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

Chapter Eighteen
(VĀSIṢṬHA-) PAULIŚA-SIDDHĀNTA — RISING AND
SETTING OF PLANETS
अष्टादशोऽध्यायः
वासिष्ठ-पौलिशसिद्धान्तौ — ग्रहोदयास्ताधिकारः
Introductory
Chapter XVIII of PS (NP’s ch. XVII) deals primarily with the star-planets according to the
Vāsiṣṭha and Pauliśa siddhāntas. The true motion of the planets is traced from one heliacal rising to
the next. The method of getting the true anomaly of the equation of the centre is similar to that of
the Moon given by the Vāsiṣṭha in ch. II and based on the same theory of the rate of motion, forming
a linear zigzag. Even the same technical term pada is used here. All these are reminiscent of the
Babylonian astronomy of the Selucid period. The text as available too is not very pure and this too
has made the interpretation of this chapter difficult for TS. In my paper on ‘Some misinterpretations
and omissions of Thibaut and Sudhakara Dvivedi in the PS of VM’ (VIJ., 11 (1973) 107-18) I have
indicated the errors occurring in the said publication. NP have improved upon TS’s interpretation
in some places, but have committed worse mistakes in other places. TS and NP have also failed to
understand how the equation of the centre has been computed and applied to the equation of con-
junction of Jupiter and Saturn. Since they had to interpret, in some way or other, the related verses,
they have altered the verses, in all sorts of ways, to yield what they thought the meaning might be.
Even in the case of Venus, where computation has been made simpler by neglecting the small
equation of the centre, they have committed several errors which have been pointed out in the
exposition of these verses given below.
[शुक्रचारः]
हित्वा ‘(जलमुनि) चन्द्रा (न्)’ द्युगणाद् ‘वेदाष्ट-भूत’हृतलब्धाः |
शुक्रोदया गुणाप्तैः सार्धाः पञ्चालिनो भोगाः || १ ||
कन्यांशा [न्] षड्विंशतिमित्वा शुक्रोऽपरेण या [त्यु] दयम् |
उदयैकादशभागान् दिनेषु दत्वा तत [श्चा] राः || २ ||
Motion of Venus

1. Subtracting 174 from the days from epoch, the quotient got by dividing
   the remainder by 584 are the heliacal risings of Venus. Its motions during the
   periods is 7ʳ 5° 30' 20" each.
2. Having gone to 26° of Virgo, i.e. at, 5ʳ 26°, Venus rises in the west (for the
   first time after epoch). Adding an eleventh of the quotient (given in verse 1)
   to the remaining days, the motions (are to be taken from the Table given in
   verses 3-5).

XVIII.3 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 313
174 days after epoch, Venus rises heliacally in the west. We have emended munijala as jalamuni
because 147 will not agree with the longitude of Venus at rising given as 5ʳ 26°. If Venus is 5ʳ 26°,
the sun must be 5ʳ 18°, to satisfy the 8° given for Venus's heliacal risng in verse 58. But since at
epoch the sun is in the neighbourhood of 358°, it can be only near 4ʳ 25° after 147 days, i.e., 38°
from Venus. After 174 days from epoch, the Sun would be near 169° according to the Vāsiṣṭha-
Pauliśa. (Vide my paper ‘The epoch of the Romaka etc.’, Indian Journal of History of Science, 13 (1978)
ii. 155-58). This gives the elongation as 7° instead of the required 8°. But this small discrepancy can
be put to an accumulated error in computing from the original. TS have not felt the need for this
emendation because they have emended kanyāṃśān into kālāṃśān and omitted the necessary kṣepa
viz. the constant equal to 5ʳ 26°, for beginning the motion. They have not understood the meaning
of the word kālāṃśān which they have brought in. It is the same as the 8° mentioned above, for
Venus. NP have interpreted kanyāṃśān correctly. But since they have kept 147 days to be deducted
intact, they find a serious discrepancy expressed by them on page 124 of Part II. However, they
derive satisfaction from the fact that the September 10th position of the sun would agree with the
time of Venus's rising and their longitudes. This at least must have shown them that the rising takes
place only more than 20 days later. They have made all sorts of unnecessary emendations, but they
have failed to do this necessary one.
We can infer from the instruction to add an eleventh of the quotient, that one synodic revolution
takes 583 10/11 days. In this period the sun has moved 1 revolution 7ʳ 5° 30′ 20″ and Venus, 2
revolutions 7ʳ 5° 30′ 20″. From this we can infer that the sun takes 365-15-25 days for a sidereal
revolution. From the methods given for the other planets also we can see that the sidereal period
of the sun used is c. 365-15-30, which is an evidence for the system given here being connected with
the Pauliśa also, as in the case of the Moon.
TS have unnecessarily emended bhogāḥ into bhāgāḥ and taking sārdhāḥ to mean ‘together with’,
instead of ‘with half’, have given a motion of 7ʳ 5° 20′ per synodic period. They are unaware that
this would make the sidereal year c. 365-22 days, so wrong. The error of 10′ 20″ in Venus would
accumulate by 1° every nine years.
NP interpret sārdhāḥ correctly, but take guṇāptaiḥ to mean ‘with 1/3 degree’ and given 7ʳ 5° 50′,
which would make the sun's sidereal year c. 365-3-0, so very wrong. By this the error in Venus
would accumulate by 1° in every 5 years.
Next the days for segments of motion in the synodic cycle are given by verses 3-5.
षष्टि[र]येण ῾वेदाग्नि-यम-᾿युतामंशसप्ततिं भुङ्क्ते |
(अर्था)ष्ट[कैर्द्वा] (त्रिंशतं) विंश(त्या) [वि]त्रिभिः सपा(दार्थान्)᾽ || ३ ||

1. A.B.C.D.मुनिजल (B. जाल). A.B. चंद्रा 2a. A.D. कन्यांशाः; B. कन्याशाः;. C. कालांशाः
   b. B. ॰ष्टमूत. A2. दूत corrected to हत; B. हत B. षड्वति
   c. A. शुक्रेन्द्या. C. गुणांशौ:; D. गुणांशाः b. A1. यात्पदयं; A2. यात्पदयं
   d. A. सार्द्धा; B. सार्धा c. B. उदैका. C.D. भागं
   B1.3. पंचलिनो; B2. पञ्चालिनो. C. भागाः; d. A.C. सतस्ताराः;. D. ततश्चवारः
   D. भोगः

314 PAÑCASIDDHĀNTIKĀ XVIII.5
वक्रमतस्तिथिभिर्द्वौ पञ्चभिरेवं ततोऽपरास्तमितः ।
दशभिः प्रागुदितः स्यान्नखैश्च जलधीन् मि(तान्) गत्वा ॥ ४ ॥
अनुवक्री परिगत्वा विपरीतमस्तमैत्यै(न्द्रयाम्) ।
षष्ट्यांशपञ्चसप्ततिमित्वाऽपरतो भृगुर्दृश्यः ॥ ५ ॥
॥ वसिष्ठसिद्धान्ते शुक्रः ॥
Days for segments of motion in the synodic cycle
3. In three periods of 60 days Venus moves 74°, 73°, 72°, respectively. In 40
days it moves 32° and in 17 days, 5 1/4°
4. From here retrograde motion (begins). In 15 days these are 2°; in 5 days
the same, i.e., 2°. Then, setting in the west, it rises in the east after ten days.
Venus is in follow-up retrograde for 20 days, moving 4°.
5. Then continuing the (direct) motion round, in the order of days and
motions reversed, Venus sets in the east. Then moving 75° in 60 days, it
becomes visible in the west.
The argument to be used in the above table of motions are the days left over, together with the
eleventh of the quotient, as mentioned. It can be seen that in my emendations of some of these I
have done very little violence to the text. I have been guided in these by the actual motion that must
have been observed, putting it to observational or other error, where the numbers are clear, but
deviate from the actual. The ratio of Venus's distance from the sun to the earth is c. 0.72 given by
all Hindu and modern astronomy, and this I have used to compute the segments of actual motion
for comparison. (See Table on next page)
Another guide is that the days and motions from the superior to the inferior conjunction must
add up to half of the whole, i.e., 292 days, and 287 3/4°, since the equation of the centre has been
dispensed with.
72° motion for the third 60 days is about 4° in excess. For the next 40 days the motion has to be
36°, and I could have filled up the lacuna by (khaissat) instead of (khairdvā) to get this. But the
Siddhānta seems to have compensated the earlier 4° excess by the 4° defect here, which of course is
an error. From this we can see that the emendation of 'dviṁśat' into 'triṁśat' is necessary lest the
motion be reduced to 26°, which is too small.
3a. A.B.C.D. षष्टित्रयेण
B. सेप्ताग्नि
B. मंशसप्त | ति; B. साप्तति भुक्ते
c. A1. अर्धाष्टकेन विंशति (A1. om केन);
B. अर्धाष्टकेन विंशातं (B2. विंशति)
C. अर्धाष्टकेन सप्त; D. अर्धाष्टकविंशत्या
d. A.B. विंशत्यैस्त्रिभिः सपादांशं;
C. सप्तत्यंशांस्त्रिभिः सपादांशम्;
D. विंशत्यं [शक्र] स्त्रिभिः सपादांशम्
4a. B. वक्रमतं क्षितिभिः
d. B. स्रान्तरवैश्च
A. मिता गत्वा; B. मिता गता;
D. सितो for मितान्
5. B. verse missing
5a. C. दन्तक्वैः for परिगत्वा
b. A1. मस्तमत्यैड्य्यां; A2. मस्तमत्यैड्यं;
C. रवशरयमानस्तमेत्यैन्द्रयाम्;
D. विपरीतं चास्तमेत्यैन्द्रयाम्

XVIII.5 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 315
TABLE I
Motion on the synodic circle (computed)

316 PAÑCASIDDHĀNTIKĀ XVIII.7
also sinning against prosody, and given only 28° for the 27½ days, after the third sixty-day
period, and 1½° for the next 3 days, thus, not accounting for 16° and 30½ days.
ardhāṣṭakaviṃśatyā cannot mean 27½, besides being an un-Sanskritic formation. Further, for the
27½ days in that part of the synodic circle the motion should be more than 26° and for the next 3
days, more than 2¼° as can be seen by examining the actual. This error of 16° and 30½ days is
doubled for the whole cycle, and the weight of this error of 32° and 61 days has been carried by
them to the 60-day period of invisibility and drawn the remark on page 121, Part II: "a rather
implausible conclusion. At any event, the description of the motion of Venus as given in our text
seems incomplete." The footnote here is uncalled for.
Jupiter and Saturn
The computation of Jupiter and Saturn follows next to Venus. This is because their treatment is
next simple, on account of their small mean motion and equation of conjunction, owing to their
great distance. Both TS and NP have expressed inability to understand the part of the computation
where the equation of the centre is obtained and applied, before the application of the eq. of
conjunction. Still, they have attempted to interpret the concerned verses, changing the wordings,
drastically, to yield their fancied ideas. In getting the eq. of conj., too, they have made several
mistakes.
As in the case of Venus, here too, the true motion is traced from one heliacal rising to the next.
The method of getting the true anomaly of the eq. cent. is similar to that of the moon given by the
Vāsiṣṭha in chap. II, and based on the same theory of the uniform increase and decrease of the rate
of motion, forming a linear zigzag. Even the same technical term, pada, is used here. All these are
reminiscent of the Babylonian astronomy of the Selucid period, as I have stated above. As between
Jupiter and Saturn, their treatment is exactly similar, so that explaining one would suffice for both.
Verses 6-13, deal with Jupiter and 14-23 with Saturn. My main aim here is to state and explain the
procedure in the computation, a thing not understood by investigators. The verification of the
epoch constants depend mainly on comparison with other systems and modern astronomy. So this
will be done separately.
[गुरुचारः]
विचतुस्त्रिंश(द्युग)णं नाडीभिस्तावताभिरपि च गु(रोः) |
(ह)त्वा 'नव नव दहनै' (रु)दया लब्धा (स्स्थि)ता दिवसाः || ६ ||
उदयनवां(शं) दत्वा दिनेषु षड्वर्गसंगुणैरुद(यैः) |
एकनवाग्नि(च्छिन्नैः) (प)दमिति साष्टादशं शेषम् || ७ ||
Risings of Jupiter
6. The days of Jupiter from epoch minus 34 days, 34 nāḍikās divided by 399,
give the number of risings. The remaining are days (after rising).
7. Add to these days a ninth of the number of risings. Multiply the number
of risings by 36, add 18, and divide by 391. The remainder here are padas.
We can conclude the following from these two verses:
(i). At 34 d. 34 n. from epoch, the first period from rising to rising begins.

XVIII.8 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 317
(ii) The interval between the risings, i.e., the synodic period is 399 − 1/9 = 398 8/9 days.
(iii) 391 padas make one full sidereal revolution of Jupiter, i.e., 360° of mean motion. In one
synodic period, Jupiter moves 36 padas. One pada = 55' 15". 36 padas = 36° 9'. At epochal days 34 d,
34 n, Jupiter's longitude is 18 padas (= 16° 35'). But NP have taken the 18 given here as degrees. This
is wrong. The difference of 1° 25' is too small to show itself in their verification, Table 32, Part II.
But for Saturn this pada constant is 89, and for Mars, 85. Taking these as degrees have resulted in
big differences which have puzzled them. See part II, page 124.
(iv) In 391 syn. revolutions there are 391 + 36 = 427 solar sidereal revolutions = 36 Jupiter's
sid. revolutions. ∴ one sid. rev. of Jupiter takes 4332-22-48 days, and one sid. rev. of the sun = 365-
15-32 days. The latter being very near Pauliśa's 365-15-30, we conclude that, there too, as in the
Moon, it is mixed up with the Vāsiṣṭha's.
TS and NP give the same interpretation, though making more than necessary emendations. In
their verification, TS use the rough syn. period of 399 days instead of the correct 398 8/9, making
the sid. period = 4333-35-0.
क्रमशो म(ध्यः) स्फुटश्च खण्डौ [कार्यो] त(यो)-श्च वि(श्ले)षात् ।
स्फुटहाना द्युषु दद्या(न्मध्यात्) सौ(र्ये)ऽन्यथा हानिः ॥ ८ ॥
8. One after another, mean and true segments are to be arranged. Taking
their difference, if the true is less than the mean, the difference is to be added
to the group of Jupiter's days (left over in the synodic cycle as the remaining
days). Otherwise, (i.e., if the true is more), the difference is to be subtracted.
'True' here means 'true as corrected for the eq. of the cent'. How to get these true positions is
given in verses 9-11, and the segments are to be got using these. So, this verse seems to have strayed
here from after verse 11. The mean positions are to be got by using the remaining padas, extending
the work done in verse 7.
Kāryau is introduced to make up for the syllables wanting. Saure would mean 'pertaining to either
Sun or Saturn'. But we are dealing with Jupiter Sūriḥ here. Saurya alone would mean, 'pertaining
to Jupiter'. Some mss. have no dvi.
Computing and arranging the mean and true segments against each other is to facilitate inter-
polation to any required day. It will also be useful to prepare an ephemeride. The example worked
will make things clear.

6a. A1. विवतु; A2. विवन् C.D. त्रिंशद्. A.B. द्विगुणं
b. A.B.C.D. तावतीभिरपि. A.B. गुरुः
c. B2.3. Unindicated om. of नव [नव....प) दमिति in 7d.
d. A1. तुद्या; A2. नुद्या; B. om. line. A. ॰स्थिदिवसाः
7a. A. उदयस्वांशं; B. a-c. missing. C.D. उदयनवांशान्
b. C.D. गुणे द्युदये. A. ॰रूदद्यः
c. A.C.D. छिन्ने
d. A. वदामिति; B. ॰मितिः. B1.3. साष्टदशं
8a. A.B. द्विक्रमशो; C. द्वि: क्रमशो; D. द्वि [हितः] क्रमशो A. प्यस्फुट
b. A.B.C. खण्डैस्तयोश्च (A. थौश्च) विशेषात्; D. खण्डस्तयोश्च विशेषात्
c. A.B.C.D. स्फुटहानौ B2. द्युयु
d. A. तथ्यत्सौरिन्यथा; B. तत्सौरिन्यथा; C. मध्यात् सौरऽन्यथा; D. त [न्म] ध्यखण्डोऽन्यथा

318 PAÑCASIDDHĀNTIKĀ XVIII.11
TS have expressed doubts about their translation, since they have not understood verses 9-11.
They have retained hi, not supplied the wanting mātrās, and not noticed the grammatical error in
Saure. NP have made three drastic emendations, quite unrelated to the lettering of the text, nihitaḥ:,
maṇḍalaḥ: and tanmadhya-khaṇḍe, though generally following TS.
‘रसविषयकृतशशाङ्काः’ क्षयखण्डे ‘(ख)धृतयः’ पदं यावत् |
‘विषय’ (रसेशा) वृद्धौ जीवः स्यात् पञ्चनवतिशतात् || ९ ||
‘षड्वसुमनवो’ हानौ तृतीयखण्डे गुरुस्तु षोडशके |
प(द)गुणिते ष्ट्यष्टकभाजिते कला पूर्वतोऽभ्युदिते || १० ||
नव सार्धाः कन्यांशाः प्रथमे खण्डे द्वितीयखण्डे (स्युः) |
चक्रार्धं च(युगां)शाः दश(च) कला देवपूज्यस्य || ११ ||
9. Jupiter being in the diminishing-motion-sector upto 180 padas, there is
the constant 1456 (to work with, in order to get the eq. cent-corrected-Jupiter).
Being in the increasing-motion-sector in the next 195 padas (i.e. 181 to 375),
there is the constant 1165.
10. Jupiter being in the diminishing-motion-sector (again) in the next 16
padas, there is the constant 1486. (After subtracting or adding the padas for
which we want computation from these numbers, in the respective sectors),
multiplying them by the padas and dividing by 24, minutes of arc are got, (as
the eq. cent. corrected total motion in the respective sector) at the rising in the
east (and also thereafter if wanted).
11. The total of such motion of Jupiter in the first sector is 5ʳ 9° 30'. In the
second sector, it is 6ʳ 4° 10'.
Briefly expressed as formulae, the eq. cent. corrected Jupiter is given by:
i. If padas are from 0 to 180, (1456 −padas) × padas' ÷ 24.
ii. If padas are in the next increasing sector, i.e. from 181 to 375, (1165 + padas) × padas' ÷ 24 + 5ʳ
9° 30', where the padas used are those given in that sector.
9a. B. रसा B. ॰ते अष्टकभा; C. ॰तेऽष्टकभा; D. ॰ते त्वष्टभा
b. B. Hapl. om: खण्डे [खधृतयः.... खण्डे] d. A.B.C.D. कलाः. A.B.1.2. ॰भ्युदिते;
गुरुस्तु (10b) D. ॰भ्युदेति
A. विधृतयः 11b. A.B2.3. खण्डे स्फुः
c. A. रसोना; C.D. रसेना c-d. B1.2.3. missing.
d. A. पचंवति c. A. चक्रार्धं व गुणाशा; C. चक्रार्धं च गुणाशाः;
10b. B. शोडशके D. चक्रार्धं द्विगुणांशाः
c. A.B.C. पञ्चगुणिते; D. पञ्चविगुणिते d. A. दश श कला; C. परशकले; D. दश सदला

XVIII.11 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 319
iii. If the padas are in the next following sector, i.e. 376 to 391, (1486 – padas × padas' ÷ 24 + 5ʳ
9° 30′ + 6ʳ 4° 10′, where the padas used are those gone in that sector.
Though the instructions are laconic, comparison with the Moon's computation makes things
clear. The increasing-motion sector is obviously the 180° from apogee to perigee, where the rate
of motion is supposed by this siddhānta to increase uniformly from a minimum to a maximum. The
apogee is at 180 padas (= 166°) and the perigee at 376 padas(= 345°). The last 16 padas, continued
by the first 180 padas form the diminishing half circle where the rate of motion diminishes
uniformly from the perigee to the apogee. Differentiating the formula, (constant ∓ pada) pada'/24,
the increase or decrease in the rate of motion is found to be 2′/24 = 1′/12 per pada. There may be
a small hiatus at the junction, apogee and perigee, owing to the unequal division of 391 into 196
and 195, to avoid half pada. But the average of the rates at apogee and perigee, (1165′ and 1486′)/
24 = 55′ 1/4, agrees with the mean motion forming one pada. (Incidentally, this justifies our amend-
ment of viṣayarasonā into viṣayaraseṣāḥ. There are other justifications also, as we shall show later).
Further, the first sector being a continuity of the third, the rate during the first pada in the first sec-
tor must follow next to the rate during the 16th pada of the third sector. Since 1486′/24 is taken as
the motion of the first pada, the motion of the 16th is (1486-30)/24 = 1456′/24. This must be the
commencement of the third sector, and this is what is given. We can also see that the fastest rate,
(at perigee), is 1486′/24 = 62′, and the slowest, 1165′/24 = 48′ 1/2, (at apogee), giving the mean
value 55′ 1/4, of the pada, already found. But the rate for the 196th pada, ending which there is the
apogee, is, (1486 − 195 × 2)′/24 = 1096′/24. But the minimum motion falling at apogee is given
as 1165′/24. This hiatus*must also be due to the fact that the 2′/24 increase in the rate per pada is
only approximate, and the actual is a little less than 2′/24. But the formulae are so given that the
total of the first sector is (1456 − 180) 180′/24 = 5ʳ 9° 30′, as given. The total of the second sector
is, (1165 + 195) 195′/24 = 6ʳ 4° 10′. The total of the third sector is, (1486 − 16) 16′/24 = 16° 20′.
These add up to 12 rāśis, exactly, as they should. Incidentally, this justifies my emendation of
guṇāṁśāḥ into yugamśāḥ, pāñcaguṇite into padaguṇite, and giving the meaning of tryaṣṭaka as 3 × 8
= 24. The justification for correcting vidhṛtayaḥ into khadhṛtayaḥ to get 180, and rasonā into raseṣā
to get 1165, are also reinforced by this perfect agreement found here.
TS and NP also give khadhṛtayaḥ:, seeing the reason for that. TS emend rasonāḥ into rasenā (= 1265),
which will give the total 6ʳ 17° 43′, far from the correct 6ʳ 4° 10′. The text itself gives 6ʳ 3° 10′, one
degree off. TS give 6 rāśis exactly, not knowing the peculiarity of this Siddhānta. Using cakrārdhe
thus, they are left with guṇāśaḥ daśa ca kulāḥ. This they interpret as 13° (wrongly, for it can mean
only 30 or 103). Emending daśa ca kalāḥ into paraśakale, they say that this 13° is the total motion of
the third sector. They do not realise that the 16 padas of the third sector is near perigee, and the
total motion must be greater than the mean motion, 14° 44′. Not knowing the nature of the method
here, they think that the total of the third sector also should be given. It has no use, and Varāhamihira
has not given it.
About pāñcaguṇite tryaṣṭakabhājite, I have emended pāñca into pada, to delete the one mātrā in
excess, and to give the agreement already seen. tryaṣṭaka is 24, as already said. TS retain the pañca,
but emend tryaṣṭaka into aṣṭaka, making it 5/8, leading nowhere.
As for NP, they generally follow TS's emendations. But, for the divisor 8 they suggest the alter-
native 83 (tryaṣṭaka). Unlike TS, they realise that the three sectors must add upto 12 rāśis and make
their own emendation of the last part of verse 11, as dviguṇāṁśā daśa sadalā, interpreting it as 20°
30′. NP have given the gist of verse correctly, but making a lot of unnecessary emendations. They
have wondered in Part II, why such small units, as padas, have been taken. This is because, they

320 PAÑCASIDDHĀNTIKĀ XVIII.13
seem to think, that the three sectors are each taken wholly to get intermediate values by interpola-
tion. An examination of the total of each sector would show how wrong it would be. The true eq.
cent. corrected Jupiter is given for the end of any pada we want. We are expected to use these to get
the true motion through any segmentation of the total padas, for correct interpolation, and the
ends of the segments may fall anywhere, from pada 0 to pada 390. Therefore the small pada seg-
ments are used. I shall work out an example at the end to make everything clear.
I shall explain the rationale of the instruction in verse 8, of adding or subtracting the difference.
The eq. cent-corrected Jupiter is subtracted from the Sun to get the anomaly of conjunction. So,
a positive eq. cent. means less anomaly of conjunction. The days left over represent the anomaly
of conj. with the 399 days of the synodic period, corresponding to 360° of anomaly. So the day is
taken as roughly equal to the degree of anomaly, and the difference in degree subtracted. Vice versa
for the eq. cent. corrected Jupiter, it being less then the mean. Varāhamihira is too astute to confuse
day and degree, as NP think. (In verses 64-81 too, there is no confusion in the author's mind, as NP
seem to think. There he has deliberately chosen the time taken by the Sun to move one degree as
the unit of time, and call it 'day', for convenience. This is patent on the face of the synodic periods
given, though TS have not even seen it, and are perplexed. We have reason to think that verses 64-
81 are by somebody else).
दिन (षष्ट्यांशान्) द्वादश 'खकृतैर्वे (दान्)' 'कृताश्विभिर्द्वौ च ।
सप्ताष्टकेन वक्री षड्[भागान्] षष्टितः षट् च ॥ १२ ॥
अनुव(क्रो) ऽशीत्यार्का (न्' द्व्यू) नार्ध (श) तेन नव ततोऽस्तमितः ।
स्थित्वा सैकं मासं स्फुटोद (योऽष्टोत्तरैरङ्गैः) ॥ १३ ॥
॥ बृहस्पतिः ॥
12. By 60 days, (Jupiter moves) 12°, by 40 days 4° and by 24 days 2°. Becom-
ing retrograde, by 56 days he moves 6° (i.e. -6°) and by 60 days, 6° (i.e. -6°).
13. Following after retrograde, he moves 12° in 80 days, and 9° in 48 days.
Then setting, staying so for a month plus one day, he clearly rises moving 6°
8'. Ends Jupiter.
The Scheme given

XVIII.13 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 321
These values agree well with actualities, considering that whole days and whole degrees are
given, excepting the last 6° 8′, given to complete the value for the synodic cycle. 6° 12′ would be
better at that region and for the whole number, 399 days. 56 days for – 6°, and 60 days for the same
– 6° must be explained by the intention to give whole degrees and segmentation.
Vargāḥ is an obvious mistake for bhāgān, and so corrected. TS have interpreted saptāṣṭakena to
mean 15, which such an expression never means. It can mean either 56 or 87. They understand
another 60 days by the word ca used. All this, to make up the wrong scheme used by them, based
on the mistaken idea that the statement of motions here begins with conjunction and ends with the
rising in the east after the next conjunction. The following is their scheme:
Days Conjunction 60 40 24 15 60 60 80 45 Setting west 30 Rising east = 414
Degrees 12° 4° 2° 0° – 6° – 6° 12° 9° (15°) = 42°
ddīnārdhamatena is emended by TS into dhyūnārdhaśatena but how can this word mean their 45? As
for the last part, sthitvā saikam māsam, they have taken it to mean 30 days instead of the correct 31
days. Let that be. They have not given any motion for it in their interpretation. It cannot be left to
be guessed and completed by an ordinary computer. They, who can be expected to know, have
guessed, quite wrongly, 15° motion for 30 days, not realising that it can be only 6° and a few minutes
more. For the 414 days from conj. to the rising after the conjunction, the total can only be about,
33° 9′ + 3° = 36° 9′, and not the 42° given by them.
As for NP, they have emended ddīnārdhamatena into dinārdhaśatena to mean 50 days. Since they
take 30 days for the setting i.e. one day less, they make the total of days, 400. They give 7° motion
for the 30 days (which they make even 29 days in the last part). They have changed the wording to
some ununderstandable form here, dyavantye māsasya. Further, the 7° is far too much for 30 days.
But there is no 7° in the text. They have corrected the text saikam into śvam, thinking that aśvam
in bhūtasaṅkhyā means 7°.
Incidentally, one other matter may be considered here, viz., the degrees of heliacal rising, for
Jupiter. During the set-period of 31 days, the sun moves about 30½ degrees, and Jupiter, about
6° 8′, and the relative motion is 30½° – 6° 8′ = about 24°, from setting to rising. This gives about
12°, for the heliacal rising of Jupiter, which is fairly accurate, especially for very high latitudes.
(Classical Hindu astronomy gives 11°). Verse XVIII. 58 gives the Vāsiṣṭha-Pauliśa’s degrees of
heliacal rising as 12°, 14°, 12°, 15°, 8°, 15° from Moon onwards, by candrādinām
dvādaśamanuravitithyaṣṭatithisaṁkhyaiḥ:. 15° for Jupiter given here is too much, and 14° for Mars is
too low. (Classical Hindu astronomy gives 17° for Mars). So, the scribe seems to have made a small
change in the order, and the correct order is “candrādīnām dvādaśatithimanuravyaṣṭatithisaṅkhyaiḥ:”,
12°, 15°, 14°, 12°, 8°, 15°, with only one change of place.
Example: Find the True Jupiter at 2415 days from epoch.
(i) The beginning of the first cycle after rising next to the epoch is 34-34 days later.
The days after this, required to find the number of cycles gone = 2415 – 34-34 = 2380-26.
Dividing by 399, cycles gone = 2380 – 26/399 = 5, with 385-26 remainder.
Adding 5 × 1/9 days, (= 0-33), we have 385-59 days left over after 5 cycles gone.
(ii) The padas at 5 cycles gone = 18 + 5 × 36 = 198.
Mean Jupiter = 198 padas = 198 × 360°/391 = 6ʳ 2° 18′.
True Jupiter:–
For the 198 padas, 180 padas forming the first sector has gone and 18 padas are left over in the
second sector.

322 PAÑCASIDDHĀNTIKĀ XVIII.15
∴ True Jupiter = 5ʳ 9° 30′ + (1165 + 18) 18′/24 = 5ʳ 9° 30′ + 14° 47′ = 5ʳ 24° 17′.
Eq. cent. = True − Mean = 5ʳ 4° 17′ − 6ʳ 2° 18′ = −8° 1′.
(iii) The padas at 399 days in the cycle, i.e., the beginning of 6 cycles gone = 198 + 36 = 234 =
180 + 54.
Mean Jupiter = 234 × 360 ÷ 391 = 7ʳ 5° 27′.
True Jupiter = 5ʳ 9° 30′ + (1165 + 54) 54′/24 = 6ʳ 25° 13′
True − mean = Eq. cent. = 10° 14′.
Eq. cent. at 0 day of 6th cycle = −8° 1′
Eq. cent. at 399 days of 6th cycle = − 8° 1′
Eq. cent. at remaining days (385-59) =
= (385-59) × −2° 13′ ÷ 399 + −8° 1′ = −10° 10′.
(iv) True Jup. is less than Mean Jup. by 10° 10′. ∴ days of Anomaly of Conj. = 385-59 + 10-10
= 396-9.
(v) True an. of conj. =
for 60 days + 12°
for 40 days + 4°
for 24 days + 2°
for 56 days − 6°
for 60 days − 6°
for 80 days + 12°
for 48 days + 9°
Total 368 days + 27° 28-9/31 × 6° 8′ = 5° 32′
for 28-9 days 5° 32′
396-9 32° 32′
(vi) True Jup. = Mean Jup. at 0 day of An. of conj. + eq. cent. + true ano. of conj.
= 6ʳ 2° 18′ − 10° 10′ + 32° 32′ = 6ʳ 24° 40′.
Note 1: The need for interpolating the eq. cent. to the remaining days in the cycle can be seen by
working for 399 days of the 6th cycle and 0 day of the 7th cycle and comparing. They must be the
same.
Note 2: The eq. cent. is computed for 0 day of each cycle, i.e., for intervals of 36 padas = 33° 9′.
Interpolation using these, as we have done, can be only rough. To get better interpolations, we can
divide the 36 padas into desired segments, find the eq. cent. of each, and use. We can form an
ephemeride, giving the values at the ends of the day segments given, 60, 40, 24, etc. and use for
interpolation. All these logically follow from the instructions, though not specifically stated.
[शनिचार:]
अध्यर्धशतं (स)त्र्यंशमपनयेत् सूर्यजस्य दिवसेभ्यः ।
'वसुमुनिगुणो' (द्भू)तेभ्यः स्थि(ता) दिनाद्या(स्स) मभ्युदयात् ॥ १४ ॥
जह्या(दु)दयदशांशं क्षुभ्यो नवसंगुणा(न् भ)जेदुदया(न्) ।
'षड्विषययमैः' शेषं पदैर्युतं त(न्र)वाशीत्या ॥ १५ ॥

XVIII.18 | XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING | 323
Motion of Saturn
As already has been said, the treatment of Saturn is similar to that of Jupiter. So there will be little
need for fresh explanations.
14. Regarding Saturn, 150-20 days are to be subtracted from the days from
epoch. These being divided by 378, the remainder are the days from the
rising gone, the quotient being the number of risings gone.
15. One tenth of the risings, (i.e., the quotient), in days, is to be subtracted
from the remainder. The number of risings got is to be multiplied by 9, and
divided out by 256. The remainder plus 89 padas form (the padas required for
using in the computation). (The idea is that 89 is to be added to (quotient ×
9), and then divided by 256, to find the padas for use).
In (15), I have emended saṅguṇād and rudayāt into saṅguṇān and rudayān to agree with bhajet
requiring accusatives; so also NP. But TS have kept them. In NP's emendation dinādyāptaṃ, āptaṃ
does not agree with the word sthitā and, the meaning also is redundant. Both TS and NP have
emended padaiḥ into pade, thinking that navāśītiḥ is degrees. Even this they doubt, as seen in the trans-
lation, because as mentioned by them in Part II, page 124, it had led to disagreement. Padaiḥ, as it
is, clearly, says that the 89 is padas. So is the 18 of Jupiter and the 85 of Mars.
We understand from the instructions that the synodic revolution of Saturn takes 378'/10 days,
that in one synodic revolution Saturn moves 9 padas, that 256 padas make nine sidereal revolutions
of Saturn, that there are 256 + 9 = 265 sidereal revolutions of the sun in 256 synodic periods of
Saturn, and that at 150-20 days from epoch, Saturn's mean longitude is 89 padas. (NP give in their
translation, “89°?”, as mentioned already. Therefore, one sidereal revolution of Saturn takes 378'/
10 × 256 ÷ 9 = 10754.84 days. One sid. revolution of the sun = 378'/10 × 256 ÷ 265 = 365-15-32.
Again, the Sun's sid. period got is Pauliśa's.
One pada = 360°/256 = 84' 22".5. The motion in one synodic revolution = 9 × 84' 22".5 = 12°
39' 22".5.
Mean Saturn at 150-20 days after epoch = 89 × 84' 22".5 = 125° 9'.4.
'षड्रूपवेदपक्षाद्' वृद्धिस्रिंशत्पदानि सौरस्य |
'नवरूपविषययमला(द्)' ह्रासः 'स्वरभास्कर'पदा(न्तः) || १६ ||
प्रचयः 'स्वराग्निखयमान्' नवनव(ति)स्त्रिघनभागलिप्तानाम् |
क्षयवृ(द्धी) द्विगुणपदैरकगुणन्नः श(नेरु)दयः || १७ ||
षोडश वृषभस्यांशा नवलिप्तावर्जिताः प्रथमख(ण्डे) |
'विषया' स्त्रिघ(ना) स्त्रिंश(त्) चतुर्युता मध्यमे खण्डे || १८ ||
14a. A. शत्र्ये; om. the two letters.
15a. A.B. जह्याद्युदय. B. दशांश
b. B. णमपानये सूर्य
b. B. नवसं०. A.B.C. भजेदुदयात्
c. A.B. गुणोष्टृतेभ्यः
d. C.D. पदं युतं. A. तन्तवाशीत्या; B. मत्तवाशीत्या
d. A.B.D. स्थितं. A.B. दिनाद्यास्स; D. दिनाद्याप्त

324 PAÑCASIDDHĀNTIKĀ XVIII.18
16. Regarding Saturn, there is an increase (of the rate of motion) for thirty
padas, from 2416. Then, there is a decrease for 127 padas from 2519.
17. Next there is an increase for 99 padas from 2037. The amount of decrease
and increase are by the padas multiplied by 2. The divisor of the total minutes
is 27, its multiplier being one.
18. The total of the first sector is 1ʳ 15° 51' and the total of the middle sector
is 5ʳ 27° 34'.
Note: The multiplication by one is unnecessary, but given to clear the doubt that may arise by the
instruction to multiply the padas by two for subtraction and additions coming before.
The meaning is clear, and no material change has been needed. I shall give what is given in the
form of formulae:
The total motion upto any pada in the first sector, viz., (1-30) = (2416 + 2 × padas) padas ÷ 27,
in minutes.
That in the second sector, viz., (31-157) = (2519 − 2 × padas) padas ÷ 27, in minutes.
That in the third sector, viz., (158-256) = (2037 + 2 × padas) padas ÷ 27, in minutes.
The total of the whole of first sector given. 1ʳ 15° 51' can be verified thus:
(2416 + 2 × 30) 30 ÷ 27 = 2751' = 1ʳ 15° 51' given.
The total of the whole second sector = (2519 − 2 × 127) 127 ÷ 27 = 10654' = 5ʳ 27° 34', given.
Being unnecessary, the total of the third sector is not given. But we can calculate it and use it to
see if all those add up to 12 rāśis, as necessary, and this will verify every instruction given.
The total of the third sector = (2037 + 2 × 99) × 99' ÷ 27 = 8195' = 4ʳ 16° 35'. Now, 1ʳ 15° 51'

• 5ʳ 27° 34' + 4ʳ 16° 35' = 12ʳ.
  Examining the constants, we find that the maximum motion per pada is 2519' ÷ 27 = 93'.3. The
  minimum rate is 2037' ÷ 27 = 75'.4. The mean rate is = 84' .35 as already found, as the mean
  motion equal to the pada. Differentiating as before, the increase or decrease in the rate is 4'/27.
  Actually it is slightly less than this, the multiplier being slightly less than 2, given. (2037 + 4 × 98)
  = 2416 shows this. The perigee falls at end of 30 padas, i.e., 1ʳ 14°, and the apogee, 127 padas later,
  at 7ʳ 13°.
  The instruction how to use the result of these verses has not been given, because it is the same as
  that given in verse 8 for Jupiter. Indeed, the un-emended reading saure there means, "with reference
  to Saturn".
  16a. B. षड्भूप. A.D. पक्षा c. A. वृद्धि; C.D. वृद्धिः
  c. A.C.D. यमला; B यमलो c-d. C.D. द्विगुणहताश्चैकगुणघ्नः
  d. A.B.C.D. पदाख्यः d. A.B. शनैरुदयः
  17a. A.B.C.D. यमा 18a. B. वृषभांशा
  b. A. नवनवतस्तिघन; B. नवनवतस्त्रिघन b. A. खण्डाः; B. खण्डा
  c. A.B.C.D. त्रिघनः. A.B. त्रिंशः

XVIII.20 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 325
As in the case of Jupiter, here too TS and NP have not understood what exactly is given in these
verses, how it is got by applying the three formulae, how the eq. cent. is got, and why the instruction
to apply this to the days remaining is given, in the manner said.
So, their emendations of the readings, done without knowing the subject matter, need not be
taken seriously. TS have emended the correct dviguṇapadaiḥ into dviguṇahṛtaḥ, meaning "divided
by 32", applied to the risings and not to the number got in the formulae. NP have kept the reading,
but given the translation as, "There is a subtraction or addition of 12 degrees and minutes, (i.e., 12°
12'). Multiply by 31 and divide (the product) by 32 (or by 32 padas). (The result is) Saturn's rising."
Where is 12° 12' mentioned? They take the 32, not as a number, but as a segment of longitude
equal to 32 padas, i.e., 45°. Again, how can this give the risings? And the risings have already been
given in verse 14. All these show that they do not understand what is said.
षड्(कृत्या त्रीनं)शान् (मुनि)भिर्लिप्ता '(श्रेषु)गुणास्सप्त' ।
षोडशभिश्चांशी(र्ति)कृतोनषष्ट्या '(वेदयम)'पक्षान् ॥ १९ ॥
वक्री वि'भूतषष्ट्या' (त्री)नंशान् षष्टितः 'कृतान्' सौरः ।
अनुगोऽर्कश(ते'ना)ष्टौ षट्कृत्या चास्त(गो) 'दहनम्' ॥ २० ॥
|| शनैश्चरः ||
19. Saturn (moves) 3° in 36 days, 35' in 7 days, 80' in 16 days, and 224' in 56
days.
20. Then becoming retrograde, he moves 3° in 55 days, and 4° in 60 days.
Then, following up direct, he moves 8° in 112 days, and setting, he moves 3°
in 36 days in the set period, (i.e., rises in the east after that). Ends Saturn.
This is the scheme given
Days Rising east 36 7 16 56 Retrograde 55 60 Direct 112 Setting west 36 Rising east = 378
Degrees 3° 35' 1°20' 3°44' - 3° - 4° 8° 3° = 12° 39'
I shall now discuss the values given, justifying the three emendations. I have made. The corrupt
ṣaḍhratāstrīṇamśān has to be emended as 3° for 36 days, considering the position, and the fact that
it must practically be equal to the rate between setting and rising, 3° for 36 days. The days must add
upto 378 days from rising to rising, also as from conjunction to conjunction. All the numbers for
19a. A.B. षट्कृताः; C. खण्डान्ये. D. परिहीनाः 20a. B. विभुत
b. A.B. स्त्रीणांशान्; C. सिंहांशा. D. स्त्रीखांशा a-b. C. षष्टाष्टरसैस्त्रीन् षष्टितः
c. A.B.D. लिप्ताश्चतुर्गुणाः A. त्रिंशान्; B. त्रिंशान्: A1. कृतात् सौरः;
c. A.B. श्वाशीति; C. श्वांशामीन् B1.3. कृत्यसौरः
d. A.B.C.D. षष्ट्या द्विगुणपक्षान् (A. पक्षात्, c. A. शतैर्नाष्टौ; B1. शतैर्माष्टौ (B2. र्माष्टौ);
B1.2. पक्षा. B. श written after पक्षा) D. शतेनाष्टौ
d. A1.2. गे दहनं || B1.2. षड्भक्त्या वास्तये दहनं ||

326 PAÑCASIDDHĀNTIKĀ XVIII.20
days are clear. Therefore, the days for the second segment must be 7. So I have emended manu into
muni. The motion given there, 28', gives the rate 2', too absurd for that position, if the original 14
days are accepted, and it cannot be that the Siddhānta does not know the absurdity. Even for the
emended 7 days, it is too low, being only 4' rate, while the rate on both sides is 5', and also consis-
tent with facts. Therefore, ścaturguṇā is emended into śceṣuguṇā. Now, these three segments can be
combined into 4° 55' for 59 days, without affecting the result. I do not know why the Siddhānta has
broken it into such bits.
Next, the total for the 378 days must be the mean motion for the period, i.e., 9 padas, equal to 12°
39'.4, roughly taken by the Siddhānta as 12° 39'. Therefore the motion for the fourth segment, 56
[

XVIII.20 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 327
I shall now give an example, to make the method clear.
Example: Find true Saturn at 5000 days gone from epoch.
i. Days 5000
To be subtracted 150-20

Divided by 378 4849-40 (12 = full cycles gone)
313-40 = (Remaining days)
Days to be deducted 10/12 : 1-12

312-28 (corrected remainder)
ii. Padas at 0 day of the 13th cycle: (89 + 12 × 9) / 256 = 197 remainder
Mean longitude = 197 padas = 9ʳ 7° 2′
197 = 30 + 127 + 40 (in the third sector)
Eq. cent. corrected mean longitude:-
= 1ʳ 15° 51′ + 5ʳ 27° 34′ + (2037 + 2 × 40) 40′ ÷ 27
= 1ʳ 15° 51′ + 5ʳ 27° 34′ + 1ʳ 22° 16′ = 9ʳ 5° 41′
Eq. cent. = 9ʳ 5° 41′ − 9ʳ 7° 2′ = −1° 21′
iii. Padas at 378 days gone in the cycle = 197 + 9 = 206
Mean longitude = 206 padas = 9ʳ 19° 41′
206 padas = 30 + 127 + 49 (in the third sector)
Eq. cent. corrected mean longitude = 1ʳ 15° 51′ + 5ʳ 27° 34′ + (2037 + 2 × 47) 47′/27 = 9ʳ 18° 0′
Eq. cent. = 9ʳ 18° 0′ − 9ʳ 19° 41′ = −1° 41′
Interpolated for days 312-28, the eq. cent = −1° 21′ − 0° 17′ = −1° 38′.
iv. Correcting the remaining days 312-28 by this, 312-28 + 1-38 = 314-6 days, to be used to find
anomaly of conjunction.
v. 36 days +3°
7 ... + 0°.35′ Mean Sat. at 0 day 9ʳ 7° 2′
16 ... + 1° 20′ Eq. cent − 1° 38′
56 ... + 3° 42′ An. of conj. + 7° 40′
55 ... − 3° ---------------------------
60 ... − 4° True Saturn = 9ʳ 13° 4′
Remaining 84-6 + 6° 1′ ---------------------------

314-6 + 7° 40′ As per Ephemeris: 9ʳ 11°.7 (sāyana)

Mars
As indicated earlier, Mars, like Mercury, needed elaborate treatment owing to certain
peculiarities about it, and so had been reserved by VM to the end of PS. The synodic period of

328 PAÑCASIDDHĀNTIKĀ XVIII.24
Mars, on which the equation of conjunction depends, is 780 days, during which there are more
than two revolutions of the Sun, and one revolution of Mars, so that one full anomalistic period of
Mars is contained within this period. This, with the large equation of the centre, and the large
equation of conjunction causes large variations in its motion from sign to sign, and even in the same
sign, according to the different types of motion governed by the anomaly of conjunction, like, fast,
slow, retrograde etc. Hence is the need for detailed treatment.
Further we have reason to think that the various motions given are all empirical, based on long
observation, synodic period after synodic period. The separation into the equation of the centre,
and the equation of conjunction is yet to come, it seems, unlike the cases of Jupiter and Saturn,
where it is easy, and done. This would explain discrepancies found in the values given.
Regarding the constants given, some can be verified by mutual comparison, and corrected where
necessary, when there is a doubt about the reading itself. But some, like the epoch constants, which
are peculiar to the Siddhānta itself, cannot be so verified and corrected when there is a doubt. Only
in such cases, where we can argue that no Siddhānta is likely to give such wrong values, and when
these values are so far from the real, that we can make some plausible corrections.
TS and NP have not understood the nature of the motion of Mars, just as they have not under-
stood Jupiter and Saturn. While TS have not even attempted translating some verses, wrongly
interpreting those attempted, NP have attempted translating all, but many wrongly. I shall point
out these after my own translation and discussion of the verses, step by step.
[कुजचारः]
द्युगणे ‘षट्(पञ्च)यमान्’ विहाय पञ्चाष्टकं च नाडीनाम् ।
‘गगनाष्टमुनिभि’रुदया लभ्यन्ते प्राङ् महीजस्य ॥ २१ ॥
उदयगुणिता विनाड्यः ‘स्वरतिथयोऽ(ब्ध्य)’न्विता दिनक्षेपः ।
‘धृतिगुणि(तांस्त्र्यग्रीन्दु)’भिरुदया(न्ह)त्वा स्थितो (तस्मिन्) ॥ २२ ॥
पञ्चाशीतिं कृत्वा प्रतिराश्य मध्यमः क्रमशः ।
राशिप्रमाणतोऽ(स्य) स्फुट(तश्च)रक्रमं कुर्यात् ॥ २३ ॥
स्फुटमध्यम(विश्लेषां)शान् क्षिपेत् मध्यमे[ऽधिके] द्युभ्यः ।
मध्यमहानौ जह्यात् गतितोऽथ चारा(न)भिधास्ये ॥ २४ ॥
Motion of Mars
21. Subtracting 256-40-0 days from the days from Epoch, and dividing by
780, the synodic risings of Mars in the East are got.
22-23. (157 plus 4) vināḍis, multiplied by the risings got, are to be added to the
remaining days. Multiply the risings got by 18, and adding 85, divide by 133.
The remainder, converted into rāśis is Mars at rising. According to the whole
or portions of rāśis, the true motions are to be taken one after another, and
pieced together.

XVIII.24 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 329
24. The difference between the mean and true degrees should be added (to
the remaining days got in 21), if the mean is greater. If the mean is less, the
difference should be subtracted from the remaining days. This done, I shall
give the true motions, according to each type of motion:
We learn from the verses the following:
. 256-40-0 days from Epoch, Mars rises in the East, after which the counting of risings begin.

1. One synodic revolution takes 780 days — 161 viṇāḍis, (i.e. 779-57-19 days). The addition of
   viṇāḍis multiplied by revolutions, is for taking the synodic period as 780 days approximately.
   '. For this period of 779-57-19 days, we get 1 + 18/133 sidereal revolution of Mars, and 2 + 18/133
   sidereal revolutions of the Sun. So, in one synodic period, Mars moves 408° 43′.3
   ∴ In 133 syn. periods = 1,03,734-3-7 days, there are 151 sid. rev. of Mars, and 284 sid. rev. of the
   Sur. From this, the Sun's sid. period got is 365-15-38 days and Mars's 686-58-50 days. The Sun's
   period is 38 viṇāḍis more than that given for it by the Vāsiṣṭha, and near the 365-15-30 of the
   Pauliśa. Therefore, like the Moon, Venus, Jupiter and Saturn, Mars also is common to Pauliśa.
2. At the first rising when calculation commences, mean Mars = 85/133 rev. = 7ʳ 20° 4′.5
3. The addition or subtraction of the difference from the remaining days has been already
   explained with reference to Jupiter and Saturn. But, here, no method is given to find the equa-
   tion of the centre. Now the true motion is affected by both the eq. of the centre and eq. of con-
   junction. The segments of motion given in verses 25-26, below, are as affected by the eq. of con-
   junction alone. By making the days given for true motion in verses 27-35 conform to the seg-
   ments, we can get the degrees, and through that the days affected by the eq. centre alone. This
   can be of use only fōr the remainder of days. But the eq. of centre at the beginning of each cycle
   is necessary. It has not been given by any rule. Since its period is about 687 days and it has its
   own rise and fall of about 11° from perigee to apogee and back, it cannot be associated with the
   synodic period of 780 days. So this is an omission.
   I have corrected the corrupt tāstrayāmṛīdubhiḥ into tāms tryagnīndubhiḥ to mean 133. This is
   necessary for agreement with the actuals, and the effect of my emendation is seen in my discussion
   (3) above. TS have made it bāṇendubhiḥ. How can bāṇa, with such different lettering, come in here?
   Further, this will give 18/15 rev. = 432° as the mean motion of Mars in one syn. period, about 23°
   wrong per period. They have made pañcāśīti kṛtvā into pañcāmśonam kṛtvā and thus shut out the
   position constant of Mars on the first day where reckoning begins, viz. the point of time 256-40-0
   days from epoch. (It will be remembered that in the cases of Moon, Venus, Jupiter and Saturn also,
   they have made this mistake). By this emendation they reduce the motion by 86° 24′, and make the
   21a. C.D. द्युगणात् b. C.D. प्रतिराशिं;
   A.B. षट्कं व यमान् (B3. षट्कं). D. षट्कैकयमान् B. मध्यतः, A.B. क्रमश
   b. B. नाडित्वं c. A.B. प्रमाणतो स
   d. B. प्राक् d. A.B.C. स्फुटता चार०; D. स्फुटिता चार०
   22b. A. योब्धान्विता; B. सोब्दान्विताः B. क्रकुर्यात्; D. क्रम[त्] कुर्यात्
   c. A.B. गुणितास्त्रयग्नी (B. त्री) दुभि; C.D. ०गुणितान् 24a. A.B. विक्षेपां
   बाणेन्दुभि b. A. ०शन् क्षिपेत्; B.C.D. शकान् क्षिपेत्
   d. A2. हत्वा; A.B. स्थितो तो समाः; c. A.B.C.D. मध्यमे द्युभ्यः
   C.D. स्थितोऽतोऽस्मात् d. A. गतितोथाचारमभि०; B. गतितोष्याचारमभि०;
   23a. A.B. पञ्चशीति; C.D. [पञ्चांशोनं] C. गतितोऽथो चारमभि; D. गतितोऽप्याचारमभि०

330 PAÑCASIDDHĀNTIKĀ XVIII.26
mean motion of Mars 345° 36' per syn. period of 780 days! They have also wrongly emended
pratirāśya to pratirāśyam.
As for NP, they have made the correct emendation tryagnīndubhiḥ, giving correctly 151 revolu-
tions of mean Mars in 133 syn. periods, and identified it with that given by the Babylonian
astronomy of the Selucid period.
But NP have not seen that pañcāśītim meaning 85, is correct as it is, and give the constant 7ʳ 20°
4'.5 at 256-40-0 days from epoch (see item 5 of discussion). They think it is the constant in degrees,
though no word meaning degrees is found here. (This kind of mistake they have made in the case
of Jupiter and Saturn also, as we have shown). But 85° would not do, so they have substituted
satrirāśim for pratirāśya and made it 175°. But even this would not do, and therefore they have
changed the days from Epoch itself into 216-40-0, by emending ṣaṭkam va yamān into ṣaṭkaikayamān.
But this has led to other troubles, leading to their remark, “For Mars this would mean a longitude
of 175° (instead of 194° derived on the basis of a₀ in table 32). This longitude would correspond to
September 27, and a solar position at 186°, hence to an elongation of 11°’ (p.124, Part II). 11° for
the first visibility of Mars is given by nobody. It is in the range of 14° to 17°.
प्रागुदये षट्(चत्वार्येक)मष्टादश(ग)स्ततो वक्रम् |
अध्यर्धं च शतं शीघ्रां [स्ततोऽस्तमितो द्यूनां षष्ट्या] || २५ ||
समतीत्य दशत्रियु(तं) निरंशगतो(ऽतस्त्रिंशतं) व्यतीत्य कुजः |
उदयमुपयाति वक्ष्ये गतिचारदि(न)क्रमे चातः || २६ ||
25-26. After rising in the East, Mars moves 146° (in quick motion) and then
18° each of (slow motion), retrograde and “follow up after” retrograde
(anuvakra), and after that 150° of quick (śīghra) motion. Then setting, it
reaches conjunction (niraṁśagataḥ) in 60 days moving 13 plus 30 (= 43)
degrees. Then it rises, (moving the same degree in the same number of days).
Beginning from here, I shall mention the series of motions with their days.
The scheme is
Rises East
Moves 146°            I                      type of motion            (śīghragati)
              18°           II                                                     (Mandagati)
          −18°      III & IV                                                (Vakaragati)
                                                                                   (Ativakragati)
              18°          V                                                   (Anuvakragati)
            150°          VI                                                  (Śīghragati)
Sets in the West
43° VII in 60 days                                                     (Atiśīghragati)
Conjunction                                                              (          ”          )
43° VIII in 60 days
Rises in the East
Total 400°

XVIII.26 XVIII. VĀSIṢṬHA-PAULIŚA – RISING AND SETTING 331
The numbers I have given in the scheme are practically what are found in the text, without
emendation excepting three. In verse 25, I have emended captasteka into catvāryeka to get 146° the
most plausible value. NP have made it ṣaṣṭāṣṭaika to get 186° which is too large. See discussion below,
150° is given by adhyardhaṃ ca śataṃ where tataḥ is emended into śatam. This is necessary to make up the
total 410° motion in 780 days. Secondly, 43° motion for 60 days given from setting to conjunction
is required to agree with the 17° usually given for heliacal rising. This is made up by emending
viṃśatam into triṃśatam with the 13° given by daśatriyutam added. I shall now show that the motion
of Mars is near 43° in 60 days, in the region of the conjunction. For its distance, nearly 1.53 that of
the Sun, given by modern astronomy and also as computed from Hindu astronomy, the equation
of conjunction at this region is 11' per day, (as can be verified) which, plus the daily mean motion
of 31'.4 gives 42'.4 per day, making 42°.4 in 60 days, roughly 43°. This also agrees with the angle
for heliacal rising of Mars, nearly 17°, given by Hindu astronomy. (In 60 days the Sun moves 59.1°.
So, the elongation is 59.1 − 42.4 = 16.7°, nearly 17°). If viṃśatim is taken as it is, we get 20° + 13°
= 33°, which is 9.4° short of the actual 42.4 and which also gives the angle for heliacal rising as great
as 26°, so far from the 14°-17° given by all.
In the mean, the motion from setting to conjunction must be equal to the motion from conj. to
rising. That is why it is not given by the text separately. That the motion segments given in the two
verses is mean is also clear, since no position of Mars from its apogee is taken into account. So the
total motion must be equal to 409°. But the total got by adding the segments is 400°. This must be
due to the defective method of the original or the empirical nature of the motions, and rounding
off to whole degrees, as seen from 43° being given for 42.4°. The opposition must fall at the mid-
point of the retrograde motion, − 18°, and divide it into −9, −9. The total motion from conj. to
opposition must be equal to that from opposition to conj. But what we actually get is 43° + 146° +
18° − 9° = 198°, and −9° + 18° + 150° + 43° = 202°. It may be that the angle segments given are
empirical and also there are errors in the apparently correct numbers giving the segments, needing
emendation. It is only in the case of Mars, does VM give these eight types of motion. In II. 12-13
of the Later Sūrya Siddhānta, a set of 8 types of motion is given. But they cannot be equated to these
each to each. So we have only to guess when in doubt. The days on the synodic cycles to pass each
type of motion must be nearly equal to the average of the days given in verses 27-35 for that type
of motion. This has been used to check the degrees of each type. But the synodic period, as also the
mean motion of 1 + 18/133 revolution in that period are very nearly correct and they must have
been got by analysis of the observed motions. So the Siddhānta must have known that the motions
and times are half and half on both sides of the opposition.
Beginning from rising type I is śīghra (quick) motion. II is manda (slow) motion. The distinction
seems to be faster or slower than the mean motion. So, the dividing point must be where the tan-
gent from the earth touches the synodic circle. Since the mean distance of Mars is 1.53 times that
of earth from Sun, this point falls about 189.5° from conjunction. Subtracting 43.5° motion from
                                                                                 d. A.B.C. द्यु (B. द्वा) नाषष्टितस्तोस्तमितः;
25a. A. षट्चप्तस्तेक॰; B. षट्वपप्तस्तेक॰;                            D. त्रिभ्रषष्टिं ततोऽस्तमितः
      D. षट्कास्तैकं; C. पदसप्तकं                                 26a. A.B.C. त्रियुता; D. त्रिहतात्
  b. A.B. ॰दशमस्तगस्ततो; C. ॰मष्टादशमासगस्ततो;             b. A.B.C. निरंशतो विशंति (B .विंशति)
      D. ॰दश वक्रगस्ततो                                                D. [निरंशगस्त्रिंशतिं]
      B. चक्रं; D. वक्रः                                                   c. B. उदयेमुपयाति
  c. A.B.C. अत्यर्धं च ततः शीघ्रात्. D. गत्यर्थं च ततः शीघ्रात्      d. A.B.C. द्यु (B .द्वा) नाषष्टितस्तोस्तमितः;