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

III.22 III. PAULIŚA-SIDDHĀNTA 63
occurs when the declinations are the same and the courses are different. The expression half-
revolution in that connection is only meant to be approximate, because by the latitude of the Moon
it may be a little more or less." Therefore we should examine whether a Vyatīpāta would occur at
the neighbourhood of the sum being six signs, because it can occur only there. But sometimes it
may not occur at all, because the definition is not satisfied (All this is expounded clearly in works
like the Siddhānta Śiromaṇi and we stop with this).
One may think that we are making contradictory statements by saying in the history of the origin
of the Vyatīpāta, that it occurs at the sum being full revolutions and here that it occurs at half revolu-
tions. There is no contradiction because the origin from which the longitudes are measured is
different in the two cases. In the former the winter solstice was taken as the origin, and, in the latter,
the spring equinox. There is a difference of three signs between the origins, which causes the same
difference in each of the two longitudes, with the result that there is a difference of six signs in the
sum. That they are the same can be shown thus: The Sun measured from winter solstice, (say, a)
= the Sun measured from spring equinox (say, b) + 3 signs. The Moon measured from winter
solstice, (say, á) = the Moon measured from spring equinox (say, b́) + 3 signs. Therefore a + a' =
b + b' + 6 signs. If a + a' = full revolution, b + b' + 6 signs = full revolution, therefore b + b' =
full revolution - 6 signs = half revolution, which proves the sameness. Spring equinox is not men-
tioned because at the author's time it was situated at the beginning of Aśvinī and longitudes are
reckoned from there.
Example 11. The Sun and the Moon at the end of the day are rā. 1-10-0 and rā. 4-23-30, and their daily
motion 57' and 783'. Taking the spring equinox to be at the beginning of Aśvinī, i.e. the winter solstice at the
beginning of Capricorn, examine the possibility of Vyatīpāta, in both ways.
Because the longitudes are from zero Aśvinī, they are the same as reckoned from spring equinox
also, both points being the same in the problem. Therefore sum of longitudes = rā. 1-10-0- + rā.
4-23-30 = rā. 6-3-30. This is 3° 30', i.e. 210', over a half revolution. The sum of the daily motions
= 57' + 783' = 840'. Therefore at 210 × 60 ÷ 840 = 15, nāḍīs before the end of the day, the sum
is equal to a half revolution or 6 signs, and so Vyatīpāta may occur in its neighbourhood.
Otherwise, if the longitudes as measured from winter solstice, the Sun = rā. 1-10-0 - rā. 9-0-0
= rā. 4-10-0. The Moon = rā. 4-23-30 - rā. 9-0-0 = rā. 7-23-30. Their sum = rā. 4-10-0 + rā. 7-23-
30 = rā. 12-3-30, and this is 210' over a full revolution. Therefore 210' × 60 ÷ 840 = 15, nāḍīs
before the end of the day. The sum is a full revolution and the Vyatīpāta may occur as its neighbour-
hood. (Note that worked in both ways, the time is the same).
Now for the readings. In the place of pāto we have taken yāto because the scribe may easily mistake
pā for yā. But the correction bhāgo of TS does not agree with the second case in arkakāṣṭhām and
deserves to be rejected. The wrong reading, śaśi-savikṣepaḥ has been corrected by us into śaśī
savikṣepaḥ, by a simple lengthening of. But TS and NP have made it śaśiravikṣepaḥ which is incorrect
and also does not agree with kāṣṭhām. The meaning which they have taken for this verse itself is
wrong. Their interpretation of kāṣṭha into 'maximum declination' i.e. 24° (or 23° 20') is not proper,
for, in his work (see Chap. IV), kāṣṭhānta is used for maximum declination and kaṣṭhā is taken to
mean only declination. Let us concede it is maximum declination and therefore means 24°. Even
this does not agree with the meaning given by them because they want and imply 23° 20' only there.
If 24° is given roughly for 23° 20', why not 23° which is nearer. They do not seem to have under-
stood at all what is sought to be conveyed by the author.

64 PAÑCASIDDHĀNTIKĀ III.24
[षडशीतिपुण्यकालः]
मेषतुलादौ विषुवं षडशीतिमुखं तुलादिभागेषु ।
षडशीतिमुखेषु रवेः पितृदिवसा ये (ऽव) शेषाः स्युः ॥ २३ ॥
षडशीतिमुखं कन्याचतुर्दशेऽष्टादशे च मिथुनस्य ।
मीनस्य द्वाविंशे षड्विंशे कार्मुकस्यांशे ॥२४ ॥
Ṣaḍaśīti-puṇyakāla
23-24. At the first point of Meṣa (Aries) and Tulā (Libra) are the spring and
autumnal equinoxes (and the sacred days thereof are when the Sun is there.)
The commencements of the sacred days called Ṣaḍaśītis are at periods of 86
solar degrees commencing with Tulā-zero point. The days in the solar months
after the respective commencement of the Ṣaḍaśītis are sacred as connected
with the manes. The commencement of the Ṣaḍaśītis are after 14 degrees of
Kanyā, (Virgo), after 18 degrees of Mithuna (Gemini), after 22 degrees of
Mīna, (Pisces) and after 26 degrees of Dhanus (Sagittarius).
The main purpose of the author in giving the equinoxes here is to indicate the sacred days con-
nected with them as can be gathered from the context. The equinoxes, i.e. the points of intersection
between the ecliptic and the celestial equator, though moving westward slowly along the ecliptic,
(this is the precession of the equinoxes), were at the first points of Meṣa and Tulā only at the period
of the author. At the present day the equinoxes have moved far into Uttara-Bhādrapada and
Uttara-Phalgunī, but the sacred days are still observed with the Sun entering Meṣa and Tulā by
blind routine.
The time taken by the Sun to move one degree is a ‘solar day’ according to Hindu astronomers.
(We have put it within inverted commas, because in English it means the ordinary day caused by
the Sun and therefore quite different). So in a solar year there are 360 ‘solar days’, and in each solar
month 30 ‘solar days’. As for counting from zero-Tulā, this is enjoined by the Dharma-śāstras. The
commencements of the Ṣaḍaśītimukha-s are, 1 × 86° = 86°, 2 × 86° = 172°, 3 × 86° = 258° and 4 ×
86° = 344°. from zero-Tulā, i.e. from rā. 6-0-0. Therefore they are rā. 6 + 86°, rā. 6 + 172°, rā. 6

• 258° and rā. 6 + 344°, and these are, respectively, 26 degrees of Sagittarius, 22 degrees of Pisces,
  18 degrees of Gemini and 14 degrees of Virgo. These sacred days are not observed in these days
  and it would be interesting to know when and how they went out of vogue. When the Sun enters
  Sagittarius, Pisces, Gemini and Virgo, we observe the sacred day, calling it Ṣaḍaśīti; and in the place
  of the last sixteen ‘solar days’ of Virgo, (these seem to have secured importance at the time of Sūrya
  Siddhānta) the dark fortnight of Bhādrapada is dedicated to the Manes, with the name of Mahālaya-
  pakṣa. The dark fortnight of Āśvina also is observed as a secondary Mahālaya-pakṣa and it is the
  belief that the Manes are sent back to their world on Naraka-Caturdaśī. Now, what is the speciality
  about 86 solar days, it may be asked. This period is three synodic months less one day. It may be
  23a. A. मेखतुलादौ A.विषुव; B.1.2. C.D.विषुवत्; B3.दिषु
  b. A. षडसीति d. A.B.विशेषा स्युः
  c. A.B.दिवसाद्ये 24b. A. °ष्टादर्शे

III.24 III. PAULIŚA-SIDDHĀNTA 65
that a section of people observed a sacred day for the manes once in three synodic months, and then
this came in its place.
[अयनम्]
उद्गयनं मकरादावृतवः शिशिरादयश्च सूर्यवशात् ।
द्विभवनकालसमानं दक्षिणमयनं च कर्कटकात् ॥ २५ ॥
Solstices
25. The Sun’s turning northward is when it reaches the zero-point of Makara,
(Capricorn), i.e. at winter solstice, and its turning southward is at the zero
point of Karkaṭaka (Cancer) i.e. at summer solstice, with the attendant sacred
days. The seasons Śiśira etc. commence with the winter solstice and each
season lasts two tropical solar months.
The precession of the equinoxes implies the precession of the solstices as well and therefore the
solstices at the zero-points of Karkaṭaka and Makara is true only for the period of the author. If the
sacred days are observed still when the Sun enters these signs, it is again blind custom.
As the seasons depend upon the position of the mid-day Sun in the sky and the length of day
time, and these depend on the Sun’s declination depending on tropical (sāyana) longitude of the
Sun, the seasonal months are different from either the solar sidereal months Meṣa etc. or the
synodic months Caitra etc., and these cannot correctly represent the seasons. That is why the Vedas
give a new set of months, (actually tropical months) for the seasons: Madhu and Mādhava are the
months constituting the Vasanta (spring) season, Śukra and Śuci are the months constituting the
Grīṣma (summer season), Nabha and Nabhasya are the months constituting the Varṣa (rainy)
season, Iṣa and Ūrja are the months constituting the Śarad (post-rainy season); Sahas and Sahasya
constituting the Hemanta (pre-winter) season; and Tapas and Tapasya constituting the Śiśira
(winter) season. (Śuklayajurveda, 13.25). Even in the Vedāṅga Jyotiṣa we have the information that
the Śiśira season begins with the Uttarāyaṇa (winter solstice). The Yājuṣa-Jyotiṣa (verse 6) says,
“When the Sun and the Moon rise together, with Śraviṣṭhā, from then commence the yuga, the
month of Māgha, the seasonal month Tapas, the light fortnight of the month, and Uttarāyaṇa”. As
Tapas is the first month of Śiśira we understand Śiśira begins with Uttarāyaṇa. By mentioning
Māgha and Tapas distinctly, we understand that the Vedas wish us not to confuse the two. But con-
fusion there has been, and still continues, with the result that people call Meṣa and even Vṛṣabha
spring months, though patently we have summer then, Kumbha and Mīna being practically the
spring months now. This confusion has resulted in Madhu, Mādhava etc. and Caitra, Vaiśākha etc.
as synonyms. People who know are amused, when in the saṅkalpa recited for Hindu rituals the
month of Vṛṣabha, which is advanced summer, is mentioned as spring.
[संक्रान्तिकालः]
षष्टिघ्ना भुक्तिहृता रविबिम्बकला भवन्ति नाड्यस्ताः ।
संक्रान्तीनां कालः पुण्योऽतोऽर्धेन चाद्यन्तात् ॥ २६ ॥
25. Quoted by Utpala on BS 2, p.23. b. A. वृत्तकशिशि; B1.2. वृवृंतकशशि
25a. A.B1. मकरादौ c. U. समाना

66 PAÑCASIDDHĀNTIKĀ III.27
Saṅkrānti-kāla
26. The angular diameter of the Sun in minutes, multiplied by sixty and
divided by the daily motion of the Sun, are total sacred nāḍis of Saṅkrānti
(literally ‘crossing’). Half this time before and after the Sun entering a rāśi, is
sacred.
The Pauliśa does not give the angular diameter of the Sun, so it must be the intention of the
author to use the angular diameter given by the Romaka or the Saura.
Example 12. The angular diameter of Sun is 31' and its daily motion 57'. The Saṅkramaṇa is 19 nāḍīs after
sunrise. Find the sacred nāḍis.
Angular diameter × 60 ÷ daily motion = 31' × 60 ÷ 57' = nā. 32-38. Half this is 32-38/2 = nā.
16-19. Therefore nā. 19-0 - nā. 16-19 = nā. 2-41 to nā. 19-0 + 16 - 19 = nā. 35-19 is the sacred
period.
The rule is proved thus: The time of the centre of the Sun’s orb crossing to the next sign is the
time of Saṅkramaṇa. At this time half the orb is in the previous sign and half in the next. The period
when parts of the orb are in both signs is the sacred period. So it begins when the east point of the
orb just enters the next sign and ends when the west point just leaves the previous sign. So, during
the interval the Sun moves a distance equal to its own diameter. This time is got by the proportion:
daily motion: angular diameter :: 60 nāḍikās: the required time. Therefore aṅg. diameter × 60
÷ 60 is the time in nāḍikās. As half this time is required for the mid-point to reach the junction of
the signs, half this period placed on either side of the time of the mid-point crossing over gives the
beginning and end of sacred period.
It must be noted that if the angular diameter is computed according to the old Hindu astronomical
works and used, the sacred period would be constant whatever be the daily motion, and the sacred
period can easily be given as so many nāḍikās before and after saṅkramaṇa. How? Let x be the mean
angular diameter in minutes. According to Hindu astronomy the angular diameter is proportionate
to the daily motion, (because the motion is taken inversely proportionate to the distance and the
angular diameter also is inversely proportionate to the distance) (See VIII. 15, IX 14-16). Therefore
the angular diameter = x multiplied by daily motion ÷ mean daily motion. The period = angular
diameter × 60 ÷ daily motion = x × daily motion × 60 ÷ (daily motion × mean daily motion) = x
× 60 ÷ mean daily motion which is constant. If to avoid this we assume that the mean diameter is
intended to be used in the rule, then the rule is unreasonable. Or we have to accept it on the injunc-
tion of the Dharmaśāstras, throwing the burden on them. We said, “according to the old Hindu
astronomical works”, because actually the angular diameter is not exactly proportional to the daily
motion.
[त्रिदिनस्पृग्योगः]
तिथ्यन्तं यदि सूर्यः स्पृशन्नुदेत्ये (षा) वासरं चाऽपि ।
योगस्तदा त्र्यहस्पृक् तिथित्रयस्पर्शनाद(वमः) ॥ २७ ॥
26a. B1. भुक्षिहताः B2. भुक्षिहता; B3. भुक्षिहता d. B1.2.पुण्यतोद्धेन
b. A1.रबिम्ब०; B.बिम्बककला For चाद्यन्तात्, B1.2. वार्धता कृतिः; B2. न नार्धतात्

III.27 III. PAULIŚA-SIDDHĀNTA 67
Tridinaspṛg-yoga
27. When a tithi extends throughout a day, coinciding with a part of the
previous day and the next day, the occurrence is called Tridinaspṛg, (literally
'touch of three days'). (If, besides a whole tithi, parts of the previous and next tithis
fall on the same day, the occurrence is called avama, literally, 'uncounted tithi').
The only thing we have done to the reading in the first half of the verse is to change śā into ṣā,
which is quite warranted. But TS have changed de into di and introduced a new word, anya. Still
their reading of the text cannot yield the meaning. To agree with tithitraya-sparśañat, we corrected
ahnaḥ into avamaḥ, because avama alone results by contact with three tithis. The word ahnaḥ is neces-
sary also, but can be understood from the context, though not mentioned, but not so, avamaḥ. If
this part is left uncorrected as TS have left, the expression would be non-sensical like Sudhākara's
meaning: "Because the day touches three tithis, it is called 'Three-day touching'. But Thibaut has
grasped the idea here, though calling it "the conjunction touching three Tithis". NP too, have
caught the idea, but since the relevant emendation to avama did not strike them, they merely say
'(there is a yoga)'."
[राहुः]
अष्टगुणे दिनराशौ 'रूपेन्द्रियशीतरश्मि'भिर्भक्ते ।
लब्धा राहोरंशा भगणसमाश्च क्षिपेल्लिप्ताः ॥ २८ ॥
वृश्चिकभागा राहोः षड्विंशतिरेकलिप्तिकालुप्ताः ।
आदितरः प्रोह्य मुखं षड्राशियुतं तु पुच्छाख्यम् ॥ २९ ॥
Rāhu (Node)
28. Multiply the days from Epoch by 8 and divide by 151. Rāhu's motion is
got in degrees etc. Add minutes equal to revolutions. (The motion becomes
exact.)
29. Deduct the motion from 7ʳ 25° 59ʹ. The remainder is Rāhu's Head (what
is called Dragon's Head, a popular name for the Ascending Node). Add 6 rāśis
to Rāhu's Head, (Dragon's Tail or Descending Node), is got.
Example 13. (a) Days from Epoch, 75,500; find Rāhu's Head and Tail. (b) Find the Head of Rāhu at
Epoch, i.e. for Zero day.
27b. A. स्पृश्यनु; B. स्पृशेत्तु॰
A. ॰देतोशावासनं; B. ॰देत्येशावासंरः; D. ॰नुदेत्येष्यं
C. ॰दितोन्यवासनं
c. A2. लब्ध्वा. B. शहोरंशा
c. B. ॰स्तदन्यहः B. स्पृक
d. A2. क्षिपेल्लिप्ताः; B1. क्षिपेछिन्त्रप्ताः
d. A.B.C.D. नादहः ||
29b. A. विंशति. C.D. लुप्ता
28a. A2.B. गुणो. B. गुणाशशौ
c. B. आदियरत. B1.2. प्रोज्य; D. प्रोज्झ्य. B. मुख
b. B. ॰भिव्यक्ते
d. A2. युतं नु. B2. पुच्छाक्ष्यं

68 PAÑCASIDDHĀNTIKĀ III.29
(a). Rāhu's motion for 75,500 days = 8 × 75,500 divided by 151, degrees = rev. 11-1-10-0. The
exact motion = rev. 11-1-10-0 + 11 minutes = rev. 11-1-10-11 = rā. 1-10-11, omitting full revolu-
tions. Rāhu's Head = rā. 7-25-59 – rā. 1-10-11 = rā. 6-15-48. Tail = rā. 6-15-48 – rā. 6-0-0 = rā.
0–15–48.
As the motion of Rāhu is zero, the Head of Rāhu is the constant itself, viz. rā. 7-25-59.
As the motion is 8° in 151 days, according to this Siddhānta, to move 360°, i.e. one revolution, it
takes 360 × 151 ÷ 8 = 6795 days. But during this period it moves one minute more, i.e. the exact
motion is 360° 1' in 6795 days, i.e. one revolution takes 360° × 6795 ÷ 360° 1' i.e. 6794 days, 19
nāḍīs.
We have seen that the Head of Rāhu for the Epoch, according to the Pauliśa is the constant itself,
viz. rā. 7-25-59. According to the Saura (condensed by the author) it is rā. 7-26-6. According to the
Vākyakaraṇa it is rā. 7-26-11. According to modern astronomy, taking the ayanāṃśa as being zero for
the period it is rā. 7-26-0. According to the Siddhānta Śiromaṇi it is rā. 7-27.13. We see that all except
the value of Śiromaṇi agree closely, verifying the Pauliśa value for the Epoch. The disagreement of
the Śiromaṇi value is only apparent, for the zero point of the Śiromaṇi zodiac is about one degree
behind that of the rest, (as may be seen by comparing with its co-ordinates of the stars or from its
Sun being one degree more than that of the rest) and if the same point is taken as the origin, the
Śiromaṇi too gives about rā. 7-26-13.
Thus the value at epoch is necessary to get the Rāhu at any moment, and it is this that is given
by Vṛścikabhāgā Rāhoh etc. But TS have not understood this need (as they did not understand
the need for the kṣepa in the case of the Moon (see II.3) and gave a wrong interpretation of
śaśimuninava-yamāśca rāśyādyāḥ) and give the following laughable explanation: "The measurement
of the limbs of Rāhu having the form of a scorpion is 25 minutes. Deducting this from the motion
of Rāhu obtained from (28), the head or face of Rāhu is to be found. This plus six rāśis is the tail.
We have to rely only on the words of the ancients to know that the scorpion-like limbs of Rāhu mea-
sure 25 minutes, there is no other reason." Now we ask: Let it be that they have not understood the
need for the kṣepa. How did it not occur to them that Rāhu can be got only by deducting the motion
from something, whether it is a cycle or some other constant, because the motion is retrograde (as
they themselves have said in other places: "Rāhu deducted from a full revolution is the Head, this
plus six rāśis is the tail", IX. 6, "deducted from the end of Pisces is the head", VIII. 8). It also appears
here that Thibaut is not satisfied with Sudhakara's explanation. Further, how did it not occur to
them that ekaliptikāluptāḥ ṣaḍviṃśativṛściakabhāgāḥ, means rā. 7-25-59, when they have correctly inter-
preted siṃhasya vasuyamāṃśāḥ as (XVI. a) rā. 4-28-0, sārdhāḥ pañcālino (XVIII. 1) as rā. 7-5-30, nava
sārdhāḥ kanyāṃśāḥ (XVIII. 11) as rā. 5-9-30, ṣoḍaśa vṛṣabhasyāṃśāḥ nava liptikāvarjitāḥ (XVIII. 18) as
rā. 1-15-51? It is really astounding what tricks the mind can play!
Incidentally, the following should be mentioned here for general information. Following the
nomenclature of the ancient Saṃhitās, the author calls the ascending node Rāhu's head, and the
descending node 'Rāhu's Tail', both being Rāhu, though generally in later astronomical works the
word Pāta is used. In recent times, somehow the term Ketu has come to be applied to the descending
node, Rāhu being retained for the ascending node, though there is no authority in astronomical
works or Purāṇas to bring in Ketu here. The ancient Saṃhitās use the term Ketu for the
Dhūmaketūs or comets and, as deities they are generally referred to in the plural. They are also
characterised by unpredictable motions and in the Bṛhatsaṃhitā too the author says so. (This is the
view of the ancients though we now know that a number of them are periodic, and their positions
can be predicted with tolerable accuracy). They are worshipped in the collective, as seated on doves,

III.29 III. PAULIŚA-SIDDHĀNTA 69
with the expression, "Salutation to the Ketus." From the singular in the mantra of their invocation,
Ketum kṛṇvannaketave, one may think that Ketu is referred to in the singular also. It is not so. Here
the word does not mean the 'Comet' Ketu at all. The mantra itself is in praise of the Sun and Ketu
here means the activity caused by the Sun in the sleeping inactive world. How then is this mantra
used to invoke the Ketus? The utterance of the word Ketu here is sufficient, as the utterance of the
many śa sounds in the Mantra śaṃ no devīḥ etc. (Ṛgveda 10.9.4) is sufficient to propitiate Śanaiścara.
though the mantra itself refers to the water-deities, or the utterance of the word mayūra in the
mantra, āmandravir intra haribhiḥ etc. (Ṛgveda 3.45.1) is sufficient to propitiate Subrahmaṇya,
though the mantra refers to Indra, as also words like, Om, atha kalyāṇa etc. cause auspiciousness by
their mere utterance. So, according to the Śāstras, Ketu refers only to the Dhūmaketus, and Rāhu
is both the nodes, the recent application of the term Ketu to the descending node being un-
warranted. Therefore, when the Dharmaśāstras enjoin the eclipses caused by Rāhu as sacred
periods, they take in both the ascending and the descending nodes, and the suggestion by some that
they do not take in the descending node on the score of some un-informed people calling it Ketu
is wrong because what they call Ketu is really Rāhu.
[चन्द्रविक्षेपः]
वक्त्रादधिकश्चन्द्रो हीनः पुच्छाच्च याति भगणोदक् ।
हीनो वदने पुच्छेऽधिकेऽ(सु) राधाति दक्षिणतः ॥ ३० ॥
भागनवत्या राहोश्चन्द्रोऽन्तरितोऽतिमहति विक्षेपे ।
लिप्ताशतद्वये [ना] त्यशी(त्याऽ)नुपातोऽतोऽन्यत्र ॥ ३१ ॥
Moon's latitude
30. If the Moon lies between the Head and the Tail it is north of the ecliptic,
(i.e. its latitude is north). If it lies between the Tail and the Head it is south of
the ecliptic, (i.e., latitude is south).
31. The latitude is a maximum equal to 280 minutes when the Moon is 90
degrees distant from either Head or Tail. The latitude is to be found by pro-
portion, at other places, using the distance in degrees from Head or Tail,
whichever is nearer.
Example 14. The true Moon is rā. 2-7-0. Rāhu's Head is rā. 5-3-0. Find the latitude of the Moon.
The Tail is Head + rā. 6-0-0- = rā. 11-3-0. The Moon is between Tail and Head. Therefore the
latitude is south. The distance of the Moon from the nearer limb, Head is rā. 5-3-0 − rā. 2-7-0 =
rā. 2-26-0 − 86 degrees. For 90° the latitude is 280'. For 86° latitude is 86° × 280' ÷ 90° = 267½',
South, as seen.
30a. B. चक्रादधिक
b. A.B1.पृच्छाच्च. B.°पोदृक्
c. A1.B1.पुंछे; C.वदनात् पुच्छाधिको
d. A.B.°धिकोमुराघाति; C.°धिकोऽसुराद्याति;
D.धिकोऽमकाद्याति
31b. B.°तोभिमहति
c-d. A. येत्यशीतिमनुपातोन्पत्र ।;
B. द्वयेत्यशीतमनुपातोन्पत्र ।;
C. द्वयाधिकसप्ततिरनुपाततोन्पत्र ।;
D. द्वय[मे] त्यशीतिमनपातोतोन्पत्र ।

70 PAÑCASIDDHĀNTIKĀ III.31
The rule is explained thus: The Moon moves in its own orbit, inclined to the ecliptic at an angle
equal to the maximum latitude. Hindu astronomy assumes this motion to be on the ecliptic itself
and gives the Moon's longitude, because there is only a maximum difference of 7'. Between the
ascending and the descending nodes the orbit is north of the ecliptic and therefore the latitude
measured on the great circle perpendicular to the ecliptic and passing through the Moon, is north.
Between the descending node and the ascending, the orbit is south and the latitude is south. The
distance between the node and the Moon, the latitude and the angle of inclination forming a spherical
triangle, we have : sin latitude = sin interval × sin maximum latitude. The maximum being small
and the latitude being generally less than this, sin latitude and sin maximum latitude are propor-
tional to the latitude and maximum latitude, and we have the formula, lat = max. lat × sin interval.
But the Paulīśa takes the latitude proportionate to the interval itself and gives the rule.
We have to consider here whether the author intends that the latitude is to be found in propor-
tion to the actual degrees of the interval or the sine of the interval in degrees. The triangle being
spherical, the correct thing would be to use the sine. But we have reason to think that the degrees
themselves are intended to be used in the proportion, for if the sine is to be used it must be men-
tioned. Also, this Siddhānta uses only the proportion by degrees in other places also, where propor-
tion by sines alone would be correct, as for e.g. in the solar eclipse, in correcting Rāhu and in
calculating valana, i.e. transformation of direction (see VI. 2-4, 8). Therefore the original Paulīśa
itself has instructed proportion by degrees, as being sufficiently accurate, which the author reiterates
here. But in computing the parallax (in time) in the case of the solar eclipse the sine is used either
by the Paulīśa itself, to avoid too much inaccuracy, or by the author VM to save the Siddhānta from
ridicule. As for TS they say that the author intends here proportion only by sine, that being the
proper thing to do.
Another thing should be mentioned here. In using proportion by degrees, the maximum error
will be in the neighbourhood of the nodes. With the maximum latitude 280', the latitude for 13°
interval would be 40½ minutes, which would be incompatible with the formula for eclipses (vide
VII. 5-6). Therefore it seems that the Siddhānta, though knowing that proportion by sine is the
correct thing for latitudes, gives proportion by degrees for the sake of ease of computation. Or by
taking liptāśatatrayeṇāśītim as the correct reading, which would make the maximum 380', the
incompatibility can be avoided. It may be argued that the error in the latitude would be great in the
neighbourhood of the maximum. But this is only erring one side while some others err on the
other. The mean maximum latitude is 309'. If some like Ptolemy give 240', which is less by 70',
what is wrong in taking that the Paulīśa gives 380', which is greater by the same amount?
Now for the readings: In the fourth foot of verse 30, mu is corrected into su because murāt is
meaningless. In the third foot of the 31st verse two syllables are wanting and nā is added for pur-
poses of syntax. For the same reason, śī-tama-nupā in the fourth foot is corrected into śī-tyā-nupā.
The correction of aśīti into saptati by TS here is unwarranted, because it is not known what it was
in the original Paulīśa. If this is done in conformity with the Saura, why not in conformity with the
Romaka, which gives 280', (see VIII. 11), and which is nearer to the Paulīśa?
Now follow six verses devoted to criticising the views of the Romaka, and an astronomer by name
Bhadraviṣṇu, the intention of the author being to create faith in his own work. (This was a custom
in those days, vide for instance the Brāhma-Sphuṭa-Siddhānta, Dūṣaṇādhyāya and the Vaṭeśvara
Siddhānta, Madhyamā-dhikāra, Chapter X). In several places of the text the readings are not clear
and we cannot be sure of what exactly the author intends to say, though the gist is clear, the matter

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:-