Bapu Deva Sastri Bhāskarācārya.

Ancient system of Hindu astronomy online

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Let ft r = breadth of water = x
tlieu bjr first obserTation



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226



Translation of the



[XI. 49.



A man standing up seoa the shadow of a bamboo in tlio
'water — the point of the water at which
""^'^^ the shadow appears is 96 digits off:

then sitting down on the same spot he again observes the
shadow and finds the distance in the water at which it appears
to be 33 digits : tell me the height of the bamboo and his
distance from the bamboo.*



4y

4:3::x:yor8x = 4yorx =

3

bj 2nd obsenraiion 11 < 8 : : :r : 5^ — 48 digiU

11 y — 628

or 8 a? = 11 y — 528 or a? =

8

4y 11 y — 628

thus a? = tnd * = ■ —

8 8

4y 11 y— 628 33 y— 1584

.'. = — or 4 y =

8 8 8



or 32 y = 38 y — 1684, or y = 1684

78 = 1512 digits = ^'

To find width of

4y 1584 >C 4



1684 — 72 = 1512 digits = 63 cubits = height of bamboo.
2iid part. To find width of water or x



8



8



= 212 digiU = 88 cubiU.— L. W.



• Let o = 96 digits
c(2 = 88
ac = 72
&o = 24
let X = distance from obser?er to
bamboo.
Now ee : ae =zj hija

12 X Sx

or 96 : 72 = or t y = =

96 4



Tlion 8 = height of bamboo

4
Again cd:b e i :j h :J h

24x

or 88 : 24 : : a; I y — 48 =

88
Sx

"" 11
Sx

then 1 = lieight of bamboo

11




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XI. 55.] SUldliankh'sirommL 227

50 and 51. Mako a wheel of light wood and in its circum-
ference put hollow spokes all having
A felfrovolvinir mstni- .

mcnt or bwayamvaha tin- boros of the same diameter, and let
them be placed at equal distances
from each other ; and let them also be all placed at an angle
somewhat verging from the perpendicular : then half fill these
hollow spokes with mercury : the wheel thus filled will, when
phiccd ou an axis supported by two posts, revolve of itself.

Or scoop out a canal in the tire of the wheel and then
plastering leaves of the tXla tree over this caned with wax,
fill one half of this canal with water and other half with mer-
cury, till the water begins to come out, and then cork up the
orifice left open for filliug the wheel. The 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

hook, nil it with water and stop up
both ends.

51. And then putting one end into a reservoir of water,
let the other end remain suspended outside. Now uncork
both ends. The water of the reservoir will be wholly sucked
up and fall outside.

55. Now attach to the rim of the before described self-
revolving wheel a number of water-pots, and place the wheel
and these pots like the water-wheel so that the water from
the lower end of the tube flowing into them on one side shall
sot the wheel in motion, impelled by the additional weight of
the pots thus filled. The water discharged from the pots as
they reach the bottom of the revolving wheel, should be drawn





8ff
11





or 2 =


8dP

4


11


X




.'. xz


= 44 % 2 = 88










Tlieii3f =


Zx
4


3)(88

= 8xM

4


= 66,


lioighi of bamboo.








u a











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228 Translatian of tlio [XI. 5G.

off into the reservoir before alluded to by means of a water-
course or pipe.

56. The self-revolving machine (mentioned by Lalla <fec.)
which has a tube with its lower end open is a vulgar machino
on account of its being dependant^ because that which mani-
fests an ingenious and not a rustic contrivance is said to bo a
machine.

57. And moreover many self-revolving machines are to be
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 becauso
ihey have been mentioned by former astroi^omers.

End of Chapter %I. called YANTRifpHYiCYA.



CHAPTER XIL
Veacrlpyion of the season^.

1. (This is the season in which) the kokilas (Indian black

birds) amidst young climbing plants,
thickly covered with gently swaying
and brilliantly verdant sprouts of the mango (branches) rais-
ing their sweet but shrill voices say, *'0h travellers! how
are you heart-whole (without your sweethearts, whilst all
nature appears revelling) iu the jubilee of spring chaitra, and
the black bees wander intoxioated by the delicious fragrance
pf the blooming flowers of the sweet jasmine I"

2. The spring-bom mallikX (Jasminum Zambac, swollen
by the pride she feels in her own full blown beautiful flowers)
derides (with disdain her poor) unadorned (sister) mIlati
(Jasminum grandiflorum) which appears all black soiled and
without leaf or flower ((it this season), and appears to beckon
her forlorn sister to leave the grove and garden with her



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XII. 7.] SiddMnta^dromanu 229

tender budding arms^ agitated by the sweet breezes from the
fragrant groves of the hill of Maulta.

3. In the summer (which follows), the lovers of pleasure
TheaR£9HHAormid-8am. &nd their sweethearts quitting their

mer i^qapoo. ^^^^^ ]^^i^ houses, betake themselves

to the solitude of well wetted cottages of the KVQAKi&A grass,
salute each other with showers of rose-water and amuse them-
selves.

4. Now fatigued by their dalliance with the fair, they
proceed to the grove, where Kama-dbva has erected the
(flowering) mango as his standard, to rest (themselves) from
tho glare of the fierce heat, and to disport themselves in the
(well shaded) waters of its bowbis (or largo wells with steps).

5. (The rainy season has arrived, when the deserted fair

one thus calls upon her absent lover :)
Hainy season.

"Why, my cruel dear one, why do you

not shed the light of your beaming eye upon your love-sick

admirer? The fragrance of the blooming ifXLATf and the

tuvbid state of every passing torrent proclaims the season of

tho nuns and of all-powerful love to have arrived. Why,

therefore, do you not liavo compassion on my miserable lot ?*

C. (Alas, cries the deserted wife, alas 1} the peacocks
(delighted by the thundering clouds) scream aloud, and the
breeze laden with the honied fragrance of the kadamba comes
softly, still my sweet one comes not. Has he lost all delight
for the sweet scented grove, has he lost his ears, has he no
pity — ^has he no heart ?

7. Such are the plaintive accusations of the wife in the
season of the rains, when the jet black clouds overspread the
sky : — angered by the prolonged absence of him who reigns
over her heart, she 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 triple meaning is attempted
to be sMpported : to cfTect tliis. several letters however are to be read dilTereutlv.
-L. W.



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230 Translation of the [XII. 8.

8. The mountain burning with remorse at the guilt of
The siRATKA'LA or leaion having received the forbidden era-

of early autumn. braces of his own pusHPAVATf daughter,

forest appears in early autumn through its bubbling springs
and streams sparkling at night with the rays of the Moon, to
be shedding a flood of mournful tears of penitence.

9. In the hkmanta season, cultivators seeing the earth

^ smiling; with the wide spread harvest,

Hkmanta or earl J winter. , ° ^ ,_ „, , , , .^,

and the grassy fields all beclockod witli

the pcarl-liko dew, and tooming with joyous herds of phuiip

kine, rejoice (at the grateful sight).

10. When the s^is'iba season sets in what unspeakable
S'lB^iHi or doid of win- beauty and what sweet and endless

^^' variety of red and purple does not

tho ' kaciimXu' grove uncoasingly present, when its leaf is iu
full bloom, and its bright glories are all expanded.

11. The rays of tho Sun fall midday on the earth, hcnco
in this s'ls'iBA season, they avail not utterly to drive away the
cold:

12. Here, under the pretence of writing a descriptive

account of the six seasons, I have
SweetB of poetry. « . , , .

taken the opportunity ot indulging

my vein for poetry, endeavouring to write something calculated

to please the fancy of men of literary taste.

13. Where is the man, whose heart is not captivated by
tho ever sweet notes of accomplished poets, whilst they dis-
course on every subject with refinement and taste? or whoso
heart is not enchanted by the blooming budding beauties of
the handsome willing fair one, whilst she prattles sweetly on
every passing topic : — or whose substance will she not secure
by her deceptive discourse ?

14. What man has not lost his heart by listening to tho
puro, connect, nightingale-liko notes of tho genuine poets ? or
who, whilst he listens to the soil notes of the water-swans on



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XIII. 2.] Siddli&nta^siromanL 231

tlio shores of large and overflowing lakes well filled with lotus
flowers, is not thereby excited ?

15. As holy pilgrims delight themselves, in the midst of
the streams of the sacred Ganges, in applying the mud and the
sparkling sands of its banks, and thus experience more than
heaven's joys : so true poets lost in the flow of a fine poetic
frenzy, sport themselves in well rounded periods abounding in
disjilays of u playful tavsto.

End of Chapter XII.



CHAPTER XIII,

Containing useful questions called PRAS^NlDHTiCYA.

1 . Inasmuch as a mathematician generally fails to acquire
Object of the Cliaptor aiid distinction in an assemblage of learned

its pnusc. ^gn^ unless well practised in answer-

ing questions, I shall therefore propose a few for the enter-
tainment of men of ingenuity, who delight in solving all
descriptions of problems. At the bare proposition of the
questions, he, who fancies in his idle conceit, that he has
attained the pinnacle of perfection, is often utterly discon-
certed and a])pa)lod, and finds his smiling cheeks 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 afiected square, i. e.
methods for the solutions of indeterminate problems of the
first and of the second degree, or by means of the armillary
sphere, or other astronomical instruments. To impress and
make them still more familiar and easy I shall have to repeat a
few.



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232 Translation of the [XIII. 3,

3. All arithmetic is nothing but tho rule of proportion :
Praiie of ingeniout per- a»d Algebra is but another name for

•**"•• ingenuity of invention. To the clever

and ingenious then what is not known I I, however^ write for
men and youths of slow comprehension.

4. With the exception of the involution and evolution of
the square and cube roots, all branches of calculation may be
wholly resolved into the rule of proportioii. It indeed assumes
many shapes, but it is universally prevalent. All this arith-
metical calculation denominated Viyi qanita, which has boon
composed in many ways by the wisest of former mathema-^
ticians, ia only for the enlightenment of simple men like
myself.

5. Algebra does not consist in the letters (assumed to
represent the unknown quantities) : neither are the different
processes any part of its essential properties. But Alpfobra ia
wholly and simply a talent and facility of invention, bocauso
the faculties of inventive genius are infinite.

6. Why, O astronomer, in finding the AitABOANA, do you

add SAUBA months to the lunar months
CHAiTBA &c. (which may have elapsed
from the commencement of the current year) : and tell me
also why the (fractional) remainders of AnniMASAS and avamas
are rejected : for you know that to give a true result in using
the rule of proportion, the remainders should be taken into
account.

7. If you have a perfect acquaintance with the^ mis^ra or

allegation calculations, then answer
Question 2nd. - .

this question. Let the place of the

Moon be multiplied by one, that of the Sun by 12 and that of

Mars by 6, let the sum of these three products be subtracted

from three times the Jupiter's place, then I ask what are tho

revolutions of tho planet whoso placo when added to or

subtracted from tho remainder will give tlie place of Saturn ?



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XIII. 9.] SidJlidnta-siromani. .233

8 and 9. In oi*dor to work this propositiou in tlio first
place proceed with the whole numbers
of revolutions of tho several planets
in the kalpa^ adding^ subtracting and multiplying them in thd
manner mentioned in the question : then subtract the result
from the revolutions of the planet given: or subtract the
revolutions of the given planet from the result> according as the
place of the unknown planet happen to be directed to be 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 the number from which it is
to be subtracted^ then add the number of terrestrial days in a
KALPA^ or if tho remainder exceed tho number of terrestrial
days in the kalpa^ then reduce it into the remainder by dividing
it by the number of days in the kalpa.*

* Bni'sKAiiA'onA'RTA bimself has giren the followuig example in his com*
mentary ta'sava'-bha'biita

Suppose Moon to haye 4 reTolutiona in a kA£PA of 60 daya

Sun, 8

Mnre, ..•••• 5««.«...**«. • .•

Jupiter, •••• 7 • .•••..••••••••

Saturn, ..•• 9

Then 4 )f 1 + 8 X 12 + & ^ 6 =: 70 and 7 X 8 rr 21.
As 70 cannot be subtracted from 21 add 60 to it = 81,

Subtract 70,

remainder 11 :

let j» = rcTolutions of the unknown planet, then by the question 11 — p ^ 9
orll — 9 = 2=;>,
but 11 + jp = 9 or p = 9 — 11 = 60 + 9 — 11 = 68 :
It thus appears tlmt tho unknown planet has 2 or 58 rerolutions in the

KALPA.

Now let us see if this holds true on the 28rd day of this ka£PA s
revolutions signs «

for Moon, if 60 : 4 : : 28 : 6 .. 12o this X 13=6.. 12%
dun, 60 : 8 : : 28 : 1 .. 24 this X 12 = 9 .. 18,
Mars, 60:5:s2d:ll.. 0this>c6 = 6.0

signs 10 •• subtracted
Saturn, 60 : 9 : : 23 : 5 ..12

Jupiter, 60 t 7 : : 23 : 8 •• 6 this >C 8= .• 18 from
for;^, 60 ! 2 : : 23 : 9 •• 6 this sub. from 2 •• 18 remainder

9.. 6
corresponding with Sutuni, 6 . . 12



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234 Translation of the [XIII. 10.

10. The algebraical learned, who knowing the sum of the

additive months, subtractive days
Question 3rd.

elapsed and their remainders, shall

tell the number of days elapsed from the commencement of

the KALPA, deserves to triumph over the student who is puffed

up with a conceit of his knowledge of the exact pulverizer

called sam'slishta united, as the lion triumphs over the poor

trembling deer he tears to pieces in play.

11. For the solution of this question, you must multiply

the given number of additive months,
subtractive days and their remainder,
by 863374491684 and divide by one less than the number of
lunar days in a kalpa i. e. by 1602998999999, the remainder
vrill be the number of lunar days elapsed from the beginning
of the KALPA. From these lunar days the terrestrial days may
be readily found.*

9 •

oriL 60:68 x: 23:2:24. Then 2 .. 24 added

to2 .. 18



■til] giree Saturn's plaoe 5 .. 12

When p = 9 — 11, then as 11 cannot be subtracted from 9 the sum of 60
is added to the 9. The reason for adding 60 is that this number is always be
denominator of the fractional remainder in finding tlie plaoe of the planets ;
for the proposition.

If days of kalpa : roTolutions : : given days give : here the days of kalfa
are assumed to bo 60 honoe GO is added.— L. W.

* [When the additi? e months and subtraotifo days and their remainders aro
gif en to find the ahabqa^a.
Let / = 1602999000000 the number of lunar days in a kalpa.
e =: 169800000 the number of additive months in a kalfa.
d =: 26082660000 the number of subtractive days in a kalpa.
A = additive months elapsed.
A' = their remainder.
II =: subtractive days elapsed.
B' = their remainder. >

a= the given sum of the ebpsed additive montlis, subtractive days and
their remainders,
and » = lunar days elapsed ;

A

tlien say As I :»::»: A + — j

I

B'
As f : <l : : « : B + j

/



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XIII. 12.] • SUldhanta-BiromanL 235

12. Givoii tlio sum of iho olapsod additivo months^ sub-
tractive days and their remainders^

KxoinpiO.

equal (according to BiiAiiMAOurrA^s
system) to 648426000171 ; to find the ahaeoana. He who
shall answer my question shall be dubbed a '^ bbahma-sid-
DH^NTA-yiT^' i. e. shall be held to have a thorough knowledge

of tho BRAIIMA-SIDDniNTA.*



A' + B' A' + B'

.'. As / : « + d : : X : A + B + ' or v -f ■

I I

.'. (u + d) a? = iy + A' + B', or (« + <0 « — Zy = A' + B',

•nd y = A + B;

.-. byaddition, (« + <0 « — (i— l)y = A + B + A'+ B',

= a;
by substituiioD, 2GG75850000 a: — 1602998999999 y = a :
now let, 26675850000 x' — 1602998999999 y' =: 1,

then we shall have by the process of indeterminate problems
0^ = 863374491684.
Again, let m = •(' <^ >^d » = 2 — 1,
tlwn m« — »y=ai(l)

and m »' — » y^ = 1 ;

and mni — mnt =iO t

,\ m {ax^ — ni) — ^^Coy^ — m£) z=zai

which is similar to (1) ;
/, a? r= a a?* — * » <

= 863374491684 « — (/ — 1) <.

Hence the rule in the text. — B. P.]

• Solution. Tho giTon sum = 648426000171 and t he lunar days in a XALPA
= 1602999000000 t

648426000171 9C 863874491684

A — =849241982386

1602998999999 and 10800 remainder :

.'. 10300 these are Innar days elapsed.

To reduce them to their equivalent in terrestrial days says

}161 subtraotiTO
, days and romajn-
" der amounting
267426000000.
.*. From 10300 Lunar days
subtract 161 Subtractive days

remainder 10189 Terrestrial days or liiABOAtiA.

19ow to find additif months elapsed.
If lunar days 1 . additive months 1 . , lunar days 1 10 additive months and
inaKALPA J • ofKALFA J ' * 10300 J remn. 881000000000.

10 additive months = 300 lunar days.

.-. 10300 — 800 = 100,00 SAiTBA days elapsed.

Hence 27 years 9 montlis and 10 days elapsed from the commencement of
XALFA.— L. W.



8 2



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236 Tranalatian of the • [XIIT, 13.

13 and H, Given the sum of the remainders of the revo-
lutions, of the signs, degrees, minutes
and seconds of the Moon, Sun, Mars,
Jupiter, the s'iqhbochchas of Mercury and Yenus and of
Saturn according to the DHfvRiDDHiDA, including the remainder
of subtractive days in finding the ahargana, abraded (reduced
into remainder by division) by the number of terrestrial days
(in a tuga). He who, well-skilled in the management of
8PHUTA KUTTAKA (exact pulverizer), shall tell me the places of
the planets and the ahargana 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 unravelling the thick mazes and wildernesses which
occur in mathematical calculations.

15. If the given sum abraded by the number of terrestrial
days in a yuga, on being divided by 4,
leaves a remainder, then the question
is not to be solved. It is then called a khila or an ^' impos-
sible" question. If, on dividing by 4, no remainder remain,
then multiply the quotient by 293627203, and divide the
product by 394479375. The number remaining will give tho
AHARGANA. If the day of the week does not correspond with
that of tho question, then add this ahargana to tho divisor
(394479375) until the desired day of the week be found.'*'

* [Aooording to the DHfTBlDDHiBi TiiiTitA of lalla the terrestrial dajs in
a YUOA = 1677917500 and the lum of all the 8G remainders for one day ^
118407188600908 : this abraded by the terrestrial days in a tuoa =r 259400968.

Let » == AHABOAtf ▲ then say

As 1 I 259400968 : : « : 259400968 >C 9

This abraded by 1577917500 the terrestrial days in a YUOA will be equal to
1491227500 the given abraded sum of the 86 remainders, now

let y =: the quotient got in abrading 259400968 x by 1577917500^ then
259400968 9 — 1577917500 y = 1491227500.

It is evident from this that as the ooeffidents of 9 and y are divisible by 4^ the
given remainder 1491227500 also must be divisible by 4, otherwise the question
will be impossible as stated in the text.

Hdnoe, dividing the both sides of the above question by 4,

64850242 9 — 894479875 y == 872806875 : (A)

And let 64850342 «' — 894479875 /^ 1, (B)



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XIII. 17.] SiMMnia-sirommi. 237

16. Tell mo, my friond, what is tlio ahabqana when on a
^^^ J Thursday, Monday or Tuesday, the

35 remainders of the revolutions,
signs, degrees, minutes and seconds of the places of the
planets, (the Sun, the Moon, Mars, Jupiter and Saturn and
the s'f GHROCHCHAS of MorcuTy and Venus) together with the
remainder of the subtractive days according to the dh£vbid-
DiiiDA, giyo, when abraded by tho number of terrestrial days
in a YUGA, a remainder of 1491227500.*

17. The place of the Moon is of such an amount^
Question 6th. that

Tli^minute^ + 10 = the seconds

tho minutes — seconds -{- 3 = degrees

the deirrees

g2 = Bigns.

/. s^ = 293627203 bj the processes of indeterminate problems.
Koir let a == 64850242, 6 = 894479376, and c = 872806876 ;
.*. we hare (he equations (A) and (B) in the forms
am — 6 jf = e !
and a a/ — ft y* =: 1,

« =: «' — hi (see the proooding note)
= 293627203 e — 894479376 1 1
as stoted in the text.— B. D.]

* Solution. Tho given snm of the 36 remainders in a Tuai =r 1491227500
according to the PHfyjimDHiDi taktba.

.-. 1491227500-7-4 = 872806876:
872808876 x 293627203

and .'. = 277496471 and remainder 10000 L e.

894479376

AHABGACrA.

10000

.•. = 1428 — 4 remainder, i. c. 10000= AUABQAtrA on a Tnosdaj, for

7
the YVOk commenced on Fridaj.
This would be the ahabqa^a on a Tuesday.

To find the ahaboa^a on Monday, it would be necessary to add the reduced
ierrostrial days in a tuqa to this 10000, tiU the remainder when diTided by
7 was 8.

10000 + 804470876 >C 2 788968760

= = 112900821 — 8 remainders

7
Monday :

10000 + 394479376 X 3

and ■ = = 169064017 — 6 remainder or =;

7
Tliorsday.— L. W.




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238 Translation of iho [XIII. 18.

And the signs, degrees, minutes and seconds together
equal to 130. On the supposition that the sum of 'these four
quantities is of this amount on a Monday then tell me, if you
are expert in rules of Arithmetic and Algebra, when it will bo
of the same amount on a Friday.^

18. Reduce the signs, degrees and minutes to seconds,
adding the seconds, then reducing tho
terrestrial days and the planet's re-
volutions in a KALPA to their lowest terms, multiply the seconds
of tho planet (such as the Moon) by tlio terrestrial days
(reduced) and divide by tho number of seconds in 12 signs :
then omitting the remainder, take the quotient and add 1 to it,
the sum will be the remainder of the bhaqanas revolutions.f





* Let X =: minutes




9 + 20

then = aeoondfl

2


9


ar-l-20
+ S = degree! .




= 130



• = 89 aeoonds.
2

68 — 39 -t- 8 = 22 degrees.

22

— = 11 signs.
2

Uenee the Moon's place = lU. . . 22o . . 68' . . 89".
t The mean pUoe of the Moon = lU .. 22o .. 68' ..89'' = 1270719^
The number of seconds in 12 signs = 1296000.

TerrestrUl days in a kalva = 167791&i60000 1 These dirided bj f 95G313

V 1G50000 become dbi- <
BcTolutions of Moon =57768300000 J diia or reduced. 185002.



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XlJi. 20.] Siddhanla-sira^nanL 239

19. Tho romainder before omitted subtracted from the
divisor will give the remainder of seconds : if that remainder
of tho 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 be found at any 8un«
rise) : but if less it may be solved. Then from the remainder
of the seconds tho AnABOANA may bo found (by the kuhaka
pulvonViOr na given in tho LfLAvXiI and BfjA-GANiTA) Or,

20. That number is the number of ahaegana by which the
reduced number of revolutions multiplied, diminished by the
remainder of the revolutions and divided by the reduced
number of terrestrial days in the kalpa, will bear no remainder.
The roducod number of terrestrial days in a kalpa should be
added to the auaboana such a number of times as may make
the day of the week correspond with the day required by
the question.

Now when the mean place of t]ie Moon wmi sought, the rul6 was


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