पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

III.31 III. PAULIŚA-SIDDHĀNTA 71
not being scientific and amenable to intelligent guess. Still, as there is much to say here too, we are
dealing with these, unlike TS who have refrained from doing so.
[भद्रविष्णुमते दोषः]
तिथिनक्षत्रच्छे(द)प्रतिपत्तिर्यदि तथा ततः साधुः |
न तथा च भद्रविष्णोस्तथाऽपि [न] विनिवर्तते लोकः || ३२ ||
Defect in Bhadraviṣṇu
32. If the tithis and nakṣatras as seen from observation of the sky agree with
those computed according to the Śāstra, then the Śāstra is correct and fit to be
accepted. It is not so in the case of Bhadraviṣṇu's work; still people do not turn
away from that and follow the correct Śāstras.
(This remark about the nature of people is true even today).
For the sake of syntax the long dā has been shortened by us. Consistent with the idea intended,
the negative particle na has been added in the fourth foot.
[पादादित्यमते दोषः]
न युगपदुदयो भानोरस्तमयो वाऽपि भवति सर्वत्र |
कस्मिन् देशेऽस्तमये पादादित्येन (नोक्त)मिदम् || ३३ ||
Defect in Pādāditya
33. Sunrise or sunset is not at the same moment in all places on the earth; (so
the place must be mentioned whose sunrise or sunset is taken as the epoch for
finding the days and doing the computation). But Pādāditya, who has placed
the epoch at sunset, has not mentioned the sunset of which place he refers to.
(So his work is faulty.)
There seems to be some error in the last foot and we are not sure-whether Pādāditya is a person
as we have interpreted, or something else or whether the word is the same at all and not an nth
incarnation of the original.
[रोमकमते दोषः]
मार्गा(द)पेतमेतत् काले लघुता न तावदतिदूरे |
'खविषयभूताष्टरसै'रब्दैः पश्याऽस्य विनिपातम् || ३४ ||
32a. A.B.C.D. छेदा 33a. A.B. भानुः
b. A.B1. प्रतिपत्तियदि; B2. प्रतिपतियदि. c. D. ॰ऽस्तमयः
B. साधु d. B. पादादित्येन; D. पादाद्विनेन
c. B. भद्रविष्टमो A. भुक्तिमिदं; B. भक्तिमिन्दुः; C. भुक्तमिदम्;
d. A.B.C.D. om. [न] D. भुक्तं विदुः

72 PAÑCASIDDHĀNTIKĀ III.34
Defect in Romaka
34. This gaṇita work had deviated from the right path handed down by a
hierarchy of good teachers and the day of its exposure is not far distant.
Witness its downfall in 68,550 years!
To agree with the fifth case in mārgāt we have corrected upeta into apeta.
It may appear strange that the author calls 68,550 years as a not far distant date. But that
depends on the outlook of people, and the Hindu mind, especially the old Hindu mind may con-
sider even this as a comparatively short period.
Or this verse may belong to the criticism of the Romaka following immediately, strayed to this
place by the mistake of the scribe. In that case the first letter kha in khaviṣayabhūtāṣṭarasaiḥ should
be corrected as sva and the expression taken to mean "6855 years, by its own measure", in which
case the following is the meaning:-
"This Romaka has not come down through a hierarchy of good teachers
because it follows the Tropical year instead of the traditional sidereal year. It
will be exposed in a period of 6855 of its own tropical years, and people will
abandon it."
How this will happen and how the number 6855 can be arrived at almost exactly, will be shown
in explaining the next verse.
(रौ)मकमहर्गणं (वा) (तद)र्कमिन्दुं च गणयतां ग्रा(ह्यम्)
चैत्रस्य पौर्णमास्यां नवमी नक्षत्रमादित्यम् ॥ ३५ ॥
35. If we adopt the days from epoch resulting from the tropical year as
adopted by the Romaka and the Sun or Moon resulting therefrom, we must
accept Punarvasu as the nakṣatra of the full moon of the month of Caitra,
instead of the expected Hasta or Citrā, Punarvasu which is the nakṣatra of
Caitra-Śukla-navamī.
This connection has been strongly established in people's mind by the observance on Caitra-
navamī as the birthday of Lord Rāma, hero of the Rāmāyaṇa, well known as born in the asterism
Punarvasu.
This is how this will happen: The months Caitra, Vaikśākha etc. are so called because the Moon
at new moon in these months is in the vicinity of Citrā, Viśākhā etc. Thus, in a given month, the full
moon, i.e. the Moon of the 15th tithi, is in a given nakṣatra or nearby, so that the other tithis also are
34a. A.B.C.D. मार्गादुपेत a-b. A.B1. ॰णं पादं मर्क॰; B2.3. C.D. ॰णं पादमर्क॰
b. B. न तावदति दूरो A. ॰यतां तां ग्राह्या; B. ॰यतां ग्राह्याः C. ॰यता ग्राह्या;
c. A. षविषय; B. om ख D. ॰यतां ग्राह्य
c. B2. पौर्णिमास्यां
35a. A.B.C.D. रोमक d. D. नवम्यां

III.35 III. PAULIŚA-SIDDHĀNTA 73
connected with particular nakṣatras. For e.g. as the Moon of the fifteenth tithi in Caitra is near Hasta
or Citrā, the Moon of navamī, six days before, is near Punarvasu or Puṣya, because the tithi, the day
and the nakṣatra, have approximately the same duration. In the same way, as the Moon of the 15th
tithi of Śrāvaṇa is near Śravaṇa, the Moon at Śravaṇa Bahula Aṣṭamī, eight days after that, is near
Rohiṇī, which is also a thing well known. Now, when the months (synodic) are 'tied' to the nakṣatras
as mentioned, there will be this conformity. But the months are kept tied to the nakṣatras if the solar
year is sidereal, and not tropical like that of the Romaka, i.e. if the solar year begins as in all other
Siddhāntas with a fixed point on the ecliptic like the first point of Meṣa, and not the movable vernal
equinox, the so called First Point of Aries, as in the Romaka or the new Indian Rashtriya Panchang.
As the difference between the two points (i.e. the ayanāṃśa) increases, the above mentioned con-
formity will gradually decrease and when the ayanāṃśa accumulates to 30° the full moon of the first
month will fall in the Phalgunis instead of near Citrā and though called Caitra the first month will
really be ‘Phālguna’. This non-conformity has already happened in the Rashtriya Panchang, with the
ayanāṃśa more than twenty degrees now. If it accumulates to 6 nakṣatras, (i.e. 80°) the full moon of
the first month, still called Caitra, will be in the nakṣatras Punarvasu or Puṣya, though the month
will really be Pausha. Thus Punarvasu connected with the Navamī of the real Caitra will occur at
the full moon of the so called Caitra of the Romaka (or the Rashtriya Panchang). The Navamī of the
new Caitra, occurring 6 days before the full moon day, will occur in Apabharaṇī, 6 nakṣatras earlier.
When a situation thus arises contradictory to their belief, people will realise that the sidereal year
is the proper thing and discard the Romaka.
Incidentally we may mention that the same confusion will arise in following the Rashtriya
Panchang also. One thing we want to say here. We do not deny that the tropical year best suits civil
purposes, but a luni-solar calendar based on the sidereal year also will suit our religious purposes
best. Therefore they have to be kept apart. (A civil calendar based on the tropical year, we already
have in the Christian Calendar we have been following, which is practically worldwide. As for the
defects in it, the ‘Calendar Reform’ will take care of it, while in making this reform we have taken
a step in isolating ourselves.) One thing could have prevented the confusion. If they had adopted
the seasonal-month-names like Madhu, Mādhava etc. preserved for us in the Vedas (vide III. 25)
instead of trying to fit in the sidereal luni-solar-calendar months, Caitra, Vaiśākha, etc., this confusion
would have been avoided and they would not have simply added one more to the one hundred
contradictory Panchangs already extant.
To continue: We shall compute after how many years the exposure of the Romaka, as mentioned
by the author, would happen. The ayanāṃśa was zero at the author's time, as we have shown on
several occasions. We must calculate when it accumulates to 6 nakṣatras. The Siddhāntas of the
author's time use a sidereal year of days, 365-15-31 nearly, and the tropical year of the Romaka is
days 365-14-48 (See VIII. 1). Therefore, the Romaka year begins earlier by 43 vināḍis, every year,
and this is equivalent to the rate of ayanāṃśa per annum, 42", nearly. If ayanāṃśa to become 42"
takes one year, to become 6 nakṣatras it will take, 6 × 800 × 60 divided by 42 = 6857 years nearly.
It is this that is given by the author as 6855 in the previous verse explained.
कालाऽपेक्षा विधयः श्रौताः स्मार्ताश्च तदपचारेण ।
प्रायश्चित्ती भवति द्विजो यतोऽतोऽधिगम्येदम् ॥ ३६ ॥
36. All the injunctions of the Vedas and Smṛtis are based on the proper time,
and by not performing the rites at those times the performer, especially a twice-born,

74 PAÑCASIDDHĀNTIKĀ III.36
acquires sin which is to be expiated. Therefore, a study of this Romaka itself is to be
expiated.
How is that sin may accrue by not performing a rite at the time enjoined by the Śāstras. But by
simply studying the Romaka, we cannot say one would also perform the rite at the improper time.
The Romaka may give the time wrongly, but one may study it not for the sake of using its time, but
for other purposes, as for instance to understand where it goes wrong, and expose its weakness to
others and save them, for which the man who studies even deserves merit. Let us understand that
these things, sin and merit, are subtle and cannot be known without a deep study of the Śāstras, and
if the author steeped in the Dharma Śāstras says a thing, let us accept it. The Vedas promise svarga
not only to the performer of the yajña but also to one who knows how to perform it properly. We
frequently meet in the Vedas the expression ya u cainam evam veda. The Vedāṅga Jyotiṣa says that
people who know astronomy know as it were the correct performance of the sacrifices themselves,
yo jyotiṣaṃ veda sa veda yajñān and they go to svarga after establishing a long line of progeny in this
world.
Now, it stands to reason that if the mere study of a good thing gives merit, the mere study of a
bad thing brings sin. It is said that even association with bad characters and sinners bring sin, as also
doing sinful things even in dreams. As for the argument that the person even deserves merit for
intending to keep off people from the improper times, he does deserve it and will get it. But that
does not mean expiation is not called for. Contact with craftsmen may be necessary to keep the
temple idols in form, but that does not mean that purificatory ceremonies need not be performed
for the idols on that account. We can say this much that in these cases the expiation is light, like the
utterances of the Lord's Name, like 'Kṛṣṇa Kṛṣṇa, Śiva Śiva! We should also take into consideration
the spirit if the times in which these statements were made.
स्फुटगणितविदिह लब्धा धर्माऽर्थयशांसि दिनकरादीनाम् ।
(कुकरणकारस्सत्यं सहते नरके कृताऽऽवासा: ॥ ३७ ॥)
37. The person having correct knowledge of the Sun, Moon, etc. gets
Dharma, which will take care of his future world, Artha which will ensure his
prosperity in this world and fame, which will perpetuate his memory. But the
bad astronomer who misleads people by his writings will certainly have to go
to hell and dwell there.
36a. A. विधय
b. B. श्रौता स्मा०
c. B1.2. प्रायश्चिती. A1. भवती
c-d. A. कुकरणविदो द्विन्यो ये कथयन्त्यस्फुटं कुकरणकारः
(A2. करः)
B. अकरणविदो द्वित्यो ये कथयसत्यं अकरणकरः सहते
नरके कृतावासाः ।
C. कुकरणविदोद्विन्यो ये कथयन्त्यस्फुटं कुकरणकरः
सहते नूनं नरके कृतवासाः ।
D. कुकरणविदो द्विजा ये कथयन्त्यस्फुट [म] सत्यं
[स गणितम्] ।
कुकरणकारसहि [ताच्च] ते क्षणं नरके कृतवासाः ॥
37. In A and B, स्फुट etc. occurs as the
second half of the verse. It is put here as
the first half to suit the sense.
a. D. लब्ध्वा

III.37 III. PAULIŚA-SIDDHĀNTA 75
It must be noted here that when even the person with correct knowledge gets so much, the writer
will get more. It must also be noted that only the writer of bad astronomy goes to hell, not the
reader, whose sin is small.
In this verse there is a jumbling of words and phrases and induction into the text extraneous
words intended as commentary. The words, sphuṭagaṇitavid etc. seem to be the first half of the verse
because in the first foot there are twelve and in the second eighteen syllables. Therefore what
comes before that is the second half. In that, there are may syllables more than the required twenty-
seven. Selecting the required words alone, we have reconstructed the third and fourth foot.
For the observations of K S. Shukla on 32-37 vis-a-vis NP, see his paper ‘The PS of VM (1)’, JIHS
9 (1974) 62-76.
[इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां
पौलिशसिद्धान्तो नाम तृतीयोऽध्यायः ||]¹

1. A.B. पौलिशसिद्धान्तः; C.D. इति पौलिशसिद्धान्तः ||
   Thus ends Chapter Three entitled ‘Pauliśa-Siddhānta: Planetary
   Computations etc.’ in the Pañcasiddhāntikā composed
   by Varāhamihira

Chapter Four
THREE PROBLEMS — TIME, PLACE AND DIRECTION
४. चतुर्थोऽध्यायः
त्रिप्रश्नाधिकारः
Introductory
Problems on Time, Place and Direction, involving spherical trigonometry, are dealt with in this
chapter. The first fifteen verses are devoted to the construction of a table of sines. As this kind of
matter does not involve constants specific to any siddhānta and is commonly found in all siddhāntas,
we cannot say which Siddhānta this belongs to, Pauliśa or Saura, the only two siddhāntas meant to be
expounded in detail by the author. Probably it is the author’s own, meant for both, or taken from
both. Two things point to this conclusion: In the part of the work dealing with the Saura, viz., chs.
IX, X, XI, XIII, XIV, XV, XVI, and XVII, no space is given to the sine tables, though required, and
to the problems dealt with here, and therefore if these are not meant for Saura, it would be imperfect
though almost full. On the other hand, certain redundant and crude rules point to this chapter’s
connection with the Pauliśa, as also its position in the chapter distribution in the PS text.
[ज्यानयनम्]
षष्टिशतत्रयपरिधेर्वर्गदशांशात् पदं स विष्कम्भः |
तदिहां(शच)तुष्कं संप्रकल्प्य रा(श्य)ष्टभागज्या || १ ||
Table of R Sines

1. Take the circumference as measured in 360 units, square it, take the tenth
   part of the square, and find its square root. The result is the diameter of the
   circle in the units taken. We assume the diameter to be 4°, (i.e., 240′) and
   hereunder give the tabular sines of angles for 3° 45′ interval.
   The rule is: diameter = √circumference²/10. It comes to this: d = c/√10. The formula, d = c/π is
   well known, and the author has taken √10 as an approximation for π which is incommensurable
   and usually represented by the approximate values, 22/7, 355/133, 3.1416 etc. The Sūrya Siddhānta
   too gives √10 as the value of π in its instruction to find the circumference of the earth from its
   diameter (I.59.): “The earth’s diameter is 1600 yojanas. Square this, multiply by 10, and find the
   square root. This is the earth’s circumference.” By thus taking √10 for π, an error of about 0.0067%
   results, and for a circumference of 21,600′, we get the radius 3415′, instead of the well-known
   3438′. But it must be mentioned here that this error does not affect the computation of the sines
   1a-b. A.B. परिधे वर्ग c. A.B. तदिहांशाश्चतुष्कं (B. ॰ष्क)
   b. A. विष्कुम्भः d. A. संप्रकल्प्य; B. प्रकल्प्य. A.B. राश्याष्ट०

IV.2 IV. THREE PROBLEMS 77
mentioned in the succeeding verses, because it can be shown that the author derives the sines from
a correct formula, (not dependent on this wrong ratio of the diameter to the circumference), based
on 120' as the radius of the circle. If he had depended on the wrong value, the first tabular sine,
i.e. sin 3° 45' would be 7' 54", (being the 96th part of the circumference, where the sine is indistin-
guishable from the arc), and not the correct 7' 51" as given by the author.
Taking the diameter as 4°, and thereby the maximum sine (i.e. the radius) as 120', is arbitrary.
In general, the Siddhāntas give the maximum sine, 3438', as arrived at from taking the circumfer-
ence as 360° or 21600'. The Vākyakaraṇa makes it 43°. In actual work, the sines enter only as a ratio
to the maximum sine, and therefore no harm, will result by taking these different maximum sines.
TS and NP have not understood the meaning of the second half of the verse, and mis-interpret
aṁśacatuṣkam as quadrant.
व्यासार्ध[स्य] कृतिर्ध्रुवसंज्ञिका कृतांशस्ततः स मेषस्य |
ध्रुवकरणी मेषोना द्वयोस्तु राश्योः पदं ज्याः स्युः ॥ २ ॥
2. The square of the radius, (i.e. 14,400), is called dhruva (karaṇī), (literally,
‘Fixed Irrational’). The fourth part of it, (i.e. 3600), is the karaṇī (Irrational)
related to the first sign, (or 30°). Dhruvakaraṇī minus the karaṇī of Meṣa, (i.e.
14,400 − 3600 = 10,800), is the karaṇī of two signs, (or 60°). The square root
of a karaṇī is the tabular sine.
Being square of tabular sines given in minutes, the karaṇīs are squares of minutes, which is their
peculiarity as given by the author, though this is not mentioned explicitly. The other well-known
characteristic of a karaṇī, viz. irrationality, is found in all karaṇīs except 14,400 and 3600, though
the author calls these also karaṇīs in a general way.
In modern terminology the word sine used in connection with the angle is defined thus:
[Figure: Right-angled triangle ABC with right angle at C and angle marked at B]
Fig. IV. 1-a
In the right angled triangle, (fig. 1-a), sine ∠ B = AC/AB, or sine ∠ A = BC/AB, i.e. as the ratio of
the opposite side to the hypotenuse. In tabulating the sines, the hypotenuse is taken as unity, and
the ratio expressed as a decimal fraction.
2a. A.B. कृते ध्रुव०; C.D. कृतिध्रुव० c. B1.3. ये योना; B3. येषोना
b. A.B. ०ज्ञिता. A.B. कृताशाःस्ततः. A.B. सशेषस्य d. A. दयोस्तु; B. दयो सु

78 PAÑCASIDDHĀNTIKĀ IV. 2
The ancients however expressed the sines in minutes-length or, more accurately, in minutes and
seconds-lengths, the maximum sine called Trijyā (meaning ‘the sine of three signs’, i.e. 90°), occur-
ring separately in the work to make up the ratio. This is the way in which they conceived the sine
(meaning ‘bow-string’ from its Sanskrit equivalent śiñjinī, synonymous with jyā).
In Fig. 1-b. A₃ E F₃ D is the circumference of the circle, centre B. A part of the circumference like
ADF, A₁D F₁, etc. is called dhanus (literally, ‘bow’) or arc.
Fig. IV. 1-b
The straight lines ACF, A₁C₁F₁, etc. forming the ‘bow-strings’ of the respective ‘bows’ are the jyās
or full sines. But in actual practice, the halves of the full sines AC, A₁C₁, etc. above are used with
the name of ‘sines’, with respect to the half-bows or arcs, AD, A₁D₁, etc. Because the arcs AD etc.
are as the angles ABD etc., the sines AC etc. are spoken of with respect to the angles ABD (= ABC)
etc. also. Thus, AC is the sine of ∠ ABD or arc AD, A₁C₁ is the sine of ∠ A₁BD or arc A₁D₁ and so on. It
is this connection of the sine with the arc that has given it the nature of a length, which is expressed
in minutes and seconds on account of the connection of the arc with the angle at the centre. It may
be mentioned here that CD, C₁D, C₂D etc., appearing like the arrows on the respective bow-strings,
are called śara (meaning ‘arrow’).
If A₃BD is a right angle, i.e. three signs, then, obviously, A₃B is the sign of this angle, i.e. it is the
sine of three signs, and therefore called trijyā. Its length is clearly half the diameter A₃BF₃, i.e. the
radius, equal to 120′.
Now, let the angle ABD be equal to one sign, i.e. 30°. ABD = DBF = 30°. ∴ ∠ ABF = 60°. AB = BF,
being radii. ∴ ∠ BAF = ∠ BFA = 60°. Thus ABF is an equilateral triangle, and AF = AB = 120′. ∴ AC
= AF/2 = 60′. Thus sine 30° = 60′. Its karaṇī is its square, viz. (60′)² = 3600, the karaṇī of Meṣa as
mentioned by the text.
Then, let ∠ A₂BD be equal two signs, or 60°. A₂BD = DBF₂ = 60°. ∴ C₂ is a right angle. So the
karaṇī of 2 signs = A₂C₂² = A₂B² − BC₂² = A₂B² − AC², (∵ △ A₂B C₂ ≡ △ BAC), = 120² − 60² =
10,800, A₂B being the radius. This also agrees with what the text says. (The square root of 10800
minutes, i.e. 103′ 55″, is the sine of 2 signs, which agrees with the value given in the table.)

IV. THREE PROBLEMS 79
Incidentally, we shall derive the karaṇī and sine of one and a half signs, i.e. 45°, mentioned in
verse 4. Let A₁BD be equal to 45°. ∠ A₁ = 45°, and ∠ C is a right angle. ∴ A₁C = C₁B. But, A₁B²
= A₁C₁² + C₁B² = 2 A₁C₁². ∴ A₁C₁² = 120²/2 = 14400/2 = 7200 = the karaṇī of one and a half signs
as mentioned in the text. Its root, 84'51", is sine 45°, agreeing with what is given in the tables.
शेषेष्विष्टेषु धनु-र्द्विगु(णं) पदात् प्रोज्झ्य शेषगुणहीना [त्] ।
[व्यासस्याऽर्धार्द्धर्गं] द्विगुणकरण्यां समायोज्यम् ॥ ३ ॥
तत्पादोऽभिमता [स्याद्] ध्रुवा तदूनाऽवशेषपिण्डस्य ।
ध्रुवकरणीदलमध्यर्धसंज्ञमन्योऽत्र विधिरुक्तः ॥ ४ ॥
इष्टांशद्विगुणोनत्रिभज्ययोना त्रयस्य चापज्या ।
षष्टिगुणा सा करणी तया ध्रुवोनाऽवशेषस्य ॥ ५ ॥
3-5. The other tabular sines, (i.e. sine 3° 45′, sin 7° 30′ etc. other than the four
mentioned of the total 24) are formed successively in the following manner:
Let the angle or arc for which the sine is required be θ.
I. Sin²θ = ¼[sin²2θ + {120 − sin(90° − 2θ)}²]
II. Sin²θ = 60 × {120′ − sin (90° − 2θ)}, where the sines are in minutes etc. Of the
24 sines, the karaṇī of the nth sine = 14400 − the karaṇī of the (24 − n)th sine. 7200
is the karaṇī of one and a half signs, i.e. 45°.
Thus, as karaṇīs 8, 12, and 16 are known, those of their halves etc. and (24 − halves) etc. can be
found successively. Thus all the sines from 1 to 24 can be found. Of the two formulae, the first is
suited to geometrical representation, and the second to computation.
Example 1. Given the 8th karaṇī (i.e. of 30°) 3600 and its sine 60′, the 16th karaṇī (i.e. of 60°) 10800, and
its sine 103′ 55″.33, find the 4th and 20th karaṇīs and sines, using each of the two formulae.
The desired sine is the 4th, i.e. of 4 × 3° 45′ = 15°. 2θ = 30°, 90° − 2θ = 60°.
3a. A. धनुर्द्वि. A.B.C.D. ॰गुणपदा॰ 4a. A. तपदो; B. पदो; C.D. त[स्य] पदो.
b. A. पदायोज्य; B. पदायोज्य्; C.D. पदायॊग॰ A.B.C.D. ॰भिमतज्या
A.C.D. गुणहीना; B. गुणाहीना b. A. तदुना. B. ॰विशेषे
c. A.B. तृव्यासपादार्द्धाद्धर्गं; (B. om तु; B2. ०र्गं); d. A. र्द्धसंज्ञा; B1.2. र्द्धसंज्ञां; B3. र्द्धं संज्ञा
C.D. [त्रिज्या तदर्धवर्गौ] द्वि C.D. संज्ञकोऽन्योऽत्र. A. विधिनुक्तः
d. A. कारथो; B. कारयो 5a. A.C. इच्छांशद्विगुणेन (C. ॰णोन)
A. समायोज्यं; B. समाप्रोज्यन्त; b. B3. त्रयंस A.B. वायज्या
C. द्विगुणज्यार्धस्य संयोज्यः; D. द्विगुण [र्ध] करणी c. A. स कारणी; B. स करणी
समायोज्यः d. B. षपा. A.B. ॰नामशेषस्य

80 PAÑCASIDDHĀNTIKĀ IV. 5
I. The 4th karaṇī = ¼ [sin² 30° + (120 − sin 60°)²]
= ¼ [60² + (120 − 103′ 55″.33)²]
= ¼ [3600 + (16′ 4″.67)²]
= ¼ (3600 + 258 97/144)
= ¼ (3858 97/144) = 964 385/576
∴ the fourth sine, i.e. sine 15° = √964 385/576 = 31′ 4″.
II. The 4th karaṇī = 60 (120 − sin 60°)
= 60′ (120′ − 103′ 55″.33)
= 60′ × 16′ 4″.67 = 964 385/576
From this, sin 15° = √964 385/576 as before = 31′ 4″.
We shall prove the first formula geometrically, and
derive the second from the first.
Fig. IV. 2
In Fig. 2, FBD = θ, and DF is its arc of which the sine wanted is DE.
DE² = the wanted karaṇī (i.e. sine² θ). DFA = 2DF.
∴ DE = ½DA. ∴ sin² θ = DE² = ¼DA² = ¼(AC² + CD²).
Now, ∵ AC = sine 2 arc FD = sin 2θ, AC² = sin² 2θ; and ∵ CD² = (BD − BC)² = (BD − AG)² =
[120′ − sin (90° − 2θ)]², sin²θ = ¼ (AC² + CD²)
= ¼ [sin² 2θ + {120′ − sin (90° − 2θ)}²]
From this, sinθ = √sin²θ.
From I, we can derive II thus:
¼ [sin² 2θ + {120′ − sin (90° − 2θ)}²]
= ¼ {sin² 2θ + 120² + sin² (90° − 2θ) − 2 × 120 × sin (90° − 2θ)}
= ¼ {120² + 120² − 2 × 120 × sin (90° − 2θ)}
(∵ sin² 2θ + sin² (90° − 2θ) = sin² 2θ + Cos² 2θ = radius²)
2 × 120′
= ──────── {120′ − sin (90° − 2θ)}
4
= 60′ {120′ − sin (90° − 2θ)}
Now, of TS, Thibaut alone proves the formula, while Sud. Dvivedi uses it. The form of the first
formula given by them differs from that given by us, and is as follows:-
sin²θ = (½ sin²2θ) + [½ {120′ − sin (90° − 2θ)}]²
Though their formula is correct, it entails more work, and to give the formula in this form they have
made many unwarranted changes in the already correct readings. We have made only one correc-
tion, and that grammatical, for the sake of syntax, viz. dhanurdviguṇapadāt into dhanurdviguṇam
padāt, which entails the occurrence of an extra syllable, which can be explained, as before, by rules
of prosody. Or, let the reading be śeṣe tviṣṭe dhanurdvi-guṇam padāt projjhya etc.
Now follow six verses giving the sines, computed by the author himself.

IV. 9 IV. THREE PROBLEMS 81
शेषज्याः 'स्वरतिथयो' 'गुण-शिव-धृति'भिश्च 'विंशतिः' सहिता ।
'पञ्चनरकं' 'शतार्धं त्रिसमेतं' 'षष्टि'रिति लिप्ता ॥ ६ ॥
सैकाऽजे पञ्चाशत् 'पञ्चाष्टकं' 'पञ्चवर्गवेदा' श्च ।
'त्रिंशच्चतुर्भिरधिका' 'षट्पञ्चाशच्छराः' शून्यम्' ॥ ७ ॥
6-7. The other sines are the following: In the first sign, the minutes parts are
successively 7, 15, 20 + 3, 20 + 11, 20 + 18, 5 × 9, 50 + 3, and 60. The seconds,
respectively, are: 50 + 1, 5 × 8, 25, 4, 30 + 4, 56, 5 and 0.
Thus we have for the first sign –

82 PAÑCASIDDHĀNTIKĀ IV. 11
Adding the minutes and seconds, and 60' for the end of the first sign, the sines are:-

IV. 14 IV. THREE PROBLEMS 83
there is one second more. By this mistake the 24th, i.e. the radius, has become 120' 1", and even
this obvious mistake they have failed to note.
The promised sine-intervals are here given:
मुनयोऽब्जे व्येकान्ते 'रसत्रयं' 'पञ्च(कौ)' 'कृता' (श्र) गवि ।
'शिखि-पक्षचन्द्र-शून्याः' द्वित्रिमिथुने कला ज्या [नाम्] ॥१२ ॥
मेषे विकलार्धशतं सैकं 'व्येकेन्द्रि(ये)-श्वरं त्रिंशत् ।
(द्वा)विंशतित्रिवर्गः ------------------------------ ॥ १३ ॥
---------------------------------------------------- ।
------ 'खगुणकृतार्णवयमनवकसमुद्रशिखिवर्गैः ॥ १४ ॥
12. Of the intervals the minute parts are, in the first sign, 7, 7, 7, 7, 7, 7, 7, 6;
in the second sign: 6, 6, 6, 5, 5, 4, 4, 4; and in the third sign, 3, 3, 2, 2, 1, 1, 0, 0.
13-14. The seconds in the first sign are, 50 + 1, 50 − 1, 50 − 5, 50 − 11, 30,
22, 9, (and here is a break resulting in the loss of the 4th foot of the 13th
verse and the first two feet with 4 mātrās of the 3rd foot of the 14th verse. From
the values of the sines we can compute that the seconds in the 8th interval must
be 55, which must have been given in the missing part. By examining the
remaining part we can construct the meaning of this verse thus). The seconds
of the intervals in the second sign are, respectively 10 × (4, 2, 0, 4, 2, 5, 3, 0)

• (0, 3, 4, 4, 2, 9, 4, 9), added each to each.
  TS have not been able to see that in what is left of the 14th verse, the digits in the unit places of
  the eight numbers giving the seconds of the intervals of the second sign are given. This is because
  they have made mistakes in the 14th and 16th sines resulting in an error of + 1" in the 14th interval,
  − 1" in the 15th and + 1" again in the 16th. This has prevented them from finding the correct
  values by comparison, so that they make many changes in the readings here, saying that the text is
  very corrupt, here. It may be seen that not a single word is wrong here.
  12a. A. मुनयोज्ये; B. गुनयोज्ये to 14c. D suggests for 13d [पञ्चाशच्च विषयसंयुक्तम्]
  b. A. त्रयं को; B. त्रयपञ्चको;. C. त्रयं (त्रिः) शशः C. indicates the gap by dots as done by
  A. कृताच्चे गवि; B. कृता-गेवि; C. कृताब्धी गवि; us. D suggests for 14:
  D. कृताग्निगवि ख[समुद्र] गुण [द्विकृताः] ..........
  c. A. शिखिपकृत चन्द्रः; B. शिखिपक्ष कृतार्णव [द्वि] यमनवे [न्द्रि] यसमुद्रशिखिवर्गे[शाः] ||
  d. C. द्वौद्विमिथुने. A-B. ज्या |; C. ज्यार्द्धे; D. ज्या [सु]
  13a. A. विकलार्द्धशत° d. A. समुद्रा
  b. B1. सैकां; B2. सैव्यं; B1.2. वर्गे; B2. वर्गे
  A1.B1. °न्द्रिय स्वरं C. खगुणकृतार्णवयमनवसकसमुद्रा शिखिवर्गैः (?) ||
  c. A1.B2. द्विविंशति
  13d. A.B. Unindicated gap of 13d. upto खगुण of 14c.

84 PAÑCASIDDHĀNTIKĀ IV. 15
‘मनुविषयतिथिरसा:’ स्युस्त्रिगुणाः पञ्चाष्टकं ‘स्वरोपेतम्’ |
सप्तदशनवपञ्चकं षोडश चेति क्रमान्मिथुने || १५ ||
15. The seconds of the intervals in the third sign, are, 3 × 14, 3 × 5, 3 × 15,
3 × 6, 5 × 8 + 7, 17, 9 × 5 and 16.
The intervals can be got by deducting the previous sines from the succeeding ones, and com-
pared with the author’s concern for correctness, they are correct. It is to be noted that the intervals
of TS also come correct in the third sign, because there is a uniform error of 1″ in all sines. The
intervals are as tabulated:

IV. THREE PROBLEMS 85
Fig. IV. 3
AK = sine AG (neglecting the first sign) because, EF, HJ, JK, GF, are all equal, as already men-
tioned. Therefore, for an arc or angle in the second quadrant, (3 to 6 signs), deduct it from six
signs and get the sine of the remainder. For that in the third quadrant, (6 to 9 signs), deduct six
signs and find the sine of the remainder. For that in the fourth quadrant, deduct it from twelve
signs and find the sine of the remainder.
Example 2 (a). Find the sine of the arc or angle equal to 4 signs.
It is in the second quadrant. Therefore, sine 4 signs = sine 6 signs − 4 signs = sine 2 signs = 103′
55″.
Example 2 (b). Find the sine of signs 7-11-15.
This is in the third quadrant. Therefore sine of sign 7-11-15 = sine (7-11-15 − 6-0-0) = sine 1-11-
15 = 79′ 7″.
Example 2 (c). Find the sine of signs 9-15-0.
This is in the 4th quadrant. Therefore sine of signs 9-15-0 = sine (12 signs − sign 9-15-0) = sine
2-15-0 = 115′ 55″.
Declination
From here to the end of the chapter, problems based on the solution of spherical triangles are
dealt with, being problems involving position, time and direction. As a preliminary, the declina-
tion of the Sun and the Moon are required, which are given first.
Stellar sphere (Bhagola)
The ancient astronomers speak of three spheres, the Terrestrial sphere or Earth sphere on
which we live, the Stellar sphere or the Sphere of the stars, and the Sky sphere or the Sphere of the
sky. We shall describe the terrestrial sphere in connection with the Sāura chap. XIII-XV. Of the
other two, we shall now describe the stellar sphere, a knowledge of which is immediately required.
It is the apparent sphere on which the stars appear to be fixed, and form a frame of reference for

86 PAÑCASIDDHĀNTIKĀ IV. 15
the motion of bodies like the Sun, Moon, planets etc. Actually the stars are at widely different dis-
tances from us and are moving in various directions at speeds of several miles per hour. But with
all this they appear to be, and can be represented, as being fixed on a sphere, because of their
enormous distances from us, so much so that we are practically viewing the same sphere as people
several generations ago did. But the ancients believed that the stars were luminous bodies fixed on
the under-surface of a sphere of radius only sixty times that of the orbit of the Sun (actually the
earth) round the earth, with the centre of the sphere at the earth's centre. This sphere seems to be
rotating about once a day, by the actual rotation of the earth, about once a day.
Fig. IV. 4
On this sphere (Fig. 4) there is an important great-circle called the ecliptic (Krānti-vṛtta) (EC),
marked by the twenty-seven asterisms, Aśvinī etc., on which the Sun (S) moves, (actually appears
to move on account of the motion of the earth), completing a revolution once a year. The Moon
and the planets move in orbits inclined to the ecliptic at small angles. Their longitudes are reckoned
along the ecliptic from a fixed point called the first point of Meṣa or Aśvinī. Latitudes are measured
on secondaries to the great circle, meeting at a point called the pole of the ecliptic (Kadamba).
Another great circle called the Celestial Equator (AOB) cuts the ecliptic at r, called the First point
of Aries or Vernal Equinox point, which, instead of being fixed, has a slow westward motion on the
ecliptic. At the period of the authors of the first Siddhāntas, this point coincided with the first point
of Meṣa, this being the reason why that particular point was taken by them for reckoning from. The
angle between the two great circles (SrR) is called the Obliquity of the ecliptic. It was about 24° at
the time of Varāhamihira, and now it is about 23° 27'. The declination (SR,) (Krānti) of a body like
the Sun, is measured along the secondary passing through the body, (NPSRSP,) called the Declina-
tion circle, all the secondaries meeting at the celestial poles, the poles of the stellar sphere, (SPNP),
on the axis joining which the sphere apparently rotates.
The declination is found by solving the spherical right angled triangle SrR. Spherical triangles
were in general solved by the Hindu astronomers using the properties of plane right angled triangles
formed by the sections of the sphere along the great circle arcs forming the spherical triangle (The

IV. 16 IV. THREE PROBLEMS 87
Greeks solved them by an extension of Manelau's Theorem to figures on the sphere.) We shall con-
tent ourselves with giving the formulae for solution and refer to them whenever necessary by way
of proof.
Let ABC (fig. 5) be a spherical triangle, right angled at C, and R the radius of the sphere.
I. Cos AB = Cos AC × Cos BC ÷ R
II. sin BC = sin AB × sin A ÷ R
III. Cos A = R × Cos AB. sin AC/sin AB. Cos AC
IV. Cos A = Cos BC × sin B ÷ R
V. Cos AB = R. Cos A × Cos B/(sin A × sin B)
These, together with the general identities, cosθ = sin
(90° – θ), sin² θ + Cos² θ = 1, will suffice for explaining the
formulae occurring in the text. [Fig. IV. 5]
[रविचन्द्रयोः क्रान्तिः]
जीवाऽध्यर्धशतां' (शघ्नै) काषष्टि (र्दि) नेशकाष्ठा (ज्या) ।
चन्द्रस्य सविक्षेपस्तदपक्र(मो) राशिपादे(भ्यः) ॥ १६ ॥
Declination of the Sun and the Moon
16. The sine of Sun's declination is found by multiplying the sine of its
longitude by 61 and dividing by 150. Its arc is the declination. The declination
of the Moon found thus is the mean declination. Its true declination is the
mean declination plus latitude. The intervals of declinations for intervals of
quarter signs are given (in the next two verses, 17-18).
This is the formula: sin dec. = sin sāyana long. × 61/150. From this the arc forming the declina-
tion is found by using the tables, and then the true declination of the Moon, using this. It must be
noted that when the longitude reckoned from r (i.e. sāyana long.) is within 6 signs, the declination
is North, and when more than 6 signs it is South. In the case of the Moon, if the declination and the
latitude are of the same direction they should be added to get the true declination. If they are of
different directions, their difference is the true declination, its direction being that of the greater.
The author has not mentioned this because it is obvious.
Example 3 (a). Find the maximum declination of the Sun.
Obviously, the maximum sine of the Sun's longitude (i.e. when the Sun is 3 signs or 9 signs) will
give the maximum declination. It is therefore given by sin dec. = 120' × 61 ÷ 150 = 48' 48". Its arc,
24° is the maximum declination.
16a. A. जीवाध्यार्द्ध०; B. जीवा व्या-र्द्ध; C. जीवाव्यर्द्ध; D. [साङ्कुलिप्ता] . A.B. काष्टान्तः; C. काष्ठान्तः;
D. जीवाव्यध्यर्ध. B. सिताशा; A.C.D. शतांशाः D. काष्टान्तः
b. A. सैकाःषष्टि; B. सैका षष्ठि; C. सैका षष्टि; d. A.B.C.D. तदपक्रम. A.C. ०पादेन्यः

88 PAÑCASIDDHĀNTIKĀ IV. 17
Example 3 (b). Sāyana Sun is rāśi 4-7-30. Find its declination.
Sin 4ʳ 7° 30′ = Sin (6ʳ-0-0 – 4ʳ-7-30) = sin 1ʳ-22-30 = 95′ 12″.
Sin Declination = 95′ 12″ × (60 + 1) ÷ 150 = 95′ 12″ × (2/5 + 2/5 × 60) = 38′ 5″ + 38″ = 38′ 43″.
Its arc, 18° 50′, is the declination. Since the sāyana Sun is within 6 rāśis, the declination is north.
Example 3 (c). The sāyana Moon is 9ʳ0°0′. Its latitude is 4°N. Find its declination.
The sāyana longitude being 9ʳ0° 0′, the mean declination is the maximum, south, i.e. 24°S. Its lat.
is 4°N, i.e. of opposite direction. ∴ the true declination is 24° – 4° = 20° S. South because South is
greater.
The author uses the word kāṣṭhā to signify declination, which is uncommon. Sometimes this word
itself is used to mean sine declination.
The rule for sin declination is explained thus: Our siddhāntas take the maximum declination to
be 24°. As the maximun declination occurs when the sāyana longitude is 3 signs, the angle between
the ecliptic and the celestial equator (i.e. the obliquity of the ecliptic) also is 24°. In Fig. 5, take AB
and AC as parts of the ecliptic and celestial equator. Then, A = 24° and AB is the sāyana longitude.
BC is the declination wanted. By formula II under the present verse, sin dec. = long. × sin 24° ÷
120′. But sin 24° ÷ 120′ = 48′ 48′′ ÷ 120′ = 61/150. Hence, sin dec = sin long × (60 + 1)/150, which
is the given rule.
Now for the direction: See Fig. 4. At r the Sun moving along the ecliptic crosses the celestial
equator, and passes from South to North. As great circles bisect one another, till the longitude is 6
sings it moves north of the celestial equator, for which declinations are reckoned, and then moves
south of it. Therefore for longitude 0 to 6 rāśis, the declination is North, and for longitude 6 to 12
rāśīs, it is South.
As the Moon and other planets move in their own orbits inclined to the ecliptic, the declinations
computed from their longitudes reckoned along the ecliptic are only approximate. To get correct
declinations, their distances north or south from the ecliptic points, called their ‘latitudes’, should
be combined in the proper manner as instructed. But the result by thus adding or subtracting will
be only approximate, because the latitudes are directed towards the pole of the ecliptic (Kadamba),
while the declinations are directed towards the Celestial Pole. The maximum error that can occur
thus is about 24′. Combined with the error in latitude due to other factors like proportion by
degrees of the argument of latitude (advocated by the Pauliśa) instead of the sine etc., the error will
be considerable.
Now, for the readings. For syntactical purposes, and getting the proper meaning, śatārmśāśsaikā
has been corrected as śatāṁśaghnaikā, ṣaṣtidineśa as ṣaṣṭirdineśa, kāṣṭhānta as kāṣṭhā jyā, apakramarāśi
as apakramo rāśi and pādenyaḥ as pādebhyaḥ. As for TS, they have not touched this verse and the next
two, saying, in so many words, that they cannot interpret them. NP’s interpretation of the three
verses has also been affected by the highly corrupt text.
लिप्ताशत [क] म (शीत्या) मेषे 'त्रिख' यु (क्त) 'मिन्द्रिय' 'मनू' (न) म् ।
गवि ('मनु') 'भव' 'मुनि' 'रूपै' - श्र [तु] गुणैः संयुतं च शतम् ॥ १७ ॥

IV. 18 IV. THREE PROBLEMS 89
नवतिस्त्रियुता षष्टिश्चत्वारिंश'च्छिवा'श्च मिथुना(न्ते) ।
मेषाद् गताऽऽगतमुदग्दक्षिणतोऽदस्तूलादिषु च ॥ १८ ॥
17-18. The (promised) intervals of declinations in minutes for intervals of
quarter-signs, are, in Sāyana Meṣa: 180 + 3, 180 + 0, 180 − 5, 180 − 14, in
Sāyana Vṛṣabha, 100 + 4 × 14, 100 + 4 × 11, 100 + 4 × 7, 100 + 4 × 1, and
in Sāyana Mithuna, 90, 63, 40 and 11. (Thus the intervals are 183, 180, 175,
166, 156, 144, 128, 104, 90, 63, 40, 11.) These are to be added successively to
get the declinations from Meṣa (Aries) to Mithuna (Gemini). Then from Kar-
kaṭaka (Cancer) to Kanyā (Virgo), these should be deducted in the reverse
order, until at the end of Kanyā, the declination is zero. These declinations
from Meṣa to the end of Kanyā are north. Then from Tulā (Libra) to the end
of Dhanus (Sagittarius), the south declinations increase in the given order,
and from Makara (Capricorn) to Mīna (Pisces) the south declinations
decrease in the reverse order, until at the end of Mīna the declination is zero
again.)
Example 4. Find the declination of the ecliptic point ending (Sāyana) Capricorn.
The end of Capricorn is rāśi 10-0-0. This falls between rāśis 6 and 12. ∴ The declination is south.
The declination ending Sagittarius (i.e. beginning Capricorn) is 24° S. The declination at the end
of Capricorn is 24° − 11′ − 40′ − 63′ − 90′ = 24° − 3° 24′ = 20° 36′S.
The author has perhaps computed the declination by applying the formula of verses 16 and got
the intervals by deducting the previous from the next. Or, these verses are taken in toto from the
original Pauliśa and given here, for there are small differences from the computed values. Or, the
differences are scribal errors. Both are given hereunder for comparison:

90 PAÑCASIDDHĀNTIKĀ IV. 19

19. (Plant a gnomon at the centre of) a circle having a diameter equal to four
    times the gnomon. Mark the two points where the shadow of the gnomon
    enters the circle and emerges from it. The line joining the points is the east-
    west line. The line drawn perpendicular to this by means of equal intersecting
    circles, is the north-south line.
    Though the east-west line is first asked to be drawn and the north-south next, as perpendicular
    bisector to the east-west, it will be better if the north-south line is first drawn by means of equal
    intersecting circles with the two points as centres. Then using the points of intersection of the
    north-south line and the original circle as centres, by the same means, the perpendicular bisector
    forming the east-west line can be drawn, which will pass through the centre as required. On the
    other hand, if the east-west line is first drawn by joining the first two points, another line parallel
    to it and passing through the centre is to be drawn as the desired east-west line.
    19a. A. संकुश्रतुविस्तारे; C. शङ्कइगुलविस्तारे in V. 9c. as noted by the scribes.
    c. A. अपेरैद्री; C.D. अपेरैन्द्री Thus, B1 adds in the margin, प्रति पत्र एक;
    d. A. यवाश्च; C.D. यवैश्च B2. adds, प्रतिपत्र एके, and B3. adds प्रति नूं
    ∙ B. After कार्ये one leaf missing upto दिवा पत्र एक. And all add अग्रो नास्ति ।