fourth 
quarter of the year, then tlio degrees found must be subtracted 
from 12 Bigns for the longitude. , 

Tho quarters of the year will be known from tho seasons, 
the peculiarities of* each of which I shall subsequently de- 
scribe/ • 

It is declared (by some former astronomers) that tlm shadow 
of the gnomon revolves on tho circle passing through tho 
ends of the three shadows made by the same gnomon (placed 
in the centre of the horizon), but this is wrong, and conse- 
quently the east and west and north and south lines, tho 
latitudes Ac. found by the aid of the circle just mentioned aro 
also wrong * 

Whether the place of the Sun be found from tho 
shadow or from tho sino of the amplitude, it will bo found 
corrected for procession. If tho amount of* precession bo 
subtracted, the Sun's true pbice will bo found, if tho truo 
place of tho Sun bo subtracted, tho amount of precession will # 
bo ascertained. * 

40. But what docs a man of genius want with instruments 

„ . about which numerous works have 

Tlio praise of instrument 

called DHivANiiiA or gcmim treated i Let him only tako a stall in 
instrument. ^ his hand, and look at any object along 

it, casting his eyo from its end to tho top, there is nothing 
of which ho will not then loll its altitude, dimensions, Ac. 
if it bo visible, whether in tho heavens, on the ground or 
in the water on tho earth. * 

Now I proceed to explain it. 

41. Ho who can know merely with tho staff in his hand, 
the height and distance of a bamboo, of which he has observed 
the root and top, knows the uso of that instrument of instru- 
ments — genius — (the rnifYAXTKA) and tell me what is tlifcrc that 

* The existence of such gross error in the principles of a calculation as ai% 
here referred to as existing in the wo'rks of Biu'bkara’s predecessors would 
seem to indicate that the science of astronomy was not of more recent' cultiva- 
tion than Mr. Bentley and others hare maintained,— L. VV, 



222 


Translation of thr 


[XI. 12. 


To find paiabkX.* 


ho cannot find out. [Here tho ground is supposed to bo 
perfectly level J • 

42. Direct tho staff lengthways to tho north polar star ; 
let drop-linefl fall from both ends of 
stuff, when thus directed to tlie star. 

Now tho spaco between the two drops is the *Bhuja or base of 
a right qngled triangle, when tho difference between tho lines 
thus dropped is the koti or perpendicular. 

43. Tho koti multiplied by 12 and divided by tho bhuja . 
gives tho PALABHA'.* 

Having in tho same way observed tho root of tho bamboo ; 
[and in so doing found the bhuja and koti], multiply tlio bhuja 
by tho height of tho man’s eye. 

41 and 45. And divide tho product by tho koti, the result 

To find tlie distance and y ou know tho distaneo to tho root 
lieight of a bamboo.' 0 f t ] 1Q bamboo 1 . 

Having thus observed the top of tho bamboo (with the 
# staff, and ascertained the bhuja and k«tO> multiply tlio dis- 
tance to tho root of tho bafhboo by tho koti, and divide tho 
product by tho bhuja, tho result is tho height of the bamboo 
abovo tho observer's oye : this height added with tho eye's 
height will give the height of tho wliolo bamboo. f 

For instaucc, suppose tho staff 1 1*5 digits long, Hie height 
of observer’s eye 08 digits; that in 

Example. making tho lower observation the 

■ bhuja = 1 14 digits = 0 cubits, and koti = 1 7 digits ; that in 
■ making the observation of the top of the bamboo, tho bhuja«*= 



o. If this nptrJA : gives tho koti 
; : 12 digits of gnomon : gives the PALABHA \ 


t The observer Erst directs a b bis staff to d, the root of the tree ; Tho staff t 

I * 



Sulilhdntas'iromanL 


223 


XI. 40.] 

116 digits and Kofl = 87 digits. Then toll me the height of 
bamboo and tho dfstanco of it. As, • 

• 68 X 144 

- ■ = 576 digits or 24 cubits distanco fo bamboo ; 

17 ■ . 

570 5 c 87* f 

nnd — a* = 432 height of tree above observer’s eye, 

116 

68 add tho eye’s height, 

500 height of treo. 

Let a man, standing up, first of all observe tho top of an 
object : then (with a staff, whether it bo equal to tho former 
or not in length), let him observe again the top of the sumo 
object whilst sitting. 

46. Then divide the two kotis by their respective iwiuas : 
take the difference of thosfc quotients, and bj it divide tho 
difference of tho heights ‘of observer’s eye — this will give the 
distanco to tho bamboo : from tlii« distance tho height of the 
bamboo may bo found as before.* 



is furnished at either end with drop lines a k, h k ; b k—a h = b c = sin .of 
/ bac. Then say 

As be i ac i ib e : d e =/ b. • 

He then observes thj top of object aid finds g /, which is easy, as / o has 
been found.— L. W. • 

* BuiflK aba font ds this rule on the following algebraic process. 



224 Translation of the [XI. 47. , 

• , 

47. There is a high famous bamboo, the lower part of 

Question ' which being concealed by houses Ac. • 

* was invisible: the ground, however, 

was perfectly level : If you, my friend, remaining on this samo 
spot by observing the top (first standing and then sitting), will 
tell me the distance and its height, I acknowledge you shall • 
have the title of being tho most skilful of observers and 
expert ip the use of tho best of instruments dhiyantra. 

Tho observer, first standing, observes the top of tho bamboo m 
and finds the bhuja, with the first staff, 

UlTiiinplfl. 

to be 4 cubits or 96 digits : ho then sits 
down and finds with another staff the biiuja to be 90 digits. 

In both cases tho Kofi was one digit. Tell me, 0 you export 
in observation, the distance of observer from tho bamboo and 
tho bamboo’s height. 

48. So also' tho altitude may bo -observed in tho surface 

^ of smooth water : but in this case the 

Observation in water. • 

height of observer s eye is to bo sub- 
tracted to find tho true height of tho object : — Or tho staff 
may bo altogether dispensed with : In which last caso two 
heights of tho observer’s eye (viz. when ho stands and sits) will 
bo two Kofis : and the two distances from tho observer to tho 


Lot * = base, distance to bamboo. Then say 
x x 

if 96 : 1 : : x : — — j then — — + 72 = height of bamboo. 

96 96 

x x 

By second observation 90 : 1 : : x : — — , thon — — + 24 = height of Ijgm- 
90 90 


boo. 

x 'x x x 6 x 

Then 72 + = 24 + ; — 48, or = 48 

96 DO 90 96 8640 

x = 69,120 digits 
= 2880 cubits. 

i x x • 

That is = 72-24 

90 96 

# 72-24 

or x ^ ■ - ■■■ that is difference of observer’s height— difference of two ioti« 
dr— bV divided by their respective buuja's.— L. W. 



XI. 49.] SiJilhan tasirom an i. 225 

, • 

places in the water where the top of the objoct is reflected, 

the bhujas. ' * 

* 49. Having scon only the top of a bamboo .reflected in 
water, whether the bamboo be near or 
, • at a distance, visible or invisiblo, if you, 

remaining on this same spot, will tell mo tho distance and 
height of bamboo, I will hold you, though appearing on Karth 
as a plain mortal, to have attributes of superhuman knowledge. 
An observer standing up first observes (with his statf) tho 
reflected top of a bamboo in water. 

Example. 

The koti = digits and imi'JA = 4 
digits. Then sitting down ho makos a second observation and 
finds the iwiiJA = 11 digits and koti = 8 digits. His eye's 
height standing = cubits or 72 digits, and sitting = 1 cubit 
or 2 t digits. Tell mo height of bamboo and its distance.* 



then by first observation 



220 Translation of the [XI. 4y. 

i 

A man standing np sees the shadow of a bamboo in tho 
' water— the point of tho water at which 

, the shadow appears is 96 digits off: 

then sitting down on tho samo spot ho again observes tho 
shadow and finds the distance in the watcr"at wlpcli it jfppcurs 
to be 33 digits : tell mo the height of the 'bamboo and his 
distance from the bamboo.* 


4:3::c: ( yor3.r = 4yorff = 

3 

by 2nd observation 11 : 8 : : x : y — 48 digits 
11 y — 528 

or 8 x = 11 y — 628 or x 

8 

4 y 11 y- 528 

thus.* = and x = 

3 8 

4 </ 11 y — 6-8 33 y — 1584 . 

3 8 or4 y— ^ 8 

or 32 y = 33 »/ — 1584, or y = 1384 
V. 1584 — 72 = 1512 digits = 63 cubits =: height of bumboo. 
2nd part. To find width of watefor x 


4,y 1581 X 4 




- = 212 digits = 88 cubits.— L. W. 


• Let e p = 96 digits 
p d = 33 
ac = 72 
b o ste 24 

let x = distance from observer to 
bamboo. 

Now e e : se=jjk a 

_ 72 x 3 x 

or 96 : 72f= x s y = 

96 4 

3 x 

3 = height of bamboo 

Again cd :5c: :j k:jb 

24 * 

or 33 : 24 : : x i y — 48 = 

, 33 ♦ 

8* 

~~ 11 
8c 

then — 1 = height of bamboo 




£1. 55.] Suldhant i-ttiromani . 227 

50 and 51. Mako a wheel of light wood and in its eircum- 

A self revolting instru ^ crenco P u ^ hollow Spokes all having 
incut or bwayakvaha yajt- bores of tho same diamoter, and let 
TBAi . tjiom bo placed at .equal distances 

from cfcch other ; tfnd let them also be all placed at an angle 
somewhat verging from the perpendicular : then half fill these 
hollow spokes with mercury : tho wheel thus filled will, when 
placed on an axis supported by two posts, revolve of itself. 

Or scoop out a canal in tho tiro of tho wheel and then 
plastering leaves of the tala tree over this canal with wax, 
fill one half of this canal with water and othor half with mer- 
cury, till tho water begins to coino out, and then cork up tho 
orilico left open for filling the wheel. Tho wheel will then 
revolve of itself, drawn round by the water. 

Make up a tube of copper or other metal, and bend it into 

■the form of an ankus'a or elephant 
Description of a syphon. j 100 ]^ ft w fth water and stop up 

both ends. • 

51. And then putting one end into a reservoir of water, 
let the other cml remain suspended outside. Now uncork 
both ends. The water of the reservoir will be wholly sucked 
up and full outside. 

55. Nqw attach to tho rim of tho before described self- 
revolving wheel a number of water-pots, and place tho wheol 
and those pots like tho water- wheel so that tho water from 
the lower end of tho tubo flowing into them on ono side ahull' 
set tho wheel iu motion, impelled by tho additional weight of 
tho pots thus filled. The water discharged from tho pots as 
they reach the bottom of tho revolving wheel, should be drawn 


8 x 3 or 8 x Sx 

’ll ""4 ~4 11 

,\ r = 44 X 2 = 88 



3 x 


Xlicn y = 


3 X 88 

= 3 x 22 = G6, height of bamboo. 


4 


4 



228 Translation of the [XI. 5t>. 

• 

off into the reservoir beforo alluded to by means of a water- 
course of pipe. 

56. The self-revolving machine (mentioned by Lalla &r") 
which has a tube with its lower end open is a -vulgar machine 
on account of its* being dependant, because? that* which* mani- 
fests an ingenious and not a rustic contrivance is said to be a 
machine# 

57. And moroover many self-revolving machines are to bo 
met with, but their motion is procured by a trick. They are 
not connected with the subject under discussion. I have been 
induced to mention the construction of these, merely because 
they liavo been mentioned by former astronomers. 

End of Chapter XI. called YantrIdhyaya. 


. UHAPTBH XII. 

Description of the seasons. 

1. (This is tho season in which) the kokilas (Indian black 

_ . birds) amidst young climbing plants, 

thickly covered with gentry swaying 
and brilliantly verdant sprouts of the mango (branches) rais- 
ing their sweet but shrill voices say, “Oh travellers! how 
'are you heart-whole (without your sweethearts, whilst all 
nature appears revelling) in tho jubilee of spring chaitra, and 
the black bees wander intoxicated by tho deliciouB fragrance 
of tho blooming flowers of the sweet jasmine !” 

2. Tho spring-bom mallika (Jasminum Zambac, swollen 
by the pride she feels in her own full blown beautiful flowers) 
derides (with disdain her poor) unadorned (sister) mIlatj 
(Jasminum grandiflorum) which appears all block soiled and 
without leaf or flower (at this season), and appears to beckon 
her forlorn sister to leave the grove and garden with her 



. XII* 7.] SitWtdnitt-siromani. 220 


i » 

tender budding arms, agitated by the sweet breezes from tko 
• fragrant groves of *tkc hill of Malaya. 

*3. In the summer (which follows), the lovers # of pleasure 
The GRfsHMi or mid-sum- and their sweethearts (putting their 
mcr eeast*. ^ • s tone built houses, betake themselves 

to the 'Solitude of well wetted cottages of the kus' akIs'a grass, 
salute each other with showers of rose-water and nmuso them- 
selves. , 


4. Now fatigued by their dalliance witli the fair, they 
proceed to tho grove, where Kama-deva has erected the 
(flowering) mango as his standard, to rest (themselves) from 
tho glare of tho fierce heat, and to disport themselves in tho 
(well shaded) waters of its bowris (or largo wells with steps). 

5. (Tho rainy season 1ms arrived, when tho deserted fair 

ono thus calls'upon her absent lover :) 

R4ny season . " # 

■Why, my cruel dear one, why do you 
not shed the light of your beaming eye upon youi* love-sick 
admirer? Tho fragrauco of tho blooming mXlatI and tho 
turbid state of every passing torrent proclaims the season of 
tho rains and of all-powerful love to liavo arrived. Why, 
therefore, do you not have compassion on my miserable lot?* 

0. (Alas, cries tho deserted wife, alas !) tho peacocks 
(delighted by the thundering clouds) scream aloud, and tho 
breeze laden with the honied fragrance of tho kaijamua comes 
softly, still my sweet one comes not. Has he lost all delight 
for tho sweet scented grove, has he lost his ears, has he no 
pity— -has ho no heart ? 

7. Such are tho plaintive accusations of tho wife in tho 
season of the rains, when the jet black clouds overspread tho 
sky : — angered by the prolonged absence of him who reigns 
over her heart, slio charges him, but still smilingly and 
sweetly, with being cruelly heedless of her devoted love. • 


* This is one of those verses in which a double or triplo meaning is attempted 
to be supported : to effect this, several letters however are to be read differently. 
— L. W. 



230 


Translation of the 


[XII. 8. . 


' e 

8. Tho mountain burning with remorse at the guilt of 
Tlio sajutk v # LA or season having received tlic forbidden cm- • 

of eturljr autumn. braces of his own PUSifi'AVATf daughter, 

forest appears in early autumn through its bubbling springs 
and streams sparkling at night with tho rays qf tho Moon, to 
bo shedding u flood of mournful tours of penitence. 

9. In tho hemanta season, cultivators seeing tho earth 

„ . smiling with tho wide spread harvest, 

Hemanta or early winter. ° 

and the grassy fields all bedecked with m 

tho pearl-liko dew, and teeming with joyous herds of plump 

kino, rejoice (at tho grateful sight). 

10. When tho s'rs'iitA season sets in what unspcakahlo 
8ViBA or closo of win- beauty ami what sweet and endless 

lor ’ variety of red and purple does not 

the f KACHNjM grovo unceasingly present, when its leaf is in 
full bloom, and'its bright glories are all expanded. 

11. Tho rays of tho Sun t fall midday on the earth, hence 
in this s 'is ira season, they avail not utterly to drive away the 
cold : 

* * * * 


12. Hero, under tho pretcnco of writing a descriptive 

account of the six seasons, I have 
8wool» of poetry. . 

taken tho opportunity of indulging 

my voin for poetry, endeavouring to write something calculated 
to plcaso tho fancy of men of literary taste. 

. 13. Where is tho mau, whoso heart is not captivated by 

. tho ever sweet notes of accomplished poets, whilst ilicy dis- 
course on overy subject with refinement and tasto ? or whoso 
heart is not cnclianted by tho blooming budding beauties of 
tho hundsomo willing fair one, whilst she prattles sweetly on 


overy passing topic : — or whose substance will she not secure 
by hor deceptive discourse ? 


14. What man has not lost his heart by listening' to the 
pure, correct, nightingale-liko dotes of tho genuine poets ? or 
who, whilst ho listens to the soft notes of the water-swans on 



XIII. 2.] Siddhanta-siromani. 231 

» 

the shores of largo and overflowing lakes well filled with lotus 
flowers, is not thereby excited ? 

*15. As holy pilgrims delight themselves, in tko midst of 
the streams of the sacred flanges, in applying the mild and the 
sparklirtg sands of ite» banks, and thus experience moro than 
heaven's joys : so 'true poets lost in tho How of a fine poetic 
frenzy, sport themselves* in well rounded periods about) ding in 
displays of a playful taste. • 

Eml of Chapter XII. 


CHAPTER XIII. 

Containing awful question* rolled i , ras / nXi)HYaYA. 

1 t 

1. Inasmuch as a mathematician generally fails to acquire* 
Objoct of tho chapter and ■ distinction in on assemblage of learned 

its praise. men, unless well practised in answer- 

ing questions, I shall therefore propose a few for tho enter- 
tainment of men of ingenuity, who delight in solving all 
descriptions of problems. At tho bare proposition of tho 
questions, he, who fancies in his idle conceit, that he has 
attained tho pimmelo of perfection, is often utterly discon- 
certed and appalled, and finds his smiling checks deserted of 
their colour. 

2. These questions have been already put and have been 
duly answered and explained either by arithmetical or algebraic 
processes, by the pulverizer and the affected square, i. o. 
methods for the solutions of indeterminate probleuw of tho 
first and of tho second degree, or by means of tho armillary 
sphere,* or other astronomical instruments. To impress ami 
make them still more familiar and easy I shall have to repeat a 

# few. 



232 Translation of the [XIII. 3. . 

3. All arithmetic is nothing but the rulo of proportion 

Praise -of ingenious per- and Algebra is but* another name for • 

*° nt - t ingenuity of invention. To the clever 

and ingenious then what is not knoprn ! I, however, write for 
men and youths' of slow comprehension. • , f 

4. With the exception of the involutibn and 'evolution of 
the square and cube roots, all branched of calculation may bo 
wholly rtjsolved into the rulo of proportion. It indeed assumes 
many shapes, but it is universally prevalent. All this nrith- • 
metical calculation denominated I’Xff ganita, which has been 
composed in many ways by tho wisest of former mathema- 
ticians, is only for the enlightenment of simplo men like 
myself. 

5. Algebra does not consist in tho letters (assumed to 
represent the unknown quantities) : neither are the different 
processes any part of its essential properties. But Algebra is 
wholly and simply a talent aVid facility of invention, because 

• tho faculties of inventive genius are infinite. 


G. Why, 0 astronomer, in finding tho ahakoana, do you 
^ ^ add sacra months to tho lunar months 

chattra Ac. (which may have elapsed 
from tho commencement of tho current year) : and tell mo 
also why tho (fractional) remainders of adihmasas and avamas 
are rejected : for you know that to give a true result in using 
tho rulo of proportion, the remainders should be tuken into 
account. m 

7. If you have a porfect acquaintance with tho xib'ra or 
allegation calculations, then answer 
this question. Let the place of tho 
Moon bo multiplied by oue, that of the Sun by 12 and that of 
Mars by G, let tho sum of these three products bo subtracted 
from three times tho Jupiter's place, then 1 ask what are tho 
revolutions of the planet whbse place when added to or 
subtracted from tho remainder will give the place of Saturn ? 


Question 2nd. 



233 


XIII. 9.] SidJhdnta-ttiroimwi. 

, 8 and 9. In order to work this proposition in tho first 
‘ place proceed with the whole numbers 
of revolutions of the several planets 
in the kalpa, adding, subtracting and multiplying them in tho 
mannei^mentiqned in the question : then subtract the result 
from tiie revolutions of tho planet given : or subtract tho 
revolutions of the given <planet from the result, according as the 
place of the unknown planet happen to be directed to bo added 
or subtracted in the question. This remainder will represent 
the number of revolutions of the unknown planet in the kalpa. 
If the remainder is larger than tho number from which it is 
to be subtracted, then add the number of terrestrial days in a 
kalpa, or if tho remainder exceed the number of terrestrial 
days in the kalpa, then reduce it into tho remainder by dividing 
it by the number of days in the kalpa.* 

• Bha'skaba'cha'rya himself has given the following example in his coni* 
merits rj va'sana'-bha'shya a ■ 

Suppose Moon to have 4 revolutions in a kalpa of GO days 

Sun, 3 «... A 

Mars 5 ..• 

Jupiter, .. .. 7 

Saturn, .. .. 9 

Then 4*l + 3X12 + BxG = 70aiid7 ><3 = 21 . 

As 70 cannot be subtracted from 21 add 60 to it ss 81, 

Subtract 70, 

remainder 11 : 

let p = revolutions of the unknown planet, then by the question 11 — p = 9 
or 11 — 9 = 2=j>, 

bum +/> = 9orp = 9- 11 = 60 + 9-11 = 58: 

It thus appears that the unknown planet has 2 or 58 revolutions in tlie 
KAL” i. 

Now let us see if this holds true on tho 23rd day of this kalpa i 
revolutions signs • 

for Moon, if 60 : 4 :: 23: G .. 12* this * 1 = 6.. 12* 

Sun, 60 : 3 :: 23: 1 .. 21 this * 12 = 9 .. 18, 

Mars, 60 : 6 : : 23 : 11 .. 0 this % 6 = G . 0 

signs 10 .. 0 subtracted* 

Saturn, 60 : 9 : : 23 : 6 .. 12 

Jupiter, 60 : 7 : : 23 : 8 .. 6 this X 8 = 0 .. 18 from 

for j>,* 60 : 2 : : 23 : 9 .. 6 this sub. from 2 .. 18 remainder 

• 9.. G 


corresponding with Saturn, 5 . . 18 



231 


Translation of the [XIII, 10. 

10. The algebraical learned, who knowing tlio sum of tha 

additive months, ' subtractive davs 

Question 3rd. J 

olapsod and their remainders, shall 

tell the number, of days elapsed fro^i the commencement of 

tho kalpa, dcseifvos to triumph over the s indent who i5 puffed 

up with a conceit of his knowledge of -the exact pulverizer 

called sam'susiita united, as the lion triumphs over tho poor 

trembling deer ho tears to piccos in play. 

1 1 . For the solution of this question, you must multiply % 

tho given number of additive months, 
subtractive days and their remainders, 
by 863374491681 and divide by one less than the number ot’ 
lunar days in a kalpa i. o. by 1602998999999, tho remainder 
will be tho number of lunar days elapsed from tho beginning 
of tho kalpa. From those lunar days the terrestrial days may 
be readily found.* 


• or if, GO : 58 : : 23 : 2 : 2L Then 2 . 24 added 
• to 2 .18 

still gives Saturn's place C . . 12 

When p = 9 — 11, thon as 11 cannot be subtracted from 9 the sum of GO 
is addod to the 9. The rosson for adding GO is tliut this number is always bo 
denominator of the fractional remainder in Uncling the place of the planets ; 
for the proposition. 

If days of kalpa : revolutions : : givon days give : here the dnya a e kalpa 
are assumed to be GO hence GO is added — L. W. 

* [When the additive months and subtractive days and tboir remainders are 
given to find the aiiauoa^a. 

Let l = 2 1602990000000 tho number of lunar days in a kalpa. • 

0 = 169300000 the number of additivo monttis in a kalpa. 
d = 25082550000 the numbor of subtractive days in a kalpa. m 
A = additive months elapsed. 

A' = their remainder. 

1) = subtractive days elapsed. 

B' = their remainder. 

a = tho given sum of the elapsed additivo months, subtractive days and 
their remainders. • 

and x =r lunar days elapsed ; 

A * 

then say As J s e : : * : A + — — • 

• 1 

B' ' 

As l : d : : X i B + — } 
l 



235 


XIII. 12.] Sirtdhanta-xironmnL 

i 

12. Given tho lum of the elapsed additive months, sub- 
* tractive days and their remainders, 
'Example. equal (according to Brahmagupta's 

system) to 648426000171 j to find tho aitargana. IIo who 
shall aftswer my question shall bo dubbed a “ bkahma-sid- 
imXNTA-viT^.i. e. Shall bo held to havo a thorough knowledge 
of tho brahma-siddhXnta.* 


A' + B # A' + B' 

Afl J:« + d::*:A + B + - — ory + 

(e + <0 x = * 9 + A # + B', or (e + cl) * — l tf = A' + B '» 
and jf = A + B ; 

by addition, (<? + rf) X — {l— l)y = A+ B + A'+ B', 

= »} 

hv substitution, 26675850000 x — 1G02998999D99 y = * : 
now let, 26675850000 x' — 1G02998999999 / =» 1, 

then wo shall have by the process of indeterminate problems 
y = 863374491684. 

Again, let m = e + d and » == l — 1, 
then ■ maf— n.y =«i (1) 
and t?i x' « y' = 1 ; 

omx' — «»iy =»a, 
and mnt — mnt = 0: 

— »0— »(«y — »0 = " : 

which is similar to (1) ; 
x — s/*"#! 

= 863374491684 a - (/ — 1) l. 

Hence the rule in the text.-— B. I).] 

* Solution. Tho givon sum = 648426000171 and t he lunar days in a kalpa 
= 1G02999G00000 : 

648426000171 * 863374491684 

... =349241932336 

and 10300 remainder : 


10300 theso aro lunar days elapsed. 

To reduce them to their equivalent in terrestrial days says 

Iflnnsr daysin 1 . Number of sub-} . . Lunardayi 
a kalpa • J * 


tractive days j : : bove found 


From 10300 Lunar days 
subtract 161 Subtractive days 


1 


161 subtractive 
days and remain- 
der amounting 
267426000000. 


remainder 10189 Terrestrial days or ahabgaba. 

Now to find additive months elapsed. . 

If lunar days! additive months 1 . , lunar days 1 . 10 additive manths and 
in a kalpa f s of KALPA J‘ * 10300 f remn. 381000000000. 

10 additive months = 300 lunar days. 

10300 — 300 = 100,00 SAUBA days elapsed. • 

Hence 27 years 9 months and 10 days elapsed from tho commencement oi 
KALPA.— L. W. 



Question 4th. 


286 Translation of tho [XIII, 13. , 

13 and 14. Given the sum of the remainders of the revo- 
lutions, of the signs, degrees, minutes • 
and seconds of the Moon, Sun, Maft, 
Jupiter, the s'foHBOCHCHAs of Mercury and Venus and of 
Saturn according to the DHfvRiDDHiDA, including the remainder 
of subtractive days in finding the ahargajja,' abraded (reduced I 
into remainder by division) by the number of terrestrial days 
(in a yfga) . He who, well-skilled in the management of 
bphuta kuttaka (exact pulverizer), shall tell me tho places of 
the planets and the aharga^a from the abraded sum just 
mentioned, shall be held to be like the lion which longs to 
make its seat on the heads of those elephant astronomers, who 
are filled with pride by their own superior skill in breaking 
down and unravolling the thick mazes and wildernesses which 
occur in mathematical calculations. 

15. If the given sum abraded by tho number of terrestrial 
days in a yuga, on being divided by 4, 
leaves a remainder, then tho question 
is not to bo solved. It is then called a khila or ail " impos- 
sible” question. If, on dividing by 4, no remainder remain, 
then multiply the quotient by 298627203, and divido the 
product by 394479375. The number remaining wil^ give the 
aiiargana. If the day. of the week does not correspond with 
that of tho question, then add this ahargaxa to the divisor 
(394479375) until the desired day of the week be found.* 


Rule. 


* [According to the BBfrapsHiDA tabtba of lalla the temstrial days in 
*V»vroA = 1577917600 and the sum of all the 86 remainder* for one day sb 
118407188600968 : this abraded by the terrestrial days in a tuga as 259400968. 

Let * as AHASQAgA then say 

Aa 1 : 259400968 : : x x 259400968 X * 

Thia abraded by 1577917500 the terrestrial daya in a yuoa will be equal to 
1491827500 the given abraded sum of the 86 remainder*, now 

let # ss the quotient got in abrading 259400968 x by 1577917500, then * 
859400968 x - 1577917500 y == 14912275C0. 

It la evident from tliia that as the coefficients of x and y are divisible by 4, the 
given remainder 1491227500 alao must be divisible by 4, otherwise the question 
will be impossible aa stated in the test. 

Hence, dividing the both sides of the rbore question by 4, 

, 64860242 9 - 894479875 y = 872806875 (A) 

1 and let 64850248 *' - 894479375 /s= 1,.. (B) 



XIH. 17.] 


Siddhdnta-stromatyi. 


16. Tell mo, rajr friend, what is the aharqana when on a 

4 Thursday, Monday of Tuesday, tho 

^Example. 35 remainders of the # revolutions, 

signs, degrees, , minutes and seconds of tho places of tho 
planets^ (the Sun, tho Moon, Mars, Jupiter *and Saturn and 
the s'faHROCHCHAS of Mercury and Venus) together with tho 
remainder of the subtractive days according to tho dhIvrid- 
diiida, give, when abraded by the number of terrestrial days 
in a yuga, a remainder of 1191227500.* 

17. The place of the Moon is of such an amount. 

Question 6th. that 

The minute. _ + 10 = the seconds 

2 

the minutes — seconds + 3 = degrees 
tho degrees 
2 

• X* = 293627203 by the processes of indeterminate problems. 

Now let a = 64860242, b = 39ltf9375*and 0 = 372806876 1 

we bare the equations (A) and (B) in the forms 

ax — by = c- % 
and 4 — b = 1, 

X = C X — b t (see the preceding note) 

= 293627203 0 — 39447U376 t : 
as stated in the text.— B. D.] 

• Solution. The given sum of the 30 remainders in a YUGA = 1191227600 

— ithe i = 372806876 . 

„ d ' = 277405471 «.d ™rf»d„ 10000 L .. 


= signs. 


391479376 


ahabgava. 

10000 


= 1128 — 4 remainder, i. e. 10000= ABABGAyA on • Tuosduy, for 
7 

the YUGA commenced on Friday. 

This would be the ahaboava on a Tuesday. 

To find the ABABGApA on Monday, it would bo necessary to add tho reduced 
terrestrial days in a yuga to this 10000, till the remainder when divided by 

7 ** 10000 + 8344783767^2 _ 7889j»760 _ ^ . 


““iSto + 831479375 * 3 1183448125 

.rf 

Thursday.— L. W. 


= 169061017 — 6 remainder or = 


7 



238 


Translation of thn [XIII. 18. 

And tho signs, degrees, minutes and Seconds together 
equal to 130. On tho supposition that the sum of these four* 
quantities is of this amount on a Monday then tell me, if yon 
are expert in rules of Arithmetic and Algebra, when it will bo 
of tho same amount on a Friday.* . • 

18. ' Reduce tho signs, degroes and .minutes, to seconds, 

# adding tho seepnds, then reducing tho 

, terrestrial days and the planet’s re- 

volutions in a KALPA to their lowest terms, multiply the seconds 
of the planet (such as tho Moon) by the terrestrial days * 
(roduccd) and divide by tho number of seconds in 12 signs : 
then omitting tho remainder, take tho quotient and add 1 to it, 
tho Bum will bo tho remainder of tho biiaganas revolutions.t 


• Let x = minutes 
*+20 

then — = seconds 

2 

* + 20 

, * + 3 = degrees. 

2 

* + 20 


» + 3 



* + 20 

* + 3 

* + 20 * + 20 2 

and * + — — + * + 3 + — = 130 

2 2 2 
. .*. * = 68 minutes. 

68+22 

- = 39 seconds. 

2 

68 - 39 + 3 = 22 degrees. 

22 

■— s= 11 signs. 

Heneo tire Moon’i placo = 11*. . . 22<> .. 58' . . 39". 
t The mean place of tho Uoon = 11*. .. 22o . . 58' . . 39" ss 1270719" 

, The number of seconds in 12 signs = 1296000. 

Terrestrial days in a Kama ss 1577916450000 1 These dindod by f 956313 
• V 1650000 become dbi- < 

Revolutions of Moon =57753300000 J i>ha or reduced. 136002. 



XIII. 20.] 


8uldltanta-8ii'omtu,d. 


239 


19. Tho rcmiflnder before omitted subtracted from the 
divisor will give i&u remainder of seconds : if* that remainder 
df tlio seconds is greater than the terrestrial days in a kalpa, 
then the question is an "impossible one” (incapable of 
solution and the planet’s place cannot bo found at any sun- 
rise) u but if less it may bo solved. Then from the remainder 
of tho seconds the aiiawiana may bo found (by tlm kuttaka 
pulverizer as given in the lIlavAtl and ufjA-GANiTA) ()r, 

20. That number is tho number of aharoana by which tho 
reduced number of revolutions multiplied, diminished by tho 
remainder of tho revolutions and divided by tho reduced 
number of terrestrial days in the kalpa, will bear no remainder. 
The reduced number of terrestrial days in a kalpa should bo 
added to tho aharoana such a number of times as may make 
the day of the week correspond witli tho day required by 
the question. 

Now when the mean place of tho Moon.was sought, the rulo was 
As tho Terrestrial \ . Revolutions in a 1 . . Given days or 1 . j ong . 9 

days in a kalpa. J 1 kalpa. ahaboa^a. J 

If any remainder existed, it, when multiplied by tho number of seconds in 
12 signs and divided bv kalpa, terrestrial days gave the Moon'a mean place in 
seconds. We now wish to find the uiiaoa^a-b'jwha or tho remainder of revo- 
lutions, from tho Moon’s given place in seconds: wo must therefore reverse tho 
operation . 

Moon’ •place in seconds X kalpa terrestrial days 

or — — BHIOASA-S'ESIIA. 

• ' seconds in 12 signs 

Tho terrestrial days, however, to bo used, must to bo reduced to tho lowe«* 
terms to which it, in conjunction with tho kalpa-bhaqa$ as or revolutions in a 
kalpa can bo reduced : the lowest terms as above stated were of the terrestrial 
days = 966313, of the Moon’s kalpa-bhaoa?? as = 33002. 

1270719 X 666313 1215205090047 . , . 

- — s= 937658 quotient — remainder 

1296000 1296000 

331017. 

937658 quotient 
1 adding one 

gives 937G59 for the bhagana-b'esha. 

Tho reason for adding one is, that we have got a remainder of 333047, which 
wo never could have had, if the original remainder had been exactly 937668, it ^ 
must luve been 1 more. This is therefore added : but the remainder of seconds 
may now be found-for it will be 1 2963000 - 331047 ss 961953. ’ 

This remainder 904953 being greater than the terrestrial days reduced to 
lowest terms, vis. 956313, the question does not admit of being solved.— L. W . * 



240 


Translation of the [XIII. 21. 

^ 21. If tlio Moon's bhaoana-s'esha or tlfe remainder after 
finding the complete revolutions admits of 'being divided by 
1650000, without leaving any remainder, the question may theh 
be solved: the reduced bhagana-s'esha on being multiplied by 
886831 and divided by 951363, then the remainder will give the 
AiiABGANA, The divisor should be added tfb this remainder 
till the day of tho week found corresponds with that of the 
question,* 


l 


22. The mean place of the Moon will never be at any 

ImponiMe question. ° q " al *° 0 5 de « reC8 > 

^ q 36 minutes and 19 seconds. 

>23. When will tho square of the ADHiitisA-s'ESHA remainder 

of tho additive months, multiplied by 

10 and the product increased by one, 

be a square : or when will the square of the adkim^sa-s'esha 

decreased by one and the remainder divided by 10 be a square? 

The man who shall tell me, at what period of the xalpa this 


Question 6th. 


* Z' 

• [To find the ahabgana from the Moon's bhagana-s'esha. 

, Let B = BHAGAlfA-B'WHA, 

T = 1677916460000 terrestrial days in a xalpa, 

M = 67768800000 the Moon's revolutions in a xalpa, 

X = AHABGAKA. 

B B 

Then, as T : M : s x i revolutions + — or y + — : 

T T 

M*-Ty = R: 

In this equation is It and T are divisible by 1650000, B must be divisible by 
.the same number, otherwise the question will be kuila or “impossible," os 
stated in the text. 

Dividing both aides of the equation by the number 1650000, we have 
35002 x - 956313 y = B' or M' x - T y = B' : 

Bow let MV-TV = 1 : or 35002 * - 956313 y' = 1 s hence we havo 
tf s=886834i 

and x = E’ i — T t (see the note on the verse 11th) 

= 886834 B* — 856313 1. Hence the rule in the text. 

And, as the reduoed bhaoavab'jbha = 837659 (see tlie preoeding note) hence 
937659 X 686834 = 831547881606 : 

This divided by 956318 will give as quotient -869555 (i. e. f) leaving a re- 
mainder of 257161 whieli should be the ahaxgana, but as the bhagavai'esba 
i. e. 937669 does not admit of being divided by 1650000 (the numbers by which 
the? terrestrial days were reduced) it ought to have been kuila or insoluble 
question! but BsisKABicnAuTA here still stated thi9 number to be the true 
ADABOAHA, — B. D.] 



XIII. 24.] Siddhdnta-ftiromani. 241 

, * 

will take place — \$ill be humbly saluted even by the wise, wlio 
generally speaking, gazo about in utter amazetnent and confu- 
sion at such questions, like the beo that wanders in the bound- 
less expanse of heaven without place of rest. 

24. • (In working questions of ki'^aka pulverizer, the ang- 

Bermrk on tho preceding mcnfc Iimst ho reduced by the samo 
question. ( number by which tho MiAJYAdivideud 

and hara divisor are reduced to their lowest terms, ajid when 
tho augment is not reducible by the samo number as tho 
bhajya and hara, the question is always insoluble.) But hero, 
in working questions of kujtaka, those acquainted with tho 
subjoct should know that tho given augment is not to bo 
reduced, i. e. it belongs to the reduced bhajya mul iiaiia, 
otherwise in some places tho desired answer will not bo 
obtained, or in others tho .question will bo impossible.* 

t 

• [TU« questions in tho 23rd verso are tlie questions of the vauo i-pbakmti or 
the atfeetcil square, i. e. questions of indeterminate problems of the second degree. 

1st question. Lot a = the adhimAsa-sVsua : 
then by question 10 j 2 + 1 = y*. • 

In such quest inns the eoelUnent of .r it ealled Prakriti, tho valuo of x KA- 
NISHTTIA, that of the augment ksiikpa and that of y JYKSUTIIA. 

Now assume y = m x + 1, 

then 10 or* + 1 = (m * + 1)‘, 

= m* x a + 2 mi x + J, 

2 in 

x 

1 ° — w»* 

Ueifce tho rule given by buaskauaciiarya in his Algebra Cli. VI., verse VI., 
for finding the kanishtiia where the kshepa is 1, is “ Multiply any assumed 
number by 2 and divide by the difference between tho square of tho number 
and the phakuiii, the quotient will be the KAJflBimiA where the xsiikpa is 1.” 

2 m 2X3 

Now assume m = 3, then * = = = 6 : 

10 — in* 10 — a 


and y = ^/lO ar* + 1 — s/wi ~ 19 : 

,\ adhimaia-s'esua = 6. 

From two aots, whether identical or otherwise, of KANIBHtra, jyfshtiu and 
ksuepa belonging to tho same pkaxriti, all others eau be derived such as 


follows. 

Let a = pbakriti, and 

J.sFi.JM the two sets of KASuaniiiA, 

^1 lit I I Olt 

wo have- ■*>+?» ; 

ax'i + ig =yi 5 

*a =3fS — aa, S ! 


JYE3HTHA and KSIIEPA, then 



212 


Translation of tho 


[XIII. 25. # 

i 

25. Tell me, 0 you competent in the spheric, considering 
it frequently in your mind for awhile/ , 
^ ut8t,ofi * what is the latitude of the city (» 

which is situated at a distance of 90° from ujJAYmf, and bears 


anil ft, X ft a = (y? — <* *?) (y2 — a ar?), 

* *! nl + • fS fi + b » = y* yl + ** f ? * 

, adding ± 2 ti x, x a y, y* to both aides , 

<* y« + + *■ jr. + •"*5 '5- 

or a (x, y a ± x a y,) a + 6, 4 a = (y, y 8 iaar, a-,)*: 
thus wo get a now set ot kanistha, jvesiitiia and kbuspa : 
i. o. now kahihhtiia = x, y a ± x 2 y l ; 

now jykshtha = y , y a ± <* #, J* a } 
and now kshkpa =: A, : ~~ 

Ilenco the Knle called buavavA given by bhaskabAchabya in his Algebra 
Ch V r L verses III. & IV. 

Now in tho present question 

x, = 6, y, = 19 and ft, = 1 , 
and also x a = G, v a = 19 and ft a = 1 s 

new lCAMSirrHA = 6 X 721 + 22S X 19 = 4320 + 4332 = fifioS ; 
now jykshtha = 721 * 19 + 10 X 6 X 228 = 13099 + 13080 = 17379 ; 
and now kshkpa =i f *1 = 1. 

Thus x = 8058 Ac., according to the Bhdvana Assumed. 

Tho second question is 

x* — 1 « * * 

10 ~ 3 '* 

or x* = 10 y* -f. 1 , 

ITore thon wo have an equation similar to the former one, but x 2 is now be 
in the pluoe ofy* andy* in tho place of x\ 
x will bo == 19 , 
or = 721 &o. 

Now given adiiimasa sksha as found by tho first case = 6. Tho proportion 
by which this remainder whh got, was 

if KALPA sauba days : kalpa-aduimabas : s x or elapsed sauba days i 

6 

:.»+ • 

KALPA SAUBA du)8 

.*. KALPA-ADUIMASAS X X = KALFA SAUBA days X y + 6 
KALPA-ADUIMASAS >C X — 6 

or y = . 

kalpa sauba days 

From this we get a now question : " AY hat are tho integer values of x and 
y in tlm equation ?” which question is one of the questions of KriTAXA and in 
which the ooellicieiit of the unknown quantity in the numerator is called bhajya 
or dividend, the denominator haha or divisor and tho augment kshkpa. 

It is dear that in this equation, if tho augment be not divisible by the same * 
number as tho dividend and divisor, the values of x and y will not be integers, 
and hone* the question will be insoluble, liut here iu order that no question 
should bo insoluble, the author has stated that the dividend and divisor 
should always be tukon, reduced to their lowest terms, otherwise tho*qae»tion 
wttl be insoluble. f 

As in the present question, if the dividend XALPA-ADHIMASIS and the divisor 
KALPA SAUBA days be taken not reduced to their lowest terms, i e. not divided by 



243 


XIII. 25.] Si<ldhdnt<t-8' irominji . 

, due east from ttfat city (iujayinI) ? Wlmt is the latitude of 
the place (II) disthnt also 90° from tlio city (’A) and bearing 
due west from it ? Wlint also is the latitude of a place (t 1 ) 
also 90° from .(B) and bearing N. J5. from (11) : and of the 
place (D) which is situated at a distance of $ 0 ° from (C) and 
bears* S. W. froul (C) ?* 


tlio number 300000, the question will bo an impossible one, because the augment 
6 is not divisible by tlio snnm number. For this reason tho dividend uud 
divisor must bo taken here reduced to their lowest terms. 

ir»'J3300000 

Honeo, dividend = reduced KALPA-AimrMASAS = = 5311 5 and 

300000 

1555200000000 

divisor = reduced kalpa sauua days = = 5181000 . 

300000 

5311 x-G 

.*. By substitution, y = , 

5181000 

which gives x — 8207 10 tho elapsed sauiu days 
or 2276 years 6 months and 6 days.— li. D.J 
* Let a = the azimuth degrees, 

d =r tlu* distance in degrees between the two cities, 
p = palaiuia' at the given city, 
k = AKSllA-KAnVA, 
and x = the latitude of the other city? 

ysin d x cos a cos d x p\ 12 

Then sin a 


Now in the 1st question, a = 00°, d = 90°, p = 5 digits, (lie I'AIjAHHa' at 
UJJ AVistf, and k = /V /l 2 *“+~ 5 » = 13 : 

• 1 /3V3S*0 ' 


( sill d X cos a cos d X ]>\ 12 

± lx—. 

Rad 12 / k 

= 90°, J = 90®, p = 

= 13: 

■ /3V3SX0 0 X Gx 12 

sin x = ( ± 1 X — 5 

\ 3138 12 / 13 

= (0±0)Xbif = lJs 

... x = 0 = latitude of (A) or of yaw ikoti. 

1 2). In the second question, a = 90®, d = 90* , p = 0 digits at ya WAkori, • 
and .-. * = 12: 

/3138X0 OX OX 12 

siu J = ( ± |X—; 

V 3138 12 / 12 

= <0±0) ^ = 0: 

... x = 0 Latitudo ot city (B) or lakka. 


(3). In the 3rd question, a = 45°, <J = 90°, p = 0 at lank* ami k =12: 
* /3438 X 2431 0 X Ox 12 

.-. sin x =| ,+ 1 X — 5 

V 3138 12 / 12 


X 2431 OX Ox 

rs~ + ‘ir) : 

= (2*31 + 0) X 1 = 2431 : 

■ ,T 2 



244 Translation of the [XIII. 26. 

26 and 27. Convert the distance of yoj/nas (between the # 
* two cities, one is given and the other 
' ' is that of which tho latitude is to b§ 

found,) into degrees (of a large circle), and then multiply the 
sine and cosine of these degrees by the cosine of the azimuth 
of tho other city and palabhX at the given city, and divide tho 
products by radius and 12 respectively. , Take then tho differ- # 
enco between these two quotients, if the other city be south 
of cast of tho given city ; and if it bo north of that, the sum of 
the quotients is to bo taken. But the reverse of this takes 
place, if tho distance between the cities be more than a quarter 
of tho earth’s circumferenco. Tho difference or sum of tho 
quotients multiplied by 12 and divided by akshakauna will 
give the sine of the latitude sought.* 


x = 46° Latitude of city (C). 


(4). In the 4th question, a = 45°, d = 90°, y = 12 at C and k = 

i . ■ 

y3438 % "V* */T °* 12 \ 12 

Bm *~V 3438 12 / 

( 3438 \ 1 3438 

T v' S -°)x^ = T . 

* = 30° Latitude of D.— L. W. 


* [I*t Z be tho Zenit h of the 
given city bearing a north latitude, 

ZHNQ the Meridian, G A II 
the ITorizon, P tho north jkjIb, S 
the Zenith of tho other oily, the 
latitude of which is to bo found 
and Z S N the azimuth circle pass- 
ing through S. Then the arc Z S 
(which is equal to the distance in 
degrees between the two cities) will Q 
be the Zenith distanco of S ; the 
arc II G, tho arc containing the 
giren azimuth degrees, and S h 
which is eqnal to the declination of 
the point S, tho latitude of the 
ot!nr .city which can be found as 
follows. • 

Let a = II g the given azimuth 
degrees, 

d = Z S the distance in degrees between the two cities, 

p =*PALABUA, 
k = AK3UA-XAK?i 




. XIII. 28.] SiiWidnias'iromani. 245 

# 28. Tell me quickly, 0 Astronomer, wliat is tlio lutitmlo 
. ^ * of a place (A) which is *distant » of the 

earth s circumference from tlio city of 
diiXrX and bears. 90? due east from it ? Wlmt also is the lutitmlo 
of a plafco distant 60s from dttXrA, but bearing 45° N. E. from 
it? What also is the. latitude of a place distant GO 1 ’ from DifXuX 
find bears S. E. from.it? What also aro the latitudes of 
three places 120° from dijAra and bearing respectively duo 
# east, N. E., and S. E. from it ?* 


and x = S h the declination of the point S i. e. the latitude of the other city. 
Then say, As sine Zg : sine Ag :: sine Z S : the huuj v i. c. the sino of 

distance from S to the Prime 
Vertical. 

or R : cos a s : sin d : bhuja 

cos a sin d 

.*. BHUJA = . 

R . 

And by similar latitudinal triangles, 12 : p : : cos d : b'an^UTALA, 
p X cos d 

b'axkutala = . 

• 12 * • 

Now when the other city is north of oast of the given city, it is evident that 
the BHUJA will bo north and consequently • 

the sino of amplitude = bituja + s'ANKifrALA : 

but when the other city is south, the BHUJA also will be south and then, the 
sino of amplitude = buuja b'ankutala, 

cos a X sin d p oos d 

or the sino of amplitude = ± . 

R 12 

And by latitudinal triangles 

jc : 12 : ? sine of amplitude : sine of declination i. o. sin x 

( cos a X sill d p x cos rf\ 

± ) 

R 12 / 

sin x = ■■■■-■ = - , 

Jc k 

hence the rule in the text. 

It the distance in degrees between the two cities be more than 90°, the point 
S will then lie below the Horizon, and consequently the direction of the biiuja 
will be changed. Therefore the reverse of the signs t will take place in that 
case. — B H.] 

( sin d X cos a cos d X p\ 

— ± . 


R 


>s dxp\ 12 

)x-. 

12 / k 


(1.) In the first question, a = 90°, d = 00°, pzs 6 digits the p^LABHA of 
duaha and k = 13. 


/2977 X 0 1719 X 5\ 12 

siu i=( ■ — + ) X — i 

V 3438 12 •/ 13 

1719 X 5 12 8593 9 

1?, .13 13 ~~ 13 



216 


Translation of the [XIII. 29. 

« 

29. Tell me, my friend, quickly, without being angry witlj, 
' me, if you have a thorough knowledge 

of the spheric, what will be the palau if.( 
of the city where the Sun being in the middle of the ardr( 
nakshatka (i. o.'having the longitude 2 signs 13° 20') lises in 
the north-east point.* 

x = 11°.. 15'.. 1" latitude of city due eart from dbara. 


(2). In' the 2nd equation, a = 45 s , d = 60 s , p = 5 & .*. Ar = IS : 

( 2977 X 2431 1719 X 5\ 12 

x )X — ; 

31158 12 / 13 

19399109 1913 


- =2604 - 


7449 7449 

. x = 49° . 18'.. 21" Latitude of city bearing 45° N. E. from duara. 


(3). In the 3rd question, a = 45°, d = 60°, p = 5 and k = 13. 

( 2977X 2431 1719 X 5x 12 

ft. lx—; 

, 3138 12 / 13 

9549239 7070 

7449 - 74 J9* ■ 

.*. t = 21* . 5J/.. 31" Latitude of city bearing the S. E. from dhaba. 


(4). To find latitude of place 120° from dhaba and duo cast. Ilorc, sin 
(2 = sin 120° = sin 6ti° = 2977, cos c2 = cos l2U° = — sin 30* = — 1719 
cos a = 0, p = 5 und k 13 : 

( 2977 X 0 1719 * 

- + 

3438 12 

9 


6v 12 
X-! 
/ 13 


x or latitude = 11 s . 15'.. 1". 

The latitudes of the places 120° bearing N. E. & S. E., will be the snmo as 
tile latitudes of those places distant 60 s and bearing S. E & N. E. lienee the 
latitudes are 21° 54'.. 31" mid 49* 18' 21".— L. W. 

•Jaw. Sun’s amplitude = sine of 4& # = 2131', 
the sine of longitude of middle uf ardra = sine of 2 signs 13° 20' = siif 73° 

20' = 3292'.. C" 40'" 

and the sine of the Sun’s greatest declination = sin 24 s = 1397'. 

Then say : As Rad: sin 24° : : sin (73° 20') : sine of declination, and as 
siue of amplitude : sine of declination : : Rad : cos of latitude, 

.-. sine of amplitude : sin 21® s : sin (73° 20') : cos of latitude. 

sin 24° x sin (73«. . 21/) 1397' X (3292' . 6" . 40"') 

eos qf latitude = .... ^r— = ■ — ■ ■ 

sine of amplitude 2431' 

= 189V 60' 48" = sine of 33* 23' 37'' : 
whence latitude will be 56* 36' 23' .*. sine of latitude = 2870* 13". 

Then say : As cos of latitude : sine of latitude : : Gnomon : equinoctial shadow 
• 1891'.. 51" : 2870 13" i: 12 

12 X (2870'.. 13") 13 

equinoctial shadow = ■ ■ - ■ — -- ■■ ■ — — 18 — digits.— L. W. 

. 1891 1 51'' W 



XIII. 32.] 


Siddhdnta-tt'i roman i. 


217 


30. 

Question. 


Question. 


Rule. 


Tell me tlift several latitudes in which the Sun remains 

• abovo the horizon for one, two, three, 
four, tivo and six months before ho 

pets again.* 

31 . »lf you, 0 intelligent, are acquaintcd/with the rcsolu- 

• tion of affected quadratic equations, 
then find the Sun’s longitude, observ- 
ing that the sum of tho cosine of declination, the sine ,of decli- 
nation, and the sine of the Sun’s longitude : equal to 5000 is 

' (tho radius is assumed equal to 3 138.) 

32. Multiply tho sum of the cosine of declination, the sino 
of declination, and tho sine of Sun’s 
longitude by 1, and divide tho product 

by 15, tho quotient found will be what has been denominated 
tho Xdya. Next square the sum and double tho square and 
divide by 337, the quotient is to be substraetfcd from 1)10078. 
Take tho square-root of the remainder. That root must then 
bo subtracted from the adya above found : the remainder will 
be the declination, when the radius is equal to 3 1-38. From* 
the declination the Sun’s longitude may be found. t 

* Anar. When tho Sun lias northern declination ho remains abovo the 
homou for one mouth in 07° N. L. 
two mouth* in Ci‘j a 
three mouths 73* 

, four mouths 7S g 
five months 
six mouths 'J0° 

These are roughly wrought s for 'BiTAf!K4n\cnAnYA , 8 rule for finding these 
Latitudes see the TUiriUB'NADiiYAYAS of tho qoladij yaya and also the OAK ITA- 
LIC YAYA.— L. W. 

f [Let a = the given sura, 

p = the sine of tho Sun's extrema declination 
x = the sine of the Sun’s declination. 

Then the cosine of declination will bo K k — a* and the sine of the Sun’s 

R x 

longitude = : 

P 

Rr 

by question i 


+ *+ =a: 

P 

P ^ R*— 4f* + (R + J>) * = a p t 
and p yfR s -jf* =z a p — (B, + p) x • 

p‘ -£*• = *> p'-iap(R + p), + (B.' + 2Tlp + p>) ll >; 

(R* + 2 Rp + ! ,*) a, (R + J>) * = — (»’ — R") p ' ; 



248 Translation of the [XIII. 33. 

33. Given the sum of the sines of the declination and of the 

• 

altitude of the Sun -when in the prime 
^ uostldh * vertical ; the taddhriti, the kujy! aifd 

sino of amplitude equal to 9500, at a place whore the paladha 




2ap (R + p) 


<«* -'Wp' 


R 8 + 2Rp + 2p 8 R3 + 2Rp + 2p 8 

2ap(R + p) a* p* (R + p) 1 

completing the square, x* — ~ x + - 


«V (R+P)' 




(R* + 2 Rp + 2p 8 ) 8 R 8 + 2Rp + 2p 8 

K*p 8 + 2K s p 8 + 2RV-«V 

■ * 


(R 8 + 2Rp + 2p 8 ) 8 


Q> p* 


R 8 + 2 R p + 2 p 8 (R 8 + 2 R p + 2 p 8 ) 8 
«P(B + P) V j R V a'p‘ 


_=± /_ 
«» V u 


R 8 + 2Rp + 2p 8 v R» + 2Rp.f 2p 8 (R 8 + 2Rp + 2p 8 ) 8 

ap(R + |») / R* p* aV 




ll 8 + 2Rp + 2p 8 ^ R» + 2Rp+2p’ (R»+ 2 Rp + 2p 8 ) * 

Now hero R = 8438 and p = 139?, 

ap'R + p) ap(U+p) a X 1397 X 4836 6734495a 

' R* + 2Rp + 2p 8 “ (R + p) 8 + p 8 ~ (4835) 8 + (1397) 8 ~~ 25328834 


— a nearly = Adya j 
15 

a 8 p* 


3808777688881 a 8 2 a 8 


and • 


(ll 8 + 2 Rp + 2 p 8 ) 8 611549831799356 337 

R 8 p“ 23«lG7713i)a89!}6 


- = nearly ; 


R 8 + 2 Rp + 2 p 8 25328834 
thor has taken the number 910678. 


= 910729, in place of this the Au- 


a- = Auya ± 910678 - rf, a 8 : * 

but of these, the positivo value is excluded by the nature of the case, becauso 
the sine Qf declination is always less thau 1397. 

Hence the Rule in the text. 


Solution. The given sum = 5000, 

5000 X 4 

Adyv= = 1333' 20" and a* = 148367' 67" 9. w 


15 


sine of declination = 1333' 20'' — 10678 — 148367' 67' 9 /''• 

* = 1333' 20" — 873' & 13''' 

— 460' 13" 47"": from which we hare the longitude of 

he Sun = 0*.. 19V. 11' 36" or 5‘ 10*.. 45' 24" or 6» 19® IV. 36'' or 

11*.. 10°.. 45'.. 24*.— H. D.J 


XIII. 35.] Siddhdnta-8' iromctni. 2t0 

ft 

,or equinoctial shadow is 5 digits, tell mo then, my clover 
friend, if quick ill working questions of latitudinal triangles 
and capable of abstracting your attention, what are, tho sepuruto 
amounts of each quantity ? 

34. 'First pssurae the sine of declination* to bo oqual to 

■ 12 times the shadow palabilC: and 

. then find the amounts of tho # remain- 
ing quantities upon this supposition. Then these on. the sup- 
position mode, multiplied severally by the given sum and divided 
by their sum on tho supposition made, will respectively make 
manifest tho actual amounts of thoso quantities tho sum of 
which is given.* 

35. If you liavo a knowledge of mathematical questions 

_ . involving thu doctrine of the sphere. 

Question. 

tell me what will bo the several amounts 
of sines of amplitude, -declination and tho kujya (where tho 
l’ALALiiA is 5 digits) when their sym is 2O00.f 


• Solution. Hero rAT.ATinJi = 5 digits • 

Hupporo the sine oi ileoliu.it ion = 5 X 12 = (10: 
and then say. If palabha : aksiiakauxa : : sine of decln. : ba if a b'ahku 

13X00 

.or 5 : 13 : s GO : sama b'a1IKU:= = 1GG , 

G 

150 X 13 

Gnomos : ajcshakaBna : : sama s'anku : tadduhiti = = lGi) , 

* 13 

GO X 5 

. 12 : PALAnnA ' s : sino of ducln. : KUJYA = = 25 , 

12 


GO X 13 

and 12 : akbuakakxa : : aino of decln. : sine of am]ilitudu = =65 . 

12 

If tin- sum : sine of deeln. supposed : : given sum : siric of decln. required, 
or -175 : GO : : iiSlJO : 12u0 . 

If 475 : 166 : : *J5U0 : 3120 bama s'anku required, 

and so on 33S0 tauuiuiti 

GOO KUJYA 

1300 siuo of amplitudo. 

Ansr. 

• L.AV. 

t Solution. Iloro also Palabtta = 5, 

tbdn suppose sino of «kt-li nation as before = 60 , 
and.*. sine of aniplit udo • = GG , 

and kujya = 25 . 



250 


Translation of tfa 9 


[XIII. 36. 


Questions. 


36. But dropping for a moment those questions of the, 
siddhXntas involving a knowledge of • 
the doctrine of tho sphere, tell me, ray 

learned friend, why in finding tho pqint of the ecliptic rising 
above the horizbn at any given time, (that is .the lagna or 
horoscope of that time,) you first calculate- the Sun's apparent 
or true place for that time, i. 0 . the Sun's instantaneous place w 
and further tell mo, when the Sun's s A van a day, i. e. terrestrial 
day, consists of GO sidereal ghatikXs and 10 pa las, the lagna t 
calculated for a whole terrestrial day should be in advance of 
tho Sun's instantaneous placo, and the lagna calculated for the 
time equal to tho terrestrial day minus 10 palas should bo 
equal to tho Sun's instantaneous place. 

37. Are tho ghajikas used in finding the lagna, ohatik/s 
of sidereal or common sXvana time? If they are sXvana 
guatikas, then toll me why are tho hours taken by the several 
Bigns of tho ecliptic in rising, i# 0 . tho rXs'yudaya which are 
^sidereal, subtracted from them, being of- a different denomina- 
tion ? If on tho other harfd you say they aro sidereal, then 
I ask why, in calculating the lagna for a period equal to a 
wholo sivANA day i. 0 . GO sidereal gjtatikas and 10 palas, tho 
lagna does not correspond with, but is somowhat in advanco 
of, tho Sun's iuBtantanoous place ; and then why the (Sun's 
instantaneous place is used in finding tho lagna or horoscope.* 

38. Given the length of the shadow of gnomon at 10 UAffs 
aftor sun-rise equal to 9 digits at a 
place where tho palabhX in 5 digits; 

toll mo what is tho longitude of the Sun, if you aro an fait in 
solving questions involving a knowledge of the sphere.f 


.Question, 


Then say as before 

as 160 : 60 s : 2000 : 800 line of declination, 

as •‘ISO 1 65 : : 2000 : 866f aino of amplitude, 

aa 160 : 25 : : 2000 : 8331 kujya.— L. W. 

• [For answers to these questions see the note on the 27th. Terse df the 7th 
CI.-B.D.] ... 

t [For solving this question, it is necessary to define some lines drawn in tho 
Annulary sphere and show some of their relations. 



&iih Ih ant a -8 ’ iromant. 251 

» 

mo, (J Astronomor, what is tho palabhX at that 
* placo whero the gnomoh’s shadow fall- 
ing due west is equal to tlyi gnomon’s 



XIII. 39.] 
39. Tell 

^Question. 


Iiet B C D E be meridian of tho given place, CAE tho diameter of I ho 
IJorizc,i, B the'Zeuith, V ami Q tne north and south poles, 11 A D the diameter 
of the Prime Vertical, FAQ that of tho Equinoctial, P A Q that of the six 
o'clock line, H/L thut of one of tho diurnal circle*, a the Sun's projected placo 
in it and /A, $ w, H * perpendiculars to C E. Then 
11 F or E P = tho latitude of the place, 

A /= the sine of the Sun's declination, 

A rt = agba or the sine of amplitude, 

fg = kujya'. (It ii called charajya' or aino of the ascensional difference 
when reduced to the radius of a great circle ) 
f 9 — kala'. (It is called b£tba when reduced to the radius of a great circle ) 
tg — isu?a hriti. (It is called taddhriti when a is at e, hriti when s is 
at H and kujya when « is at/.) 

The miTA h*iti reduced to the radius of a great circle is called ishta antya , 
but a coincides with H, it is called antya' only. 

It is evident from the ligure above described that 

(1) ISnTA H$ITI = KAIiA' + kujya', 

(2) isej-A antya' = SUTRA ± charajya', 

(3) hriti = dyujya' or cosine of declination dt kujya', 

(4) Antya' = radius ± charajya'.* 

Here the positive or negative sign is to be taken according as the Son is in 
the northern or southern hemisphere. 

" . u,2 



252 


Translation of the [XIII. 39. 

height when tho Sun is in the middle of flio sign Leo, i. o. a 
when his longitude is 4 signs and 15 degreed.* 

• 

Now nt a gitfen hour of tho day, tho isiita hkiti and others can be found 
as follows. 

Half tho length df tho day diminished by tho time from noon (or^ho nata 
Xa'la properly so cullod) is tho unnata kala (or elrvrfed time). Subtract from 
or add to tho unnata ka'la tho ascensional difference according as the Sun id 
in tlio northern or sunthprii hemisphere i reduce the remainder to degrees : tho 
sine of the degrees is BC/tra. Tho b(jtra multiplied by the cosine of declination 
and divider! by the radius gives tho kala'. Then from the abore formula) wo 
can easily find the ibitta hkiti and others. 

Now to find tho answer to tho present question. 

Square the length of tho Gnomonic shadow and add it to the square of tho • 
Gnomon or 144 : and square-root of the sum is called tho liypotbcnuse of the 
shadow. From this hypothenuse find the Maha's'anku or the sine of the Sun’s 
altitude by tho following proportion. 

As tho hypothenuse of the sliadow 
: Gnomon or 12 
: ; Radius 

: Tho MAiiAfl'Ainrv or the sine of the Sun’s altitude. 

Then by similar latitudinal triangles, 

as the Gnomon ot 12 digits 

: AK8HA xarna fpuud from given palabha' 

: : MAItAS'ANKU • 

: isiita nmn (see verses from 45 to 40 of tho 7th Chapter). 

Reduce the given unnata ka'la to f degrecs and assume tlio sine of the degrees 
as iBuyAHTYA (for this will always bo very nenr to tho isiirA'MTYA). Then 
t casino of declination = isuta nmn 

Radius ihhta'ntya 

From this the cosino of declination will nourly bo found, and thence tlio 
declination and ascensional difference cun also bo found. From the ascensional 
difference, just found, find the ibuta'ntya' of two kinds, one when the Sun is 
supposed to be in tho northern hemisphere and the other when the Sun is 
supposed to be in the southern hemisphere. Of these two ISBtantya's that is 
nearly true which is nearer to the rough ibuta'ntya' first assumed (i. e. tlio bine 
of the unnata ka'la). From this new isiita'ntya' find ugain.the declination 
and repeat the process until the roughness of declination vanishes. Frdin tho 
thrlinnlion, last found, the longitude of the Sun can bo found.— 1). D.] 

* Tlio hypothenuse of the shadow is first to be found. Then say 
As hypothenuse of the shadow 
' : Gnomon 

x : Rud 

: the maba' b'anku or the sine of the Sun’s altitude. 

Here wo shall flud sine of 45°. This is the sama b'ank u. 

It is 2431' signs 

Sine of dcolination of the Sun when in 4 . . 15 s = 987' 48'' 

243?'! * — 987' 77 48""'| * = (taddbeiti — kujya')* 

or 5909761 — 975749 .. 9 = 4 93 10 11 51L 

TADDHKITI — mcjya'= ^4931011 .. 51 = 2221' .. 15" 

Here wti have 3 sides of the latitudinal triangle consisting SAVA B'ANKU, 
declination and taddhkiti — kujya'. Hence wo may find the latitude. 

Then by similar latitudinal triangles e 

Alt taddhkiti — kcjya' 2221* . 15' , 

: sine of declination 987' .. 48" 

• : : Gnomon 12 
S PALABHA'Oi digits.—!. W. 



XIII. 40.3' Siddhdntas'iromarii, 258 

40. When th<£ Sun entors the prime vortical of a person 

- * at cjjayin! either at *5 gratis after 

Question, 

sun-rise or 5 gratia before or after 
midday, what are his declinations ? 'If you will answer mo this 
I will hold you to 1*> the sharp ankusa (goad)'for the guidance 
of tlio intoxicated elephants, the proud astronomers.* 

* First of all assume IT*N tlio tad- 
Dhriti = sine of tlio givm devilled 
timo that is = sin 30". From this 
find the b'anktj or the sine of ullitudq 
by similar triangles. 

Tt aksha k SUVA or hypothenueo s of 
equinoctial shadow. 

: Gnomon 13 
: : TADDUSITI 

12XTADDHMTX 

: SAMA 8'AWKU = =S 

13 

ON 

From 0 N, to find O B the sifto of declination sny 

PALABIIA' 5C o N 

as AKfliiA kahn a : falaiiiia 7 : : ON: OB = ■■ ■— - = sine of do- 

• . 13 

clination. * 

From O B we may now find tlio longitinlo of tho Sun and 0 D tlio a«oen- 
atonal difference: Now deduct this iisc«ibioiu 1 dilleronce from tlio sine <ff 
elevated time converted into degrees, Hence 

U D — 0 1) = C 0. 

Now reduce 0 0 to terms of a small circle on the supposition that tho Sun 
lias the declination now found. 

As Rad : C O : : cosine of declination : N B. 

Now find also B A by thu sumo proportion. 

Then N B +,11 A = N H* a new value of taddiirttt. 

Iffil N : gave OB:: 1IN': OB' corroded value of 0 B. 

Hence a corrected longitude of the Sun. 

The operation to be repeated till rightness is found. 

2nd. — To find the declination from tho bata ka'la or timo from noon =3 
ain 30°. , 

Let a = the sine of kata ka'la : R* — o* = stiTBA*, 

.nd x = tho sine of declination : R* — x 3 = c<>» k of declination. • 

The uutra reduced to value of diurnal circle will give kala' 

The proportion is. As R : stiTBA : : cos of declination : kala', 
but 1 do not know what cos of declination ia but only Us square. 

I must therefore malto this proportion in squurcs 

(B> -«»)(»>-*») 

As R* : stirs a* : : cos* of declination : kala ) ! = ■ 

K* 

• Nowhy sim ilar la titudinal tri angle s • 

As u) * : * : : kala^I 1 : «n«* of declination 

’ SHISiiy 25 (R’-«*)(R*-**) 

A sine* of decliuation = •* x kala ] 1 — • ; — X 

* 141 R a 



254 


Translation of the 


[XIII. 41, 


41. In a place of which the latitude is Unknown and on 

Question.' " a da ? which is ™known, the Sun was 
. observed, on entering the prime ver- 

tical, to givo a shadow of 10 digits from a gnomon (12 digits 
long) at 8 ghatikXs after sun-rise. If yofi will tell ifie the 
declination of the Sun, and the palabhX I .wiR hold you to be 
expert without an equal in the great expanse of the questions 
on directipns space and time.* 

42. 0 Astronomer, tell me, if you have a thorough know- 

Quoition. Iedg0 tlie la^ibudinal figures, tho 

palabuI and the longitude of tho Sun 


Now R*— <s* = 8884883 

25 (R* — a*) = 25 x 8864883 = 221622075 
and 144 R* = 144 (3438)* = 1702057536 
221622075 (R* — x*) 

— — = i 1 , 

1702057536 * 

• 1702057536 

E* — ** x* = 75 Nearly 

221622075 


8 B* 

.*. 26 a?* = 3 R* : ** = = 1363828 

26 

and x = ^1363828 = H67' = sine of 19“ .. 61 # 

Hence the Sun's place may be found.— L. W. 

9 To find the sine of altitude or maha' b'avkv 

(16)* + (12)* = (20)* ... hypothenuse of the shadow = 20. 

Thou say 

4s 20 : 12 : : 3438* : 2062' .. 48" = the maiia' b'ankit. 

Now supposo the siue of ukitata ka'la or 8 guatika'b to be the taddh^iti 
•— 2655. ' 

Then by similar triangles 

2655' X 12 

2062' .. 48" : 2655' 1 : 12 : AES BA XASpi ss — 

20624 

From this find the palabha'. 

To find deelination says 

As AESBA KA1*A j palabha' : : 2062 / .. 48" : sine of declination. 

From this find the coiiite of declination, the kum, the ascensional difference, 
Ac. The ubnata Xala diminished by the ascensional difference gires the time 
from 6 o'clock : the sine of this time will be the sOtba and hence the kali : 
thenle (BUJia/ being added) the taduhbiti : and thence the AKsna kabna and 
dcolination. The operation to be repeated till the error of the original asaiimp- 
i turn rauiiWs.— L. W, r 



. XIII. 43.] Siddlidnta-sfiromani, 255 

• » 

.at that place, whero (at a certain time) the ku rue is equal to 
• 245 jand the TADDft^LETi is equal to 3125.* * 

* 43. Given the sum of the 3 following quantities, viz. of the 

A .. . sines of declination, and of the alti- 

Questipn. * i 

* tude of the Sun (when in the prime 

vertical) and of the taddhrtti decreased by tho amount of tho 
„ kujyA equal to 6720, and givon also tho sum of tho kujya, the 
Bines of amplitude and declination (at tho same time) equal 
to 1960. I will hold him, who can tell mo the longitude of 
tho Sun and also palabh^ from the given sums, to bo a bright 
iustructor of astronomers, enlightening them ns tho Sun makos 
the buds of tho lotus to expand^by his genial heat.f 


, • Ansr. Let x = the palabha 

then say. Ai x : 12 : : 245 : sine of deolination = 

Now find the taddk jm minut kujya'. 

* 2910 * . 35280 

As * : 12 : : — : taddheiti — kujya = 

X X * 

But TADDHEITI — XUJYA zp 3125 — 215 = 2880. 
35280 35280 49 


2910 



.-. 2880 = and x' = = — 

*» 2880 4 

... a? = $ = 3 | PALABHA. 

To find declination say 
As 3k : 12 :: 245 i 840 sino of declination, 
llenje the lorigitudo of tho Bun may bo discovered as before.— L. W. 
f This question admits of a ready solution in eonscqueuoe of its peculiarities. 
The sine of declination ) 

sama8'avxv> =6720 

and TADDHRITI — KUJIA ) 

are all three respectively perpendiculars in the three latitudinal triangles. 

And the kujya 

the sine of amplitude 
and the tadduaiti — kujya 
are bases in the same 3 triangles. 

lienee we may take tho sum of tho 3 perpendiculars and also the sum of the 
three bases and use them to find tho palabha. 

As the sum of the ) Bum of the 3 bases Gnomon palabha 
3 perpendiculars) in the same triangles 

1960 X 12 s 

6720 s 1960 :: 12 : = 3J. * 

. 6720 

Now the kujya, sine of amplitude^nd sino of declination are the threo tides 
of a latitudinal triangle. Tlieae throe 1 may compare with tho threw Gnomon, 
palabha and aksua KABpA to find tho value of any one. • 




Question. 


Question. 


256 , Translation of tho [XIII. 44, 

« 

44. Given tho sum of tho sine of declination, sine of tho, 

Sun's altitude in pfime vertical and 

Question. , *• 

the taddh$iti minus kujya equal to 

1440', and given also tho sum of the sine of amplitude, tho 

sine of tho SunValtitudc in prime vertical *«and the taAdhriti 

equal to 1800', I will hold him, who having observed tho 

given sums.* , f 

45. Given the equinoctial shadow equal to 9. Wliat longi- 
tude must the Sun have in that lati- 
tude to givo an ascensional difference 

of threo OTiAfis ? I will hold you to be the best of astrono- 
mers if you will answer mo this question, f 
40. Hitherto it has been usual to find tho length of tho 
Sun's midday shadow, of tho shadow 
of tho Sun when in tho primo verti- 

« 

But tho aksha kabna must be first found to complete the sum of thoso three. 
AKSHA KAUNA = y = \/~ = J 
Gnomon = 12 

fa lamia = 3$ > = 2P sum of tho 3 sides of a latitudinal triangle. 

AKKIIA KAUNA = 124 3 

flow if 28 : 12 : : 19fi0 : 840 the sine of declination. 

Hence the plnce of tho Sun as before. — L. W. 

• This question is similar to the preceding. 

In tho first sum wo have the sum of three perpendiculars -in threo dilFercnt 
latitudinal Triangles. In the second we have tho sutn of tho throo J i)' pot lie* 
mines of those same throe Triangles. Houce wo inn)- say. • , 

sura 3 per. sum of 3 corresponding hy. .Gnomon aksuakabka 
A s 1440 : 1800 :: 12 : 15 

Now from axsiia kauna to find PAXAnnX 

„ PALADIlX = == y/bi = 9. 

Now sine of amplitude, sine of the Sun’s altitude in tho Frimo Vertioul, and 
' the TADDHurri are the three sides of a latitudinal.— L. \V. m 

f Let x zz sine of the Sun’s declination, 
then 12 : 9 £j kujyX = $ r. 

Again a* = cosine of declination. 

Then as R s cos of declination : : sine of ascensional difibrcc. : kujya 
S ine of asceusl. diflVe. or ciiabajya = sine of 3 GUatis == sin lb* = 10G2'. 
cosin of dccli). % ciiabajya 

— — zz KUJTA r 

B 

# 1002 . 

dr = ^ * , 

B 



Siddhdula-s'iromani. 


257 


, XIII. 46.] 

. » 

eal, and wlicn in ha iutermodinfco circle (i. o. when lie has an 
, * azimuth of 45*) by three different modes of Calculation : now 
Bo who will by a single calculation toll me the length of thoso 
tlireo shadows and of tho shadows at any intermediate points, 
at tho Arish of the querist] shall bo hold to bo a' very Sun on the 
Earth to expand the lotus-intcllects of learned ^tronomers.* 

, • [tTere the problem it 'this Giron tho Sim’s declination or, amplitude, 
tho Equinoctial shadow of tho placo and tho Sun's azimuth, to find tho Sun’s 

^ FoTsolring this problem BhXskarXciiAkya has stated two different Rules 
* in the Ganit£dhy4ya. Of them, wo now shew hero tho second. 

“Multiply tho square of the Radius by tho square of the equinoctial shadow, 
and the square of the cosine of the azimuth by 14k The sum of the products 
divided by the difference between tho squares of tho cosine of tho azimuth and 
the eino of the amplitude, is called the pratuzma (lirsl) and the continued 
product of the Radius, equinoctial shadow and tho smo of the sinphtuilo 
divided by the (same) difference is oalled the anya (second). Take tho square- 
root of the square of tho ANYA added to the PUATIiama: this root decreased 
er increased by the ASYA according as the Sun is m tho northern or southern 
hemisphere gives tho hypothenuse of tho shadow (of the Sun) when tho Sun 
is in any given direction of the compass.” , 

“But when the cosine of rtie azimuth is loss than tho sine of the amplitude, 
take tho square-root of the square of tho ANYA diminished by tho pratiiami : 
tho anya uocreased and increased <sepa*iteh) by tho square-root (just found) 
gives tho two values of tho hypotUunuwi (of tho Sun s shadow) when ihu Sun 
is in the northern hemisphere." • 

This rule is proved algebraically thus. 

Lot a =P the Bine of amplitude, 

A = the sine of azimuth, 

and x 5 the hypotlmu use' ot Idio* shadow when the Sun is in any given diroc- 
tion of tho compass. 

Then suy |g ft 

as^j? : 12 : : R : the MAUA s'anktj or tho sino of tho Sun’s altitude =: 

and tho siue of tho Sun’s zenith distance ~J R*-^— ^ 

12 R ' R 

ITOW, as 12 ! « = s S'ANKUTALA = • 

X x 

... BiHU ortho sino of an are of a circle of position contained between tho 
Sun and the Prime Vertical = a ? ~ s (»co Ch. VIL V. 41) here .lie sign- 
or + is used according as the Sun is in tho uorthern or southern Hemisphere. 
Then say 

a «R 

as — 144 : » + — : s R : A s 

jf x 

t R 



2o8 


Trnnshdion of the [XIII. 47. 

47. no who, knowing both the azimuth anil the longitude, 
_ • of the Sun . observes buo shadow of the 

Question. 

gnomon at any time, or he who know- 
ing the azimuth observes two shadows and can, find the pala- 
bha, 1 shall conoeive him to be a very (Uku^a in destroying 
conceited simkqjf of astronomers. . ‘ 

[On this IIhXskabachatiya has given ^n example in the-GA^i-, 
tXijhyXya, as follows. 

“ Given the hypothenuse of tho shadow (at any hour of tho 

. day) equal to 30 digits and tho south 

Example. ' , . 

bhuja* oqual to 3 digits : given also 


or A \/ a 4 — 1 It = ur +«R; 

A 8 -r 8 -144A 8 = a 8 x a *2Reax + e 8 R 8 ; 

(A* — a 8 ) x* ± 2 K e a x = e 8 It 8 + lift A* ; 

K e a e* K* + 14i A* 

** i 2 * = i 

A 8 -o 8 A 8 — « 8 

or * s + 2 ANYA X ±Z PliATHAMA (1) 

.*. X 3 ± 2 AN YA X + ANYA 8 = PRA T1IAMA + ANYA 8 

and x = \/ pkathama + anta 8 £ anya. 

But when A xZL a and the Sun is in the uorthorn hemisphere, the equation 
(1) will be x 3 — 2 anya * = — jmiathama , 

and then x = anya ± anya 8 — first : 

i. e. the value of the hypothenuae of tho shadow will be of two kinds here. 

llouco the Buie. 

Bhaskauachahya was the first Ilindu who lias given a general rnlo for 
finding the Sun’s shadow whatever be the uximutli ; uud he was the first who 
lias shewn that in certain cases tho solution gives two diilcreut results,— B. 1).] 

* [On a levelled plane draw east and west and south uud north, lines and on 
their intersecting point, place Gnomon of 12 digits: the distunco betweef. the 
end of the shadow of that Gnomon and the east and west line is culled tho 
BUVJA. 

It is to he known here that tho vnluo of the great hhuja (as stated in 41st 
yerse of the 7th Ch.) being reduced to the liypothenuso of tho shadow becomes 
equal to the bhfja (above found). 

Or as tho Radius • 

: the great bhfja 
: : the hjpothenuse of the shadow 

: the reduced buvja or the distance of the end of the shadow from tho 
oast and west line. 

This reduced bhuja is called north or south accoiding as tho aid of llio 
shadow falls north or south of the eu^ and wcot line. 

It is very clear from this that the reduced bhcja will be tho cosiuo of tho 
aximuth in ft small circle described by tho radius equul to the shadow. 

Or as tho shadow 

: the reduced bhuja • 

«. i : radius of a great circle 
: the cosine of tho azimuth. 

, This is fhe method by which all Hindus roughly determine tho azimuth of 
the Sun from llie bhfja of his gnomonie shadow.— B. D.]* 



Sithlhdnta-s'ii'oHMQt. 


259 


, XIII. Id.] 

i % 

Jlic hypothenuso equal to 15 digits, and the north ihiim a equal 
, to 1 digit, to fiiftl the PATiAWTi. Or, given •the declination 
efyual to 81(5 and only one hypotlicnuso and its corresponding 
imuja at the time, to tind the PAijAiwX.”] 

48. First of all jnultiply ono biiuja of tlio* shadow by tho 

. • . hypotlicnuso of tho other, and tho se- 

4 cond biiuja by tho hypotlicnuso of tho 
first: then take the difference of those two biiuja* thqs multi- 
plied, if they aro both north or if both south, and their sum 
if of different denominations, and divide the difference or tho 
sum by tho difference of the two hypothenuses j it will be tho 
l'AiiAinrX.* 

49. How should ho who, like a man just drawn up from tho 

bottom of a well, is utterly ignorant of 
^Question. palabitX, the place of the Sun, tho 

points of tho compass, the number of the years elapsed from 


* Tho nilo mentioned hero for finding the YUMIM* when tho two shadows 
and their resjiectivo DnuJAfl arc given, u proved thus 
Let A, = tho first hypothenuee of the shadow, 
b x = its corresponding BIIUJA, 

A a = tho second liypothenu&e, 
and b a = its corresponding biiuja, 

Then 

12 R 

As A, : 12 : : R : — the first maiii r'aHki- j 

. //, 

4 12 R , 

and in the same manner — — ■ = the second mama s'aitku j 
h a 
h x R 

and also as h x : h % : : R : = the first great BiirJA , 

h 

b a R 

., == the second great rum , 

& 2 

b x R __ h H R 
h a 

Then the r ala Bn a' (■» Ch. XI - v * M ) 

12 R 12 K 

~ K 

l x h a -h b a h x 
h x — h 

Hence tho Rule.— L. \V . 



Question. 


2(50 Translation of the [XIII. 50. 

« 

tlio commencement of the yuua, the mouthy tho tithi or lunar, 
(lay and -the day of week, being asked by others to tell quickly 
tho points of the compass, tho place of the Sun, &c., givc*a 
correct answer ? lie, however, who can do so,, has my humblo 
reverence, and what astronomers will not acknowledge him 
worthy of admiration ?* ' 

50. He, who can know merely with tho staff in his 1 hand, 

. . tho height and distance of a bamboo 

QuoBtion. 

of which ho has observed the root and 

< 

top, knows tho use of that instrument of instruments — Genius 
(the dh(yantra) : and tell mo what is there that he cannot 
find out ! 

51. There is a high famous bamboo, the lower part of 
which, boiug concealed by houses, &£. 
was invisible: the ground, howovor, was 

perfectly level, it you, my friend, remaining on this same spot, 
by observing the top, will tell mo the distance and its height, 
I acknowledge you shall havo the titlo of being the most skil- 
ful of observers, and exporfr in tho use of the best of instru- 
ments, dhIyantra. 

52. Having seen only tho top of a bamboo reflected in 

water, whether tho bamboo bo near or 
at a distance, visible or invisible, if 
yon, remaining on this samo spot, will tell me the distance and 
height of the bamboo, I will hold you, though appearing on tho 
.Earth as a plain mortal, to have attributes of superhuman 
, knowledge, t • 

53. Given the placos of the Son and tho Moon increased 
by the amount of the precession of the equinox, i. e. tboir 
longitudes, equal to four and two signs (respectively) and tbo 
place of the Moon decreased by tho place of the ascending 
node equal to 8 signs, tell me whether the Sun and the Moon 
have the same declination (either both south or one. north 


* This refer* to the 84tl> verse of the Ch. XT.— L. W. 
t [Answers to these queetious will be found in the 11th Ch. — B. D.] 



201 


XIII. 59.] SiihlMnta-tfmmani. 

t nnrt ono south), it you have a perfect acquaintance with tlio 
DiiIveiddhida Tantra. 

9 54. If tho place of the Moon with the nmouijt of tho pre- 
cession of the equinox bo equal to 100 degrees, and the place 
of the' Sun increased by tho same amount to 80 degrees, and 
the place of the* Moon diminished by that of tho ascending 
.node ^equal to 200 degrees, tell me whether the Sim and tho 
Moon havo the same declination, if you have a perfect acquain- 
tance with the DhIveiddiiida Tantra. 

55. If you understand tho subject of the 1 ’ata i. e. tho 
equality of the declinations (of the Sun and the Moon), tell 
me tho reasou why there is in reality an impossibility of tho 
i’Xta when there is its possibility (in the opinion of Lama), and 
why there is a possibility when there is an impossibility of it 
(according to tho same author). 

50. If tho places of the Sun and tho Mooh with tho amount 
of tho pveccssion of tho equinox bo equal to 3 signs plus and 
minus 1 degree (i. ,o. 2 8. 29® and 3*. 1° respectively) and 
the place of the Moon decreased by that of the ascending 
node equal to 11s. 28 f , tell me whether the Sun and tho Moon 
have the same declination, if you perfectly know the subject. 

57. (Tn the DnfvRiDDiiiDA Tantra), it is stated that the i*Xta 
in to ram cm some places when it has already taken plaec (in 
reality), and also it has happened whoro it is to come. It is 
a strange thing in this work when tho possibility and impos- 
ibility of tho pXta are also reversely mentioned. Tell mo, 
O you best of astronomers, all this after considering it well.* . 

58. I (Bh^skara), bom in tho year of 1030 of tho S'ali- 

Dtf e of tlie Author*, birth viflANA era, have composed this SlD- 

and his work. dhInta-s'iromani, when I was 30 

years old. 

59. He who has a penetrating genius like the sharp point 

of a largo darbiia straw, is qualified 

Author's apology. ^ compose a good work in maflie- 

* [Answers to these questions will be found in the lost Chapter of fho Oaxita- ■ 

DAYAYA.— B. D.] 

• •• 



262 Translation of the Siihlhanta-s iromuni. [XIII. GO. 

matica : excuse, therefore, my impudence, <5 learned astrono- 
mers, (in- composing this work for which I anf not qualified). 

60. I, haying lifted my folded hands to my forehead, beg 
the. old and young astronomers (who live -at. this time) to 
excuse mo for hating refuted the (erroneous) rules prescribed 
by my predecessors ; because, thoso who fix their belief in the 
nilcs of the predecessors will not knqw what is tho truth , 9 
unless I refute the rules when I am going to state astronomical 
truths. 

61. The learned Maiies'waka, the head of all astronomers, * 

tho most good humoured man, the 

Author’s birth-place, Ac. .? . ........ 

store of ml sciences, skilful in tho 

discussion of acts connected with law and religion, and n duaii- 

mana descended from S'antmlya (a muni), flourished in a city, 

thickly inhabited by learned mid dull persons, virtuous men 

of all sorts, and nfcn competent in the three Vedas, and situated 

near tho mountain Sahya. 

62. Ilia son, tho poet and intelligent Biiaskara, mado this 
clear composition of the SiddjiXnta by the favour of tho lotus- 
liko feet of his father ; this SiddhXnta is the guidance for 
ignorant persons, propagator of delight to tho learned astro- 
nomers, full of easy and elegant style and good proofs, easily 
comprehensible by the learned, ami remover of mistaken ideas. 

63. I havo repeated hero some questions, which I fiavo 
stated before, for porsons who wish to study only this Pkas'na- 
PHyXya. 

6t. The genius of tho person who studies these questions 
becomes unentonglcd, and flourishes like a creeping plant 
watered at its root by the consideration of the questions and 
answors, by getting hundreds of leaves of clear proofs, shoot- 
ing from the Spheric us from a bulbous root. 

n ■ * 

End of tho 13th and hist Chapter of the (IolXdhyaya of the 
Sid'dhanta-s'iroma^i. 



.APPENDIX. 


ON THE CONSTRUCTION OF THE CANON OF 
SINES. 

1. As tlio Astronomer can acquire the rank of an A'citahya 
iti tho science only by a thorough knowledge of tlio inode of 
constructing tlio’ canon of sines, Huaskaka therefore now pro- 
ceeds to* treat upon this (ifitcrosting and manifold) subject ill 
the hope of giving pleasure to accomplished astronomers. . 

2 and 3. Draw a eircle with a radius equal to any number 
of digits : mark on it the four points of tlio compass ami 300". 
Now by dividing 90° by the number of sines (you wish to draw 
in a quadrant), you will get tlio arc of the first sine. ^ This 
arc,* when tnultiplied by 2, 3 Ac., will successively bo tlio arcs 
of other sines. Now set off the first arc on the circumfcrenco 
oil both sides of one of the points of tlio compass and join tho 
extremities of those arcs by a transverse straight lino, tho half 
of' which should be known tho sine of the first arc : All tho ‘ 
other sines are thus to be known. 

4. Or, now, I proceed to state those very sines by mathe- 
matical precision with exactness. The square-root of tho dif- 
ference between the squares of the radius and the si no is cosin o. 

5. Deduct the sine of an arc from tho radius tho remainder 
will hb the versed sino of the complement of that arc, and jtlie 
cosine of un arc deducted from the radius will give the versed 
sino of that ore. Tlio versed sine has been compared to the " 



204 AjipcmUjc. 

arrow between the bow and tlio bow-string : but here it has # 
received tho nauio of versed-sine. 

6. The half of the radius is the sine of 30° : the cosine of 

00°. will then bo the sine of 60°. The square-root of half 
square of radius vfrill bo the sine of 45«. « * 

7. Deduct the square-root of five times tho fourth power 
of radius, from fivo times the square , of radius and divide m 
the remainder by 8 : the square-root of the quotient will bo 
tlio sine of 3f> 0 . 

= sino 360.* 

v 8 

8. Or tho radius multiplied by 5878 and divided by 10000 
will give tho sino of 30°, (wlioro tho radius = 3138.) Thp 
cosine of this is tho sino of 5 1°.+ 

9. Deduct the radius from tho squarO-root of tho product of 


• [Tina is proved thus. 

Let a = sine 18° ; and R — a = covers IS 0 or vers 72°. 

Then = sine V° : (>ce tho lOtli verse.) 

or y/ R = sine 36° i 

a/6 K* — R 

but a = ~ (see tho 9th torse) 

, .!». 3G» 1),J 

K X 5878 

t The Rule in 8tli verse vie., seems to be the same ns above ami 

10000 

to be deduced from it t 


</l - 2.237411 &c. 


— j/£ 
8 


and.'. 5 — i/b = 2.702589 which divided by 8 = .315323 • 

R X 5878 

sine 4 g° = R ^/.aiiaaa = K X .W>78 = L. W. 


10000 



AppenilU, 


2(55 ' 

tho square of radius and live and divide the remainder by 1 : 

( * the quotient thus* found will give the exact sine of 18°.* 

• 10. Half tho root of the sum of tho squared of the sino 
and versed sine of any arc, is tho sino of half tfiat arc. Or* 
the sine of half that arc is tho square-root of? half tho product 
of the radius and tlio versed sine. 

11.. From tho sino of auy arc thus found, tho sino of half 
tho arc may bo found (and so on with tho half of fliis last). 
In like manner from tho complement of any arc may bo ascor- 
• taiued the sino of half tlio complement (and from that again 
. tho sino of half of tho last arc). 

Thus tho former Astronomers proscribed a modo for deter- 
mining the other sines (from a given one), but I proceed now 
to give a mode different from that stated by them. 

‘ 12. Deduct and add tho product of radius and sine of 
miuja from and to the square of radius and extract the square- 
roots of {he halves of the results (thus found), tlieso roots will 
respectively givo tho sines' of flic half of 00° decreased and 
increased by tho buuja. . 

In liko manner, tlic sines of half of 00° decreased and in- 
creased by the koji can bo found from assuming the cosine 
for the sine of bhuja. 

13. Take the sines of mifJAS of two arcs and find their 
difference, then find also the difference of their cosines, square 


• [This is proved linn. 

Let O be centre of tho circle ABE 
and ^ C = 3G<\ then A B = 2 mu 
18°, an«l • (C A 13, C13A) each of 

tlioin sss 2 Ua 

Draw AD bisecting the Z. CAE, 
then AB, AD, CD will be equal to 

each other. , . 

Now let x = «n 18°, then by simi- 
lar triangles CB : AB = AB .• BD or 
V — 2-r; 

/. 4 x 1 = B* s — 2 B x which gives 
i 



V 



266 Appendix. , 

these differences, add these squares, extract their square-root 
and halve it. This half will be the sine of half the difference , 
of the sines.* Thus sines can bo determined by several ways! 

« 14. The square-root of half the square of the difference of 
the sino and the <cosine of the bhuja of an, arc is cqual*to tho 
sine of half the difference of the bhuja and its complement, f 
I will now givo somo rules for constructing sines without^ 
having recourso to tho oxtraction of roots. 

15. Divide tho square of tho sine of the bhuja by the half 
radius. Tho difference between the quotient thus found and * 
tho radius is equal to tho sine of the difference between tho . 



t Let be ss sine of any are and bg = its 
cosine. 


Draw the sine ad = cosine bg, then ah its 
sine will be caual to be and a/zzfl : 

+/»* = aW\ but as a/* 

. ah* of* ah* J 

af*zz—t and — = — ; ' 

2 2 4 



ah 

L. W.l 

2 



Appendix. 267 

, degrees of BHUJA*and its complement.* In this way several 
sines may bo foufid hero. ' 

[As these several rules suflico for finding only the sines 
of arcs differing ■ by 3 degrees from each other and not the 
sines 8f tho intermediate arcs, tlio author therefore now pro- 
ceeds to detail *the mode of finding tho intermediate sines, 
.that is the sine of evqry degree of tho quadrant. This mode, 
therefore, is called Pratibhagajyaka-vidhi.] 

16. Deduct from the sino of bhuja its part and divido 

Buie, for finding U„ .in. tLo ton - fold sine of K0 T' S73 ' 

• of every degroe from 1° to 17. Tho sum of these two results 
90°. 

will give tho following sino (i. c., the 
sine of bhijja one degree rnoro than original nnujA and tho 
difference botween tho same results will givo tho preceding 
sine, i. c., tho sino of bhuja ono degree loss than original 
bhuja). Hero the first sine, i. c., tho sino of 1°, will bo 60 and 
the sines of the remaining fires may bo successively found. 

18. The rule, however, supposes that tho radius = 3138. 
Thus the sines of 90° of tho quadrant may bo found. 

Multiply tho cosine by 100 and divide tho product by 1529. 


Buies for finding tho 21 
sines vis., of 3°i, 7°J, 11°}, 
15°, &o. 


19. And subtract tho part ot 
the sino from it. Tho sum of thoso will 
bo tho following sino (i. o., tho sino 


of hre of 3°f degrees more than original arc) : and tho difler- 


• Let ab be any arc, and at = ah, 
then ad = its complement, 
ed = their difference, 
and bo = 2 ab. 

Now = sin — or sin ab, 

V a a 

B % vers bo 

or . ' = sin* ab, 

2 

2 sin* ab 
or wrikto as - ■ - , 

B 

tin* ab 

then B — vers 6c or sin rrf = R — 

*» 






2C8 


Appendix. 

• 

cnco of them will bo procoding Bine (i. e., the sine of arc 3°J 0 
degrees less than original arc). 

20. Bat the first sino (or the sine of 3 0 $) is Lore equal to 
22'1# (and not to^225 as it is usually stated to. be). Bjr this 
rule 2 1 sines may*bo successively found.* r 

21 and 22. If the sines of any two arcs of a quadrant bo 

K»l« tor finding tiro ...... multi P liod ^ thoir COsineS «cipr8c*11y . 

of sum nml diilercnce of any (that is the sine of tho first arc by 
w wc§ * the cosine of the 2d and the sine of , 

tho 2d by tlio cosine of tho first arc) and the two products 
divided by radius, then tho quotients will, when added to- 
gether, bo tho sino of tho sum of tho two arcs, and the differ- 
ence of those quotients will bo tho sino of their diffcrenco.t 
This excellent rule called jya-bhXvanX has been prescribed fo&. 
ascertaining tho other sines. 

23. This rulo 'is of two sorts, tho first of which is called 
samXsa-bitavanX (i. e., tho rule for finding tho sine- of snm 
of two arcs) and tho second ANTAHa-miavanX (i. c., the rulo 
to find tlio sino of difference of arcs). 

[If it bo desired to reduce tho sines to the value of any 
other radius than that above given of 31-38.] Find the first 
sine by tho aid of the above-mentioned rule pratibhXuajyaka- 
vidiii. 

24 and 25. And then reduce it to tbe value of any now 
radius by applying tho proportion. After that apply the jyX- 
bhXvanX rulo through tho aid of the first sine and tho cosine 
4hus found, for as many sines as are required. The sines vtfll 
thus be successively eliminated to the valuo of any new radius. 

The rulo given in my Pat! or LfoXvATf is not sufficiently 
accurate (for nice calculations) I liavo not therefore repeated 
hero that rough rale. 

* 

# [Time rales given in the verses from 16 to 20 are ensilj deduced from the 
rules given in the rones 21 and 22.— B. D.] 

t BiiabxavXohArta has given these rules iu his work without any demon* 
stration.— B, 1).] 



INDEX. 


Age, birth, &o. of tho Author, page 2C1. 
Arinillary Sphere 151, 210. 
Astronomical Instruments, 209. 
Atmosphere, 127. 

Mandapliala, 109. 

Mumloclichn, 109. 

Mouth, 129. • 

Moon, Eclipses of, 176. 

Celestial latitude, 200. 

Clopsydra, 211. 

Chukra, 212. 

Canon of Sines, 263. 

Fhalalca-Yantra, 213. 

Phases of the Moon, 206. 

Planets, 128, 135. 

Pay of Brahma, 163. 

Day of the Pitris, 163. 

Days and nights, 161. 

Pclngos, 125. 

Drikkarnyi, 110. 

Driyantra, or genius instrument, 321. 

Earth, 112. 

Earth's diameter, 122. , 

Eclipses, 176. 

Epicycles, 114. 

Equation of the centra, 141, 141. 
Errors of Lalla, Ac. 169, 165, 205. 

Gnomon, 212. 

• 

Quoslions, 231 . 

Rising and setting of tho heavenly bo- 
dies, 196. 

signs, 161. 

> 

Season*, 228. 

# Sovi*n Winds. 127. *' 

Sighroclicha,109. 

Sigus, rising of the, 161. 

Sphere, 107. 

Sun, Eclipse* of, 176. 

Svrayauvaha Yantro, or self-revolving 
instrument, 227. 

Syphon, 227. 

Horoscope, 166, 211. 

Time, 160. 

Kalpa, 108. 

Kendra, 109. 

Winds, 127. 

Lagna, 166, 211. 

Ldngitudos, 212. 

Year, 129. 

Yugas, 110. 




•GENERAL INDEX 


Page 

Additive months, ... ... ... ... 0 

Airs, Different kinds of, ... ... ... 127 

Amplitude, The sine of, and the udayasta-sutra, ... ... 171 

Apogee, ... ... ... ... ... G 

Apology, The Author’s, ... ... ... ... 2G1 

Apsis, The reason for the invention of the liigher, 143 

Arc of sines, How to find the, ... ... ... ... 17 

Armillary sphere, On the construction of the, 87,8$ 106, 151, 209, 

; Uses of the, ... ... 210 

Ascensional difference and its placo" ... ... ... 160 

— ■■■■ of the sun, «... ... 1(?1 

— The sine of the, . ... ... ... 89 

Astcrisms, The Bliogas of the, ... ... 62 

Astcrism, Rohini, crossing the cart of a planet, ... ... 64 

Asterisms or principal stars of the Yoga-Tdrds,... ib, 

Astymomic&l Instruments, On the constuction of the, ... 87 

Astronomy, Origin of, ... ... ... 1 

■ ■■■—. - The science of, ... ... ... ... ib, 

■ — Originally, taught by the sun, ... £ 

Atmosphere, Extension of, ... ... ... ... 127 ' 

Aurigac, The longitude and latitude of the star Proj&pati, ... 65 

Author’s birth-place, See., ... ... ... ... 262 

Autumn, The sdratkdla or season of early, ... 230 

Avaha or atmosphere, The, ... ... ... ... 127 

Azimuth or vertical circle, ... ... ... .. .• ... • 152 

Bhfiskfra, ... ... ... ... ... ... 107 

Brahmd, Length of a day and night of, ... 9 4 

— , Period of his life, ... ... ... ib. • 

Brahmagupta, Prate of, ... ... ... 203 



INDEX. 


-*v, 

Brahmdnda, Dimensions of the, ... ... ■ ... ... 120 

Canon of sines, The construction of the, ... 2€5 

.Circle, ... ... ... ... ... • 209, 212 

■ To find the'fongitude of planets by the, .. ... *, ih 

Centre of, • ' , ...' ... 150 

Circles, diurnal, ... ... ... ... ' 159 

— of the 12 signs, ... ...' ... ... 8 *f 

Clepsydra, Kapala Yantra or, ... ... ... 91, 209 

Compass, the points of, ... ... ... ... 120 • 

Day, Determination of the place whero the, becomes of 60 Ghati- 

kds, 83 # 

Terrestrial, ... ... ... ... ( 2 ) 6 , 132 

Subtractive, ... ... ... ib. 

of the week begins, To find the instant when, ... ... 12 

To find the lunar, at a given time, ... ... ... 2 .) 

Period of, ' ... ... ... ... 2 

of the gods, ... ... ... ... 3 

— To find the Ruler of the present, ... ... ... 9 

— Subtractive, called Avama^the reason of, ... 130 

— of the pitris, Longth of.the, ... ' ... ... 163 

— of Brahma, The explanation of the, ... ib. 

and night, Definition of the artificial, and the day and 

night of the pitris, ... ... ... ... 162 

— and night, Determination of place and time of perpetual,... , ib. 
Days and nights, Cause of increase and decrease in the length of, 161 

— from tho time the planetary motions commenced to the 

present midnight, 8 

•— Sidereal and terrestrial, their lengths, ... ... *128 

Lunar, ... ... ... ... 132 

Degree, measure of, ... ... ... ... ... 5 

Degrees of latitude are produced from the distance in Yojanas 

.and vice versil, ... ... ... ... ... 121 * 

Deluges o* dissolutions, ... ...* ... 125 

— Fourfold, ... ... ... ... ... ib. 

Demons, The day of the, ... .... ... 3 

'The night of the, ... ... ... ... ib. 

The year of the, ... ... ... ^ ib. . 



INDEX. 


3 


, . P'W 

Earth, Description 1 of the, ... ... •... ... l 12 

Refutation of the supposition that the earth has suc- 
cessive dhpjrorters, ... ... ... JJ& 

—* - Refutation of ( tho objection as to how the«barth has its 

own inherent 4 >owcr, ... ... ... ... ib. 

-Attraction of the, „. ... ib. 

9 Bauddhas, opinion of the,... ... ... ' ... ib, 

— Jainas, opinion of the-, ... ... ...’ ... ib. 

«. — — Refutation of the opinion of the Bauddhas, regarding the, J 14 

Refutation of the opinion of the Jainas, regarding the,.., ib. 

* — - Refutation of the supposition that it is level,... ... ib. 

— Reason of the false appearance of plane form of the, ... ib. 

Proof of the correctness of alleged circumference of the, 115 

-j— ■ - Confirmation of the alleged circumference of the, ... ib. 
— - Questions regarding the, ... ... ... 70, J<>7 

—— Superficial area of the, ... ... * ... ... J22 

— Middle line of the, ... , ... ... 1 1 

■ circumference of the, ... ... ... ... 122 

— its diameter and circumference, ... ll 

— - — diameter of the’ shadow of, at the moon, ... ... 41 

Eclipse, Given the quantity of the eclipsed part to find its cor- 
responding time, ... ... ... 45 

■ To find the Valanas used in the projection of an, ... ib. 

— ■ To find the Angelas or digits contained in the moon’s 

latitude, diameter, eclipsed part, &c., at a given time during an, 40 

of the sun, ... ... ... 48, 111, J70 

— of the moon, ... ... ... ... 41, ib.' 

— — the science of, very secret, ... ... 50’ 

— To find the magnitude of an, ... ... ... 42 

To ascertain the occurrence of a total, partial or no, ... ib. 

— To find the half duration of the, and that of tho total 

darkness, ... ... ... ... ... , 43 

■■ 1 To find the time of the phases of an, ... ...• ... • ib. 

- — f To find the quantity of the eclipsed part at a given time 

. during the first half of an, .. ... ... ... *4t 

— To find the quantity of the eclipsed part at a given time , 

• during the latter half of an, ... ... ib. 



INDEX. 


Paje, 

Eclipse, To mark "the latitudes found at the Jbeginhing and end • 
of an, ••• mi 5? 

; To find the magnitude of an, ... ' 54 

* The limit <#“ the magnitude of the feclipsed portion whidh 

is invisible in the solar or lunar, ... . ...' • ... ib. 

To find the path of the coverer in an, ... % # ib. 

To find the direction of the beginning of total darkness " 

by the projeotion of an, ... .. ... ... 55 

— To find the direction of the end of the total darkness,... ib . • 

— The cause of the directions of the beginning and end of 

a solar, ... ... ... ... 170 * 

The cause of the directions of the beginning and end of 

the lunar,* ... ... ... ... ... 177 

The determination of the coverer in the, of the sun and # 

moon, ... ... ... ... 178 

Eclipses, What covers the sun and the moon in them, ... 42 

— Projection of solar, .., ... 62 

‘ The directions of the beginning and end of the lunar 

1 and solar, ... ... ... ... ib. 

— — To find the probably times of the occurrences of, ... 41 

Ecliptic, the, .. ... ... ... ... 38, 89, 153 

— - Variation of the, ... ... * ... 184 

— To find the sine of the zenith distance of the culminat- 
ing point of the, ... ... ... ... 1 ... t 48 

■ - Pour common points of the, .. ... 92 

* To find the Horoscope or the point of the, just rising 

at a given time from sunrise, ... ... ... 39 

— — The Madhya Lagna or the culminating point of the,.,. «89 
Epicycle, Construction of diagram to illustrate the theory of,... 144 

* — ■ Construction of the mixed diagrams of the excentric 

and, ... ... ... ... ... ... 146 

Epicycles of the sun and moon, ... ... ... ... 17 * 

Equation, «The reason for assuming the ftianda-spashta planet as 

a mean in finding the 2 ends, ... ... 142 

■ of centre, The principle on, which the rule for finding 

the amount of is based, ... ... ... 141 

Equator, The four cities placed at the, ... 80 • 



INDEX. 


5 


Equator, There is nt> equinoctial shadow at the, 

*» — ■ To find the rising periods of the signs of the ecliptic 

at the, * ... ... ... 

Equinoxes, the precession of the, ... ... ... 29, 

Equinoctial, the* .*. ... ... ... ... 152, 

■ — — how to make anj ... 

■ * — shadow, from the latitude to find the, ... * ... 


« shadow to find the co-latitudc and latitude, 


•Ganges, Source of the, , v ... ... 

Geographical, Anomaly, as curious fact rehearsed, ... 

* Golddhydya, ... ... ... ... 

Globe, equatorial, circumference of the, ... 

Gnomon, the, ... ... ... ... *91, 209, 

— vertical, 

Gnomonic shadow, ... ... 

Gods, the Mina of tho, • ... ... * ... 

— the day of the, ... , ... ... 

— — the night of the, ... ... 

the year of tho, ... «.. ... 

Grammar, in praise of,...’ ... .... ... 

Heavenly bodies, Rules for finding tho times of the rising and 
setting of the, ... ... 

Himalaya mountain, The, ... ... ... ... ... 

Horizon, the,*.., ... ... ... 

Horoscope, the, ... ... ... ... 

Hypothenuse, Given tho shadow to find its, ... ... 

i, . to find the, and the equation of the centre, 

- — explanation of the reason of omission of, in the 
manda process, 

Instrument, the self-revolving Spheric, ... 

for measuring time,... 

— a self-revolving or Swayanvaha yantra, 

the praise of Dhiyantra or genius, ... • ... 

Instruments, astronomical, ... ... ... 

Jambudvipa, position of Mountains in the, ... ... 

Juga, The number of days in a, ... ... ... «... 

• Jupiter, years of, ... ... ... ... ••• 


Page 

80 

38. 

i57 

209 

210 
31 
30 

119 

120 
105 
123 
212 

20 

27 

95 

3 

ib. 

ifc 

107 

19G 

117 

90 

89 
29 

144* 

147 

90 
ib. 

227 

*221 

209 

?17 

0 

94 



6 


INDEX. 


* a 9\ 


Jupiter, The nodt of, ... ... ..." 

••• 

7 . 

revolutions of, ... ... ... ... 

... 

.5 , 

Jupiter’s apogee, .. ... ... ... 



7 

Kalpa, The length of, ... ... *... 

?.. 

4 

Krita yuga, solar years elapsed from the time when the planeta- 
ry motions commenced to the end of the last, ... 

7 

Lagna, etymology of the word, ... ... ... 

... 

icB 

Lalla, the error of, 

;. ... 

122 

* The wrongness of the ltule given by, 


123* 

— -an error of, exposed, ... 



1G9 

— - another gross error of, 


ib. m 

— - — cause of error in, and others stated, ... . 



205 

Latitude of a place, to iind from the gnomon’s shadow, 


30 

rectified, ... ... ... 



2^1 

celestial, ... 

201, 

203 

Latitudes, determination of, in which different signs arc 
above and below the horizon, 

always 

169 

Lokas, arrangement of the seven, 


320 

Longitude of the sun, how to find, ... ... 



31 



1L 

Lunar Mann, the, ... 



03 

— - M6na, use of the, ... 

... 

ib. 

Mathematics, in praise of, ... 



106 

Mathematical calculations, two kinds of, ... 

... 

> ib. 

Matters, Cosmographical, ... ... ... 



7<i 

Mars, 2nd equation of,... ... ... ... 


19 

— - epicycles of, ... 



18 

— — - nodes of, ... ... 

... 

• 7 

— revolution of, 



5 

Mars’ apogee, ... ... ... 

••• 

7 

Mauu, the length of, 



3 

Mercury, node of, 

... 

7 




5 

Mercury's apogee, 


7 

Meridian, The, how to determine, .... 



PG 



27 

Meru, why due north of all places, ... 



120 • 



INDEX. 


7 


Paijfl 

’Meru, place of, . ... ... ... ■.. 79, 162 

— in Ilavrita, position of the mountain, ... 117 

Minute, measure of* ... ... ... ... * ... 

Month, S6vana, ‘ ... .».. ... ... • ‘ 2 

1 ■■■*" solar, * , ... ... ... ... ft. 

• ■ ; lunar, ... ... ... ... ... 2, G, 131 

2 period of, ...* ... ... ... • ... 2 

intercalary, ... ... ... 8 

' ■ to find the ruler of the present terrestrial, ... ... 9 

— length of lunar or lunation, ... ... 129 

• ■ ■ ■ the reason of additive called Adhimasas, ... ... ib. 

Months, seasons and year, ... ... ... 93 

Moon, node of the, ... ... ... ... ... 7 

eclipse of the, ... ... ... 41 

on the phases of tlio, and the position of the moon’s cusps, G9 

— diameter of the, . ... ... ... • 41 

— thc.colour of the eclipsed portion of the, ... ... 65 

rules for finding the time at which the declination of the 

sun become equal to that of the*.. ... ... ... 7Sf 

— - to find the true places of the, . ... ... ... 19 

to find tho time of daily setting of the, G9 

— - to find the time of daily rising of the, ... ... 70 

— - — — to find the phases of the, ... ... ib. 

— j cause bf the phases of the,.., ... ... Ill, 206 

• to find the true diurnal motions of the, ... 21 

— revolution of tho, ... ... .„ ... 5 

— epicycles of the, ... ... ... 17* 

Moon’s parallax, ... ... ... ... ,. 49 

Moon’s apogee, ... ... ... ... 6 

Motion of planets, ... ... ... ... , tf 80 

■ heliocentric, ... ... ... 21 

— — — different kinds of, ... ... ... ... 14 

— decreasing retrograde, ... ... ... * ... -ft. 

■ ^increasing retrograde, ... ... ... ... ft. 

— direct, ... ... ft. 

mean, ... ... ... ... *.. ft. 

, stationary, ... ... ... ft. 



8 


INDEX. 


ify* 

Night, determination of the place where the, becomes of CO * 
Ghatikud, ... ... ... ... ... gg 

r Nodc, ... ... ... ... ... • G 

Nonagesimal, to 3nd the sine and cosine of the zenith dis- 
tance of the, ... ... ... 48 

Ocean, situation of the great, ... ... ... 80 

Orbits of Planets,... ... ... ... 154 

Parallax in longitude and that in latitude, ... 48, 177 

— — — not being necessary in lunar eclipses, ib^ 

what is the cause of, and why it is calculated from the 

radius of the earth, ... ... ... 179 • 

Parallel sphere, ... ... ... ... ... 121 

- — ■■ - and Bight spheres,.., ... ... 82 

Pardvaha, ... ... ... ... ... ... 127 

Parivaha, ... ... ... ... tb. 

Persons, praise of, ingenious, ... ... ... ... 232 

Pitris, day and night of, ... ... ... 102 

Planet, to find the conjunction of a, with a star, ... ... 64 

■ rectified mean placo of \ ... ... 20 

to find the motion of a minor, ... ... ... 21 

■ 1st equation of, ... ... ... 18 

- — and star, to know whether the timo of conjunction is 

past or future, ... ... ... ... ... 64 

to find the mean placo of a, at a given time, : t 12 

to find the dimensions of the rectified periphery of the 

epicycles, ... ... ... ... ... 18 

■ to find the time at which a, rises or sets heliacally, ... 66 

Beason of correction which is required to find the true, ( 

from the mean place of a, ... ... ... ... 137 

Planets, to fiud the mean places of, ... ... ... 10, 12 

to find apogees and nodes of, ... ... ... 12 

... ■ — an easy method for finding the mean places of, tb. w 

— determination of the dimensiohs of the orbits of the, 

and their daily motion in Yojanas, ... ... .. 86 

,■■■?— of their duly motions in miputes or angular motions, jb. 

■. to find the radius of the diurnal circle of, ... ... 23 

■ ■ to find the ascensional difference of, ... ib. . 



INDEX. 


9 


1 Page 

•Planets, cause of the motions of, .. ... ... 13 

Tl ap»w of, ib. 

observation of, .. ... ... ... . ... 59 

— the fight* and association of, ... ... t .GO' 

which i^ conquered in the fight, ... “... ... ib. 

r which is the conqueror, ... ... ib. 

% 1 rules for finding tjic true places of, ... t ... 13 

« — ' deflection of, ... ... ... 14 

attraction of, ... ... ... ... ... 14 

* on the conjunction of the, with the stars, Cl) 

, — order of the orbits of the stars and, ... ... 70 

»■ — — - and stars, on the heliacal rising and setting of the, ... C5 
« • to find the length of a day of the, ... ... 23 

— — number of risings of, ... ... C 

rules for finding the mean places of, ... ... 1 

— ■ motion eastward of, ... ... 4 

— to find the’longitudc of,... ... ... ... 212 

-■■■■■ an illustration of the motions of, ... 12 S 

— - the minor 5, Why they require both the 1 st and 2nd 
equations to their true places, ' ... *... ... 147 

■ ■■■ ■■ how the 1 st and 2 nd equations arc to be applied, ... 20 

*■ - to find the true place of, ... ... 19 

— — which set hcliacally in the western horizon and rise 

hcliacally in the eastern horizon, ... ... ... GG 

— why their mean and true motions coincide, 149 

■■ manner of observing the retrogression, &c. of, ... 719 

— ■ - ■ on the principles of the llules for finding the mean 

places of, ... ... ... ... ... 127 

— on the principles on which the Rules for finding the 
true places of the, are grounded, ... ... 135 

■ ■ - ■■■ the cause of variation of apparent size of, the discs of,... 143 

■ their conjunction with the sun, ... 56 

— 1 -- - on the conjunction of, ... ... ... ... . ib . 

kinds of conjunction of, ... ... ...* ... * ib. 

• "" to find whether the time of conjunction is past or future, ib. 

*■< ■'— to find the time of conjunction from a given time, ... ib. 

conjunction the correction called the aktha-drik-karTna, 57 * 

. j » • 

» 



30 


INDEX. 


• 

• 

Planets, to find the distance of two, in the same circle of posi- 

Page 

tion, ... * ... ... ... * 

... ... 

58 

the apparent diameters of the, in minutes, 



89 

Poetry, sweets of, 

•V * 

230 

Points of the compass, ... ... * 

• 

120 

Pole, North, of the Earth, ... ... t « 

* • 

... a ... 

134 

Poles, the inhabitants of tho two, ... 


80 

Projection'of Eclipses, 

... 

52 

Quadrant, 

... • 209, 

213 

Questions, Pras'nddhy&ya containing useful, 

... 

231- 

Miscellaneous, ... 

... 

260 

Radius of the diurnal circ-lo, ... ... 

... 

310 # 

Rainy season, .. 

... ... 

229 

Retrogression of planots, 

... ... 

22 

Samv&ha, 



137 

Sandhi, height of, 

... ... 

3 

Saturn, revolutions of, ... ... * ... 

... 5, 

6 

Saturn’s apogee, ... ... .„. 


7 

node, ... ... ... 

... ... 

ib. 

Seas and Dvipas, positions of tho, 



116 

Seasons, months and year, . ... 

••• 

93 

— * description of the, ... ... ... 

... ... 

228 

Seconds, measure of, ... 

... 

5 

Semi-circles, ... ... ... ... 

... 209, 

213 

Shadow, dotormination at noon, of tho direction of the gnomonic, 

. 84 

Sphere, oblique, ... ... ... 

,,,■ ... 

85 

Sidereal month, 


2 

1 Jay and night, 


ib. 

■■ revolution, 

... ... 

• 5 

— — — • Mfina, the, 


9 k 

Signs, positions where same are always invisible, 

... ... 

83 

Sine of amplitude, 



'28 

— — ■■ Rules for finding the, of every degree from 1 

1° to 90', J5, 

267 1 

• ■— way of refutation, of using the tersed, 

... 390, 

193 

Sines, versed, 



36 

— Rules for finding the 24, viz. 3° f, 7° i, 31° 

i 15°, Ac..!. 

2$7 


( —— Rules for finding the, of sum and difference of any two arcs, 208 



. INDEX. 


11 


Sines, on the canon of, ... ... ... 136, 

Solar Mfina, use of the, ... ... ... ... 92 

Sphere, oblique, ... ... ... , 121 

•— — parallel,... • ... ... ... ... ... , it. • 

Spherfb, iu praiso o£ the advantages of, ...' 105 

Spring, *..!.• . ... ... ... ... ... 228 

Star, the longitudes and latitudes of the Agustya, Mrigavyddha, 

Agni and Brahmahridaya, ... ... ... ... 03 

Stars, thetyosition of the polar, ... ... ... 80 

i to find longitudes of the principal of the Aster is ms or, ... 01 

Summer season, the Grishma or, ... ... 229 

Sun, revolutions of tho, ... ... ... 5 

— rules for finding the time at which the declination of the 

sun and moon become equal, ... ... ... 72 

- — to find the true places of the, ... ... J9 

to find the true diurnal motion of the, ... 21 

— epicycles of the, ♦ ... ... .?. 17 

— duration of the eclipse of the, ... ... ... 51 

— Eclipse of, vide Eclipse. 

— Rules for finding the time at jvhich the declination of the * 

moon become equhl, with that of the, .. ... ... 72 

— and Moon, to find the time of their true declination, ... ib. 

— question about the revolution of tho, ... ... 70 

— diameters of tho Moon and the, ... ... 41 

— Apogee of tho, ... ... ... 7 

— Revolutions of the— in a year are less than the revolu- 
tions of stars by one, ... . ... ... 129 

— the reason for finding the exact place of the, in order to 

find Lagna, ... ... ... ... ... 167 

— declination and longitude of, how to find, ... ... 37 

— the prime vertical of the, ... ... 171 

— mean place of tho, to find, ... ... ... ... 82 

— zenith distance of the, at noon, ... ... ib. 

— shadow of the, and its hypothenuse, ... ... a .... 33 

— amplitude and the sine of amplitude of the, reduced, ... ib. 

j ■■ - altitude, ... t ••• ••• ••»* 34 

— altitude, zenith distance, Ac, at given time from noon, ... 38 



12 


INDEX. 


• 


Page 

.Sundial, a new, # ... 

... 

213 , 

— ■ how to use a, 



217 

Supreme Being,* the excellence of the, 

... 

11?" 

STivaha, ...^ 



127 

Syphon, description ftf a, ... ... f ... 

• 

227 

Syxygy, to reduce the places of the sun, the mofiu and her 


ascending node as given at midnight to the instant of the, 

•’ 42 . 

Terrestrial liina, its use, 

... 

95 

Terrestrial and Lunar days in a yuga, 

« 

6 

shadow equinoctial, ... 

... 

30 « 

Time, rules for resolving the questions on, 

... ... 

20 

— , kinds of, 

... 

2 

, measurable (Murta), ... 



ib 

, immeasurable (Amurta), ... 

... 

i b. 

, number of kinds of, 



9L 

Triangles arising from latitude, ... 

... 

173 

Tropic, Terrestrial, 



84 

Udvaha, ... ... ... , 


127 

Universe, the, ... ... t 



126 

Unmandala or six o'clock line/'.. • 

... 

152 

Venus, resolution of, • 

... 5, 

6 

— node of, ... 



7 

, Apogee of, 

... 

ib. 

Virginia, of the stars Apamvatsa and Apa or, ... 



Go 

Water, Observations in, 

... 

2TC4 

Winter, hemanta or early, ... ... ... 



230 

Winter S’isira or close of, 

... 

ib. 

Year, to find the ruler of the present terrestrial, 

... ... 

9 

, solar, ... 

... 

•3 

— , the season and months of the, ... 



93 

, two halves of a tropical, ... 

... 

ib. 

, Length of the solar, ... 



129 

ITuga, length of the great, 

... 

3 

Ifugas, length of the four small, 



ib. 

fuga, number of months and days in a subtractive and additive, 6, . 

37 

* 



INDEX OE SANSKRITA TERMS, 


Abhijit, ... Cl, 88, b3, 
AcbArya,. ... 
Adhara-Kakslia, 

^ Adbara-Vrifctas, 

AdbimAsa, 6, 130, 8, 131, 

' 120 , 

AdhimAsa-soaha, 131, 132, 
240, 241, ... 

Aditya, 

AdbyAtma, ... 

Agastya, ... 63, 08,. 

Aggi, 63,68,119, 

Agra, 172, 111, 174, 251, 
17J, . ... 

AgrAdi, 

AgrAdi-Kbanda, *174, 
AgrAgra-Khanda, 174, 
Abargana, 8, 131, 237, 230, 
232, 109, 236, 8, 234, 
235, 236, 29, 20, 130, 
133, 240, 108, 

AborAtra, ... 2, 

AborAtra-Vtittas, 

Aintlra, 

Aksba, 188, 201, 100, 58, 
67, 191, 46, 

Aksha-Drikkarma, . 110, 
201, 69, ... 

Akshaja, 

Aksbajyfi, ... 
Aksha-Karna, 262, 253, 
254, 265, 256, 243, 244, 
217, 249, 256, 215, 220, 
178, 

Akslia-Ksbetras, 
Aksha-Valana, 199, 198, 
194; 46, 193, 195, 2oO, 
*192, 188, 197, 47, 190, 
189, 187, 194, 185, 189, 
Akaha-Vritta, 

Alakananda, • • • ••• 


Pago 



Page 

09 

Ain&vaflyA, ... 

* J30, 

163 

203 

Angulas, 

* ... 

46 

88 

Amurta, 

... 

2 

153 

Aniruddha, ... 

77, 

78 


Ankusa, 

227, 

253 

232 

Ansuvi-Marda. 


60 


Antara-bhavana. 

268 

242 

Antyajas, ... 


120 

77 

Antya, ... 

251,36, 37, 

89 

77 

Anuradha, ... 

68, 94, 

05 

88 

Anvala, ... 

123, 

126 

118 

Anya, ^.. 

257, 

258 


Apa, 

65, 

68 

28 

ApAm-vatsa, ... 

65, 

68 

175 

» Apaaavya, ... 

••• 

00 

175 

A^ashtasAras, 

... 

202 

176 

•Apsuras, 

... 

lid 


Ardra, 

21G, 68, 

94 


Aruiia, 


118 


Aiyabliatta, ... 

122, 

209 

107 

AshAd 1 

8, 94, 

129 

3 

AhIobIiA, ... 

75,65,04, 

68 

159 

Aspasbta, 

202, 

200 

120 

Aspabbta-sura, 

... 

202 


Asia, 198,197,201, 

89 

198 

Astalagna, 20i, 

198, 197, 



89, 

... 

166 

61 

AstamAna, ... 

... 

56 

196 

Abu, ... 

... 

128 * 

194 

Asuddha, 

167, 39, 

40 


As'uras, 84, 82, 

79, 85, 81, 



116, SJ, 95, 80, 83, ... 

76 


Asus, 197, 198, 

194, JG8, 


170 

215, 161, 173 

, 133, 129, 

128 

172 

Atiwina, 129, 8, 

9*,6J},64,* 



75, 

... 

' 63 


Atisigbra, 


15 

• 

Ativakra, 

15, 

• 14 

190 

Atyantika, ... 

• ••• 

126 

193 

Ansa, 


5* 

119 

Avama, 

130, 181, 

132 



INDEX OF SANSKRIT! TERMS. 


Page 

Ayana, 190, 191, f>9, 196, 

203, ... ... 72 
Ayana-Drikkarma,110,202, 58 
Ayana-Valana, 198, 200, 

202, 197, 199, 18c5, 187, 

47, 195, 46, 19k, 189, 
188,192,193, ... 194 

Baku, 257, 172, 194, 71, 72, 70 

Bais&kha, * ... ... 8 

Balva, > ... ... 25 

Bfipudeva, ... ... 97 

Bauddha, ... 113, 114 

Bava, ... .. 25 

Bhd-Bhoga, ... 24, 25, 75 

Bhfidra, ... ... 119 

Bh&drapada, ... 8, 64, 94, 129 
Bhadrfiswa, ... 82, 84 

Bhadrdswa-Varsha, 80, 117, 119 
Bhdgola, 153, 156, 159, 

160, ... ... 210 

Bhaganas, ... * 5, 238 

Bhagana-sesha, 239, 240 
Bhajya, ... 241, 242 

Bhanji, ... ... 143 

Eharagi, ... 65, 94, 68 , 

Bharata, ... 85, 84 

Bhdrata-varsha, 120, 80, 

117, 120, ... ... 119 

Bhdsandhi, ... ... 75 

Bh&kara, 263, 223, 114, 

221, 213,... ... 262 

Bhaskardchdrya, 138, 186, 

126, 124. 149, 142, 125, 

148, 119, 122, 108, 87, 

182, 179,240,169, 158, 

241, 258, 242, 247, 268, 

233, 257, 209, 215, 205, 

203, 201, 188, ... 194 

Bhavana, ... ... 242 

Bheda* ... ... 60 

Bhoga, ... 24, 25, 62, 61 

Bhogya, 167, 39, 168, 40 

Bhogya-khanda, ... 195 

Bhogydfisas, ... ... 167 

Bhuja, 135, 16, 30, 18, 31, 

27, 44, 28, 29, 27, 85, 17, 

22, 176, 173, 194, 172, 

245, 2G6, 267, 265, 258, 

222, 225, 217, 174, 224, 


Page 

258.259.219.209, 215, 1 

220, 208, 195, 194, 207, 

176, 225, 219, 224, 223, # 

219, ... ... 259 

Bhuja-phala, 148,146, l^ 

145, '. 18 

Bhujantara, ....* 150/20, 109 

Bhukta, ... 39,40, 167 

Bhumi, ' ... 129, 79* 

Bhur, ... ... 79 

Bhurloka, ... * 120 

Bhuvaloka, ... ... 120 « 

BhuvanakoB'a, ... 127 

Bhuvar, ... ... 79 < 

Bowtia, ... ... 229 

Brahmd, 4, 79, 78, 112, 127, 

139, 117, 95, 125, 118, 

91, 110, 116, 88, 178, 

164, ... ...* 163 

Brfihraaga, ... ... 262 

Brahma-Mfina, ... 95 

Brahm&nda, ... 112, 126, 79 

Bran man dee, ... 87 

Brahmagupta, 203, 201, 

148.122.202.209, ... 208 

Brahma-hpdaya, 69, 68, 


65, 

... 

63 

Brahma-siddhdnta, 

235, 

97 

Brahma-siddhdnta-vit, 

... 

235 

Brischika, ... 

... 

8 

Buddhitattwa, 

... 

112 

Chaitra, 131, 129, 228 

,8, 

6 

232,94, ... 


9 

Chaitraratha, 

... 

118 

Chakra, 

... 

212 

Chakshu, 

... 

119 

Ch&pa, 

... 

248 

Chara, 109, 215, 134, 

133, 

109 

Cliarajya, ... 256, 
Chara-kdlas, ... 

161, 

251 

... 

161 


Chara-khanda, 169, 161, 

264, 165, 160, 1, 88 

Chatushpdda, ... 25 

Ch&turdnana, ... 112 

Chaturyuga, ... ... 8 

Chheda, ... 87, 49, 86 

Chhedaka, ... 176, 170 

Chitrd, ... 94, 68, 65 

Painandina/... ... 125 



INDEX OF SANSOITA TERMS. 


s 



> 

Page 


Page 

Daityas, 110, 115.162, 122, 

112 

Garaja, ... 

25. 

DaksninAyana, 

93, 

163 

Garuga, 

258 

Barbha, 

261, 

150 

Gata, ... 9 25, 

17 

Danavas, • ... 

... 

126 

Gatiphala, ... 

149 

Danuias,. • 

•i* 

112 

Gatis, ... 

■251) 

DcsAntara, 11, 134, 109,133, 

12 

Ghati, 211, 209, 164, 167, 


Dhanishfhl 6l 68, .65, 69, 


2LL, 256, 216, 214, 130, 


94, 


63 

159,168,218, 168,184, 


Dhanu, 

129, 

8 

256 

253 

DhArA, 

245, 

246 

Ghatika, 2, 128, 25*, 25, 


Dhivriddttida, 

... 

236 

11, 134, 24, 49, 93, 250, 


Dbivriddhida, tantra. 123, 


254, 210, 48, 57, 43, 49, 


287, 205, 236, 

|M 

261 

68, 76, 25,218, 130, 128, 


Dhriti, 

IM 

118 

83, 1,40,129,44,12,45, 


Dhiyantra, ... 

221, 260, 

224 

49, 74, ... 

82 


Dhruvas, ... ... 202 Golabandha, ... ... 151 

Dhruva-yashti, ... 151 Goladhydya, 105,262,37, 217 

Digvalanaja, ... ... 202 Gomedaka, ... ... 116 

Digvalanaja, tryasra, ... 202 Grahana, ... ... 176 

Diks, ... 118 Grahayuti, ... ... 61 

Di^ya, ... ,91, 95 Gfishma, 229, 93 

Drcshknas, ... ' ... 210 Guhyakas, ... ... 67 

Dridha, 1 ... ... <238, Hara, ... 242, 241 

Driggati, ... ... 49 Harivarsha, ... ... 117 

Driggola, ... 153, 160 Hasta, ... 94, 65, 63 

DrigjyA, 35, 171, 37, 36 * Hemanta, ... 93, 230 

Drigyfi, ... ... 219 Ifomakuta, ... ... 117 

Drikkarma, 66, . 58, 196, Him&laya, ... ... 117 

110, 204, 205, 203, 197, Hiraijmaya, ... ... 117 

200, ... ... 203 Hiragya-garbha, ... 77 

Drikkarama-vasana, 190, 206 . Hpti, ... 251, 252, 176 

Dpkshepa, 60,49,183, 181 Mvpta, ... ... 117 

Drikshepa-vritta, 152, 181 Ilavpta-varsha, ... 117 

Driksutra, ... ... 181 Indra, ... 80, 119, 118 

Drinmarujala, ... 171 Tsd, ... ... 119 

Dwap&ra, ... 110, 108 Ishta, ... ... 167 

Djvipas, ... 116,80, 117 Ishta-hpfci, ... 251, 251 

Dyujya, 110, 159, 189, 197, Iahtantya, ... 25 L, 252 

200,251, 201,24, ... 196 Jaina, ... 114, 113 

Dynjyd, ... 194, 123, 195 Jambu, ... ... 118 

Gabhastimat, ... 120 Jambddwipa, 116 117 

Gamya, ... 25, 17 Jambunadf, ... ... 118 

Gand&nta, ... ... 75 Janaloka, ... 120 

Gandhamfidana, ... 117 Jina-vptta, ... • ...' 193 

Gandharva, ... ... 120 JyA-bbavana, ... 268 

Ganes'a, ... ... 105 JyautUhopanisbat, .... 91 

4taitfdhy&ja» 208, 258, ( Jyeahtht, 24 L, 242, 129, 8, 

257, 144, 261, 247, 145, 75, 68, 65, * ... 

, 125, 148, 158, 126, 156, 108 KachnAr, ... 


230 



4 


INDEX OF SANSKRITA TERMS. 


Page 


.Kadaruba, 229,100,118, 191 
Kadamba-bh'rama-vj-itta, 


192, 

... 

191 

Kakshavptta, 

137, 

139 

Kil», 254, 252, 

25p, 251, 


133, 

« 

215 

Kalansas, ... 

66, 67" 

68 

Kali, 

110, 3, 

108 

Kalpa, 235, 238, 

, 242, 240, 



130, 233, 234, 23S, 4, 
157,29,95,7,4,83,118, 

131, 132, 130, 127, 108, 239 


Kalpa-adhimdsas ... 242 

Kalpa-bhaganas, 239, 234 

Kalpa-saura, ... 213, 242 

Kama-deva, ... ... 229 

Kanyd, ... 8, 129 

Kanishthd, ... 241, 242 

Kdpala-yantra, ... 91 

Karran, ... 132, 1G7 

Karana, ... 93, 26, 25 

Karga, ... ... 142 

Karagi, ... 34, 35 

Karka, ... 8, 129 

Karkyddi, ... 142,141, 140 

Karkyddi-kendra, 142, 144 * 

Kdrtika, 9, 4, 129, 8, 94 

Kaseru, ... ... 120 

Katahas, ... ... 79 

Kaulava, ... ... 25 

Kausa, ... ... 131 

Kendra, 21, 144, 159, 143, 

158, 18, 19, 160, 142, 

148, 16, 145, 14J, 109, 

'140,20, 109,16, ... 22 

Kendra-gati,... 146, 144 

Ketumdla, ... 84, 82 

Ketumdla-varalia, 80,119, 117 

Khagola, 151,152, 153, 160, 210 
Khakaksha, ... .. 127 

Khanda, ... 175, 195 

Kliandas, ... 117,217, 120 

Khandakas, ... 215, 214 

Khecbaragolc, ... 160 

Khila, ... 240, 236 

Kinnfrayaraha, ... 117 

Kinstughna, ... «■• 25 

?okilaa, • ... ... 228 

Kona-sadku, ... 171, 34, 35 


Page 

Kona-vpttas, , 152, 117 ' 

Koti, 135, 16,142,70, 135, 

141, 27, 265, 222, 223, • * 

267, 208, 209, 144, 207, 

202, 203, 18, 175, ■ 174a 
44, 172, 173, 194, 224, 
45,225,176,44/17,71, 223 

Koti-jihala, ... 19,145,.) 18 

Krama, • ... . ... * 39 • 

Krduti-pdta, 157, 154, 153, 172 
Kranfcijyd, ... 1 ... 192 

Krauncha, ... ... 116 , 

Krishna, ... ... 163 

Kpshga-paksha, ... 163 

Krita, ... 108,3, 110 

Krifca-yuga, ... 4, 8, 1, 10, 7 

Krittikd, ... 65, 68, 94 

Kshepa, ... 241, 167, 242 

Kshcpa-vfitta, 157, 155, 154 
Kshepa-pdtaa, .. 157 

Kujyd, UO, 23, 174, 175, . 

176, 160, 215, 255, 254, 


256, 252, 243 

i,25J, 2411, 

259 

Kuldehalas, ... 

Ml 

120 

Kumariku, ... 

Ml 

120 

Kumbha, . ... 

8, 

129 

Kuru, 

84, 117, 

119 

Kurukshetra, 

134, 

11 

Kuru-varsha, 

... 

80 

Kusakasa, ... 

..i 

229 

Kuta, 

i •• 

69 

Kuftaka, 

242, 236, 

ail 

Lagna, 2S0, 1B7, 200, 210, 


89, 197, 166 

, 211, 201, 


167, 


198 

Lakshmf, 

... 

116 


JUBMUIIll, ... ... iiU 

Lalla, 108, 128, 149, 205, « 

188, 169, 205, ... 236 

Lambana, 111, 183, 182, 

181, ... ... 170 

Lambana-kalas, ... 181 

Lamba-rekhd, 217,218, 214 
Lankd, 89, 109, 80, 115, 9, 
134,10,133, 132,82,11, 

20, 243, 184, 118, 120, ( 

117, 108, 8, ... 3a 

Lahkodayas, .. ... 17Jr 

Lildvati, ... 136, 268 

Loka, ... 116,126,120. 



INDEX OF SANSKRITA TERMS. 


• . * Page 

LokAloka, ... ... 126 

Madhya, ... * 12, 89 

Madhya-gati, 134, 1, 209 

Madhya-gati-vasana, ... 127 
Madlwajya,- • 48, 49, 216 

Madhya-lagntL • ... 166 

Madhyama, ... • ... 89 

Madhyarekha, ' 131 

<*Madhya-sanku, • ... 171 
MAgha, 94, 129, 8, 68, 65, 

212, i ... ... 94 

Mahan, ... ... 78 

MahAhrada, ... ... 118 

Maharloka, ... ... 120 

Mah&sanku, 257, 252, 254, 259 
Mahattattwa, ... 112 

MahAyugas, ... ... 164 

Mahendra, ... ... 120 

aheswara, ... ... 262 

akara, ... 129, 8 

Malatf, ... 228, 229 

MallikA, ... * * ... 228 

Malaya; ... 120, .229^ 

Malyavan, ... ... 117* 

Mana, 94,96, 95, 93, 91, 92, 77 

MAnasa, ... ... 118 1 

Manda, 17, 142, 147, 21, 

15, 20, 14, 164, ... 19 

Mandakendra, ... 109 

Mandaphala, 109, 19, 150, 

137, ... ... 19 

Manda-prativritta, 142, 137 
Mandspaahta, ... 142 

Mardardha, 44, 43, ... 51 

Manddra, ... 118, 119 

Manda-spaahta, 157, 138, 

J54, 155, 21, 137, 142, 

143, ... ... 20 

Mandatara, ... 14, 15 

MandochchA, 13, 14, 10, 16, 

109, 146, 143, 137, 16, 

109, ... ... 0 

Manu, ... 108,7,3, 95 

Manus, ... 7, 4 

Manwantara, ... 3 

MArgasirsha,... 129, 9-4, 8 

^fAsha, ... • g 

Matsya, ... ... 120 

MAyA, ... 77,112, 06 

f , • •• 


Page 

MAya-Asura, .. ... 76. 

Meru, 119, TO, 162, 121, 

1J7, 120, 119, 81, 134, 

118, 88, 163,81,169, 85, 
117,114,80, ... • 115 # 

Mesha, 131, 129, 8 

Mina, ... 8, 129 

Misra, ... ... 232 

Mithuna 8, 129 

Mithnna-sankranti, ... 130 
Mriga, ... 68, 64, 94 

MrigAdi, ... 140,141, 144 , 

Mpgasirsha, ... ... 94 

MpgavyAdha, 63, 68 

MulA, ... 94, 65, 68 

Muni, ... ... 96 

Munjala, ... ... 157 

Murta, ... 2, 3 

Naduna, ... ... 118 

Nd<Ji-vulaya, .. 209, 210 

NAga, • ... 25, 120, 79 

Nairrita, ... 118, 119 

Nakshatra, 246, 75, 24, 94 

Nakshatra Aliordira, ... 2 

Nakshatra mdsa, ... £ 

Nakshatra-vfitta, ... 78 

Naraka, * ... ... 119 

Nata, 187, 195, 189, 190, 253 
Nataghatijya, ... 194 

Natajyd, ... ... 194 

Natakarma, ... ... 148 

Nati, 111, 182, 183, 181, 184 
Nati-kAla, ... 183, 181 

Navfinsas, ... ... 210 

Nichochcha, ... ... 147 

Nichochcha-vritta, 143, 

138, ... ... 15a 

Nila, ... ... 117 

Nishadha, ... ... 117 

Paksha, ... ... 163 

Pala, ... ... 2 

Palas, 129, 214, 215, 250, 2 

Palabha, 251,249, 248, 245, 
244,243, 250, 256, 246; 

254, 252, 253, 258, 220, 

222, 255, 259, 219, 215* 

161, 80, 173, 31, ... 168/ 

PalAtmakas, ... • ... 2 \3T 

Para, ... ... Ir2 



6 


INDEX OF SANSKRIT* TERMS. 


Page 

. Pariyatra, ... ( ... 120 

Parama-lamb&na, * ... 139 

Parvata, ... ... 126 

IMta, 159, 26i; 154, 158, 

• 155, 13, 153, 75, 72, 73, 74 

Pdtfidhikdra, ... * 75, 72 

Pdta-kdla, 75, 74 

Patas, ... 159, 158, 14 

Patala, , ... 116, 79 

Pdtdla-bhumia, ... 76 

Pati, ... 268, 282 

Pattika, 215, 214, 218, 217 

Paurdnika, ... 114, 126, 115 

Pauslia, ... 8, 129, 94 

Phala, 34, 35, 138, 144, 142 
Phalaka, ... 214,209, 216 

Phalaka-y antra, 216, 213 

Ph&lguna, .. 94, 129, 8 

Pippala, ... ... 118 

Pitps, 110, 163, 76, 110, 85, 

94, 162, ... « ... 163 

Pitrya, ... ... 91 

Prdehyapara,.,, ... 219 

Pra'iapati, ... Op, 68 

Prajiipatya, ... 95, 91 

Prdjdpatya-mdna, ... 95 ' 

Prakdsa, ... ... 126 

Pr&kritika, ... ... 125 

Prakpti, ... ... 241 

Pralaya, ... ... 163 

Prana, ... ... 2 

Prdpas, 23, 40, 37, 39, 68, 

66, 2, 70, 38, 24, ... 69 

Prasnfidhydya, ... 262 

Prathama, ... 257, 258 

Pratibhagrjyaka-vidhi, 207, 208 
JPrativptta, ... 137, 143 

Pravaha, 14, 72, 13, 160, 85 

Pravipalas, ... ... 129 

Pnnarvasu, ... 68, 94, 65 

Purdoas, ... 96, 178, 119 

Pnr&imd, ... ... 163 

Pnrusba, ... ... 77 

Purva, «... ... 64 

Purva-Bbddrapada, 94, 64, 68 

Purva-phfilguni, 68, 64, 94 

Purvdabddhd, 68, 94, 03, 04 

sfiwhkara, , ... ... 117 

imshpavati, ... ... 230 


Page 

Pushyd, ...,212,68,05, 94 4 

Rahu, ... 178, 13 , 

Ramyaka, ... ... Ilf 

Rdsi, .... « ... 5 

Rdais, 9 5 

Rdsi-vptta, *. 78 

Rdsyudaya, ...# * . •... 250 
Revatf, 2J2, 5, 3, 65, 75,/ 

94, . ... ... * 68* 

Rjg-veda, ... ... 78 

R}kshaka, .... * ... 120 

RohinI, ... 68, 64, 65, 94 4 

Rom aka, ... ... 80 

Romakapattana, 118, 117, 

115, ... ... 120 

Rujya, ... ... 110 

Sahya, ... 120, 262 

S'aka, ... ... 1J6 

S'akuni, ... ... 25 

S'dlivdhana, ... 108, 261 

Sdlmala, ... ... 11G 

Sdma, 193, 252, 255, 14, 15, 191 
Sdmas, ... V.. 193 

‘Samasa, ... 53, 52 

Samaaa-bhavana, ... 268 

Samagama, ... ... 60 

Samaaanka, 111, 171, 253, 

175, 174, 249, ... 175 

SambMi-Horaprakdsa, ... 67 

Samputa, ... ... 79 

Sama-kdla, ... , ... 42 

Samvatsaras,,.. 92, JO 

Sdma-yeda, ... ... 78 

Sarasliahta, ... ... 234 

Sanhitikas, ... ... 163 

Sanhitd, ... ... 178 

SanaUchara, Mi *••• fib 

Sandhis, ... 3, 108, 7, 4 

Sandhyd, ... 108, 8 

Sandhydnsa, ... 108, 8 

Sdndilya, ... ... 262 

Sankrantf, 130, 93, 8, 92, 131 
Sankrdntis-karka, ... 130 
SankarshaQa, ... 77 

SafLku, 36, 37, 28, 171, 49, 
172,178,170, 175, 176, * 

274, 252, 220, 219, 255, 253 
Sankutala, 39, 194, 176, 

245,257,172,28,172,35, 176 
• » 



INDEX Of 6ANSKRITA TERMS. 


Sanmya, ... • ... 1 

• Sara, 107,200,192,202, ] 
tfarat, 

SaratkAla, ' ... 5 

Sarasflktf, * ..! .a J 

Saura, 129, 191, 132, 243, 
2^5,232,242,' • ... J 

S&vaiia, 2, 170, 2.50, 130, 
131,129,95,0,170,109, 

171, # | 

S&rana-gnatia, ... - 

* Savita, 

Sashadbha, ... 

> SasL 

Satatard, ... 

Satatdraka, ... 212, 

Satyaloka, ... 

Sesha, 

Betha, 

Sha^asiti-mukhas, 

Shadvarga, ... . 

Siddha, 

Siddhapura, 115, 120, 118, ’ 

»2 

Siddha-puri, ... 

Siddhas, ... 79,* 118, 

SiddhAnta, 202, 175, 165, 
105, 170, llo;.156, 187/ 
96, 185, 182, 100, 250, 
207, 180, ... 

Siddhantios ... 
Sildhanta-Siroraani, 262, 
Sighra, 147, 142,18, 19, 15, 
Sighrochcha, 138, 158, L40, 
230, 169, 5, 0, 22, 13, 
10, 109, 87, 14, JO, 149, 
*143, 

Sigbrochchas, 21, 158, 138, 
Sighra-karga, 154, 

Sighra-kendra, 150, 159, 
109 

Sighra-phala, io9, 19, 19, 
158, 

Sigbra-prativfitta, 138, 157, 
SigHratara or Atisighra, ... 
Singra-dhivriddhida, 
•^Binha, ... 129, 

Siromagf, ... 120, 

/Sisira, * . . 03, 


Sitd, .a ... 1 19 ’ 

Siva, ... 118» HO 

Slokas, 8, 13, 81, 61, 29, 
92,07,75,79,92,28,80, . . 

42, 81, 5J, 94, 19, 85, ' 

148, ... ... 84 

Soma-SiddhAuta, ... 97 

Somamanijiala, ... 152 

Spash^a, 142, 138, 47, 155, 

40, 159, 158, 154, 182, 

205, 197, ... ... 201 

Spashta-paridhi, ... 134 

Spashia-pAtas, 154, 153, 

156, ... ... 159 

Spashta-s&ra,... 197, 200 

Spaahta-valana, 191, 185, 

190 187 

Spasliia-valana-Butra, ... 2^3 

Sphuta, 71, 11, 230, 141, 

142, 41, ... ... 18 

Sphuta-gatP, ... ... 13 

Sphuia-kakaka, ... 150 

, Sphuja-koti, ... 146, 144 

Sphuio-lambana-lipta, ... 183 
, SrAvaqa, 8, 03, 01, 08, 65, > 

129,94, ... ... 69 

SfidharfiuhArya, ... 122 

Sringavan, ... ... 117 

Sriugonnati-vasana, 209, 206 

Sripati, ... 188, 140 

... 61 


Sthityardba, ... ... 61 

Suchi, ... ... 41 

Sugandba, ... 119, 111 

Sukla, ... 1*7, 163 

Sukla-paksha, ... 183 

Sukti, ... ... 120 

Sumeru, ... ... 116 

Suparswa, ... ... H8 

S'urya, ... ... 77 

S'urya-siddbAnta, 209, 167, 
96,21,20,97,76, 12,... 129 
Sutra, 254,252,220,261, 263 
Sutaka. ... ... 95 

Swarloka, ... * ...' 120 

Swati, ... 68, 09, 94 

SwayanvAha, .. ..* 227 

SwayanvAha-yantra, ... 22^ 
Swetajala, ... * ... Ip 

Taddhriti, 250, 249,. 256, 



8 


INDEX OF SANSKRITA TEEMS* 


?age 

• 248, 254, 253, 255, 252, 

256, 175, ... ... 374 

Taddliriti-Kujya, ... 174 

r &itila, ... ... 25 

Tala,' ... * ... 227 

Tamraparpa, ... ... 120 

Tantra, 237, 261, 108, 236, 205 
Tapoloka, ... ... 120 

Tatkilika. '... ... 12 

Timi, <... 26,71,27, 54 

Tithis, 131, 132, 25, 24, 
03,260,26,24,184, ... 181 

Tithi-kshaya, ... 6 

Tret*, ... 108, 110 

Trotayuga, ... ... 10 

Trijya, ... ... 194 

Trinsanoas, ... • ... 210 

Triprasna, ... 40, 26 

Triprasnddhydyas, ... 247 

Triprosna-vasana, ... 160 

Trita, ... ' 8 

Trysra, ... ••. 202 

Tula, ... 8, 129 

Tulddi, ... ». 9? 

Thriya, ... 213 

Tdrti, ... ... 2 

Uchcha, ... .» 13 

Uchcha-rekhd, 139, 144, 

141, 146, ... ... 140 

Udaya, 48, 198, 200, 197 
Udaya-lagna, .. 198, 200 

UdaW 220 

Uday^ta-sutra, 172, 219, 
172,220, ... ... J71 

Ujayantara, 333, 132, ... 109 

Udita, ... 171, 170 

Ujjayinf, 134, 243, 11, 253, 

U5, ... - 242 

Ullekha, ... 60 

Unmandala, 175, 352, 161, 

165, 162, 104, 174, ... 176 

Unnata, 252, 254, 46, 370 
Upavritta, ... — J94 

Upavritta-trijya, ... ■ 194 

XJttara, ... 04, 119 

UtfcarM-BhAdrapada, 94, 64, 

** 68 , ... - 00 

IWra-pMlguni, 04, 68, 94 

Uttardshidiia, 68, 61, 61,94, 63 


Paqe , 

Uttara-ydna, ...• 93, lift 

Vadavanala, ... 116,122, 1 If 9 - 
Vaibhrdia, ... , ... 118 

Vaidhrita, . . ... 72 • 

Vaia'dkha, . *94? 129 

Vaivaswata, • .*4, 7 

Vayu,. ’... * ... .118 

Vakra, ... 15,* • J4 # 

Valana, 205, 208, 394, 197, 

195, 46, 203, 196, 192, 

193, 387, 190, 191, 189, 

184, 46, 47, 58, 51, 54, 1 

188,45,185, 186,46,47, 

52, ... ... 53 

Valanas, 205, .45, 196, 188, 191 

Valana sutra, ... 203 I 

Vanija, * ... ... 25 

Varga-prakjifci, ... 241 

Varshas, ... 85, 71, 93, 117 

Varuija, ... H8, 120, 1J9 % 

Vasana, ... 176, 334 

Vasana-Bhdshya, 142, 170, 
p 233, ... ... 130 

Vasanta, ■ ... ... 93 

r Vasishtha, ... ... 139 

Vaskara, ’ ... ... 197 

Vasudeva, ... . 77, 116 

Vata, ... ••• ll*® 

VAyu, ... »•• 119 

Veda, ... ••• 1 

Vedas, 178, 107, 77, 79, 

78, 202, ... ... 152 

Vedavadana, ... ... 107 

Vidy&dhara, ... ... 79 

Vigraha, ... ••• 

Vijaya, ... ... JJ 

Vikala, ... ... *4 

Vikalas, ... jj 

Viksliopa, ... 14, 169 

Vikshepa-kendra, 156, 155, 

154,158,... .» 159 

Vikshepa-vyitta, ... 156 

Vimaadala, ... 15Q, 142 

Vindhya, ... ... 120 

Vipalas, ... - 129 

A Z «,* s 

Vishkambhas, 119, 75 

Visbpu-padi, i«. 


93 



\ 

INDEX OT SAN8KRITA TERMS. v 

It 0 


. Pago 

Vialigu, ... 118, 119 

4Truchika, .... ... 129 

Vjiihabhv , ... . 8, 129 

Vjisl&abhasaiArAnti, v . 180 

- ... 147 

... 119 
72, 75 

• ... 79 

167, 89 

... 150 
... 78 

Yuma, ... 118, 119 

Yama-kotf, 120, 248, 80, 

118, 117, ... ... 115 

Yamvottara-yfitta, ... 152 

Yantra, ... 216, 927, 91 


▼ fivw, 

Vyf 


1 Vyomakaksha, 
Vyulkrama, ... 
Yajanas^ ... 
. Yajuiweda, •H 


• Page . 

YantrAdhyAya, 228, 209 • 
Yaahti, 217, 215, 201, 202, 
160,209, 220,219, ... 2J8. 
Yashti-chinba, ...' 218 

Yoga, ... 25, 75, 24, 90 

Yoga-tArA, ... 65, 62, 64 

Yojanas, 244, 183, 88, 11, 
126,87, 86,49,125, 84, 

184, 122, 207, 127, 86, 

121, ... ... 41 

Yuga, 260, 286, 108, 7, - 

m 6, 5,9, 8,29, 180, 
109,110,287,8,2,110,4, 108 
Yugdnghri, ... 110, 108 

Ynjanaa, ... ... 115 











