सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

C O N T E N T S
Page
Foreword ... xv
Preface ... xix

1. सिद्धान्तशिरोमणौ गणिताध्याये मध्याधिकारे
   कालमानाध्यायः ... 1
2. Madhyādhikāra –
   Section II Bhagaṇādhyāya ... 18
3. Grahānayanādhyāya ... 29
4. Madhyādhikāra - Kakṣādhyāya ... 45
5. The Adhyāya known as Pratyabda Śuddhi
   in Madhyādhikāra ... 53
6. The Section known as Adhimāsādi-
   nirṇaya in Ch. I ... 70
7. The Section Bhū-Paridhi-Mānādikara
   — Circumference of the Earth ... 83
8. Spaṣṭādhikāra - Rectification of Planets ... 102
9. The Tripraśnādhikāra ... 223
10. Parvasambhavādhikāra ... 331
11. Lunar Eclipses ... 346
12. Grahacchāyādhikāra ... 449
13. Grahodayāstādhikāra ... 471
14. Sṛṅgonnatyadhikāra ... 482
15. Grahayutyadhikāra ... 501
16. Pātādhyāya ... 521
    Appendix ... 545

ओं श्रीमहागणपतये नमः, श्रीगुरुभ्यो नमः
सिद्धान्तशिरोमणौ गणिताध्याये मध्याधिकारे कालमानाध्यायः
Verse 1. May the Sun at once give expression to
our tongue in meaning ful words-that Sun, who rises
to protect this world, whose duty it is to expel darkness,
who is reported to be the Spouse of the lotus-creeper, who
purges out the sins of all those that supplicate him, and
on whose rising take place Vedic sacrifices and thereby
the gods in heaven headed by Indra get feasted.
Commentary (Comm.). Needs no Commentary.
Verse 2. Excels that blessed Brahmagupta, the son
of Jishnu, who is hailed as the crest-jewel of Mathemati-
cians ; excel also those like Varāhamihira who were the
authors of well known works, who were adepts in reason-
ing, and u age of beautiful expression and on a study of
whose works, one like me even of lesser intellect, will be
able to produce monumental works.
Comm. Needless to comment. However, we may
state one thing. Bhaskara’s quoting the name of Brahma-
gupta at the very outset, and that too with reverence,
informs us that he is going to accept the Āgama of
Brahmagupta in preference to that of others like Arya-
bhata. This means that he is going to adopt the number
of revolutions of planets in a Yuga and such other astro-
nomical constants as were adopted by Brahmagupta. This
point will be clarified later.
Verse 3 This blessed author Bhaskara is now writing
the work named Siddhānta-Siromani, the crest-jewel of
all astronomical works, for the pleasure of good minded
astronomers, after having bowed to the lotus-feet of his

2
father, from whom he has derived his knowledge; this
work, containing good metres, will be easy to understand,
besides being flawless and clear, and will enable intelligent
readers to develop their ability to understand things.
Comm. Not neeessary.
Verse 4. Ancient astronomers did write, of course,
works abounding in intelligent expression; nonethe-
less, this work is started to give expression to some
lacunae in their works. I am going to make amends for
the deficiencies of the older works and these improvements
will be found here and there in their respective places; So,
I beseech the good-minded mathematicians to go through
this entire work of mine also (for, otherwise, they may
not locate my own contribution).
Comm. Not necessary.
Verse 5. May the good people be pleased with the
particulars of my contribution! May the ill-minded people
also derive pleasure out of ridiculing me with ignorance
unable to understand my contribution!
Comm. Not necessary.
Verse 6. A Siddhānta work ie an astronomical trea-
tise is such a one which deals with the various measures of
time ranging from a Tṛti (to be explained shortly) upto
the duration of a Kalpa which culminates in a deluge;
planetary theory, arithmatical computations as well as
algebraical processes, Questions with respect to intricate
ideas and their answers, location of the earth, the star
and the planets, and description and usage of instruments
Comm. Not necessary.
Verse 7. Though one knows astrology and that par
of the science of Jyotisha which is known as Samhita (an
which deals with various subjects like Muhurtas i.e. auspi

8
cious moments to be prescribed for various functions,
Desarishtas ie Calamitous occurences to the countries etc.)
which form a part of the Science, he cannot answer so
many intricate problems pertaining to Astronomy. Such
a person, who does not know the astronomical part of the
science, which abounds in innumerable reasonings, is one
like a king depicted in a drawing, or a lion fast tied to a
pole.
Comm. Not necessary.
Verse 8. The Science of Jyautisha without Astronomy,
is like a king's army without roaring elephants though ex-
celling in horses etc; is like a garden without mango trees,
or like a lake without water, or again like a lady parted
with her newly married lover.
Comm. Not necessary.
Verse 9. The Vedic lore prescribes Sacrifices to be
performed; these sacrifices are based upon a knowledge
of appropriate time to perform them. This science of
astronomy gives a knowledge of time; hence it has been
reckoned as one of the six Vedāngas or limbs of the Veda.
Comm. Not necessary.
Verse 10. (Out of the six Vedangas) The science of
grammar is like the face of the person of the Veda, the
science of Jyautisha takes the place of the eyes, the
Nirukta that of the ears; the Kalpa that of the hands; the
Siksha that of the nose and the Chandas the place of the
feet.
Comm. Not necessary.
Verse 11. This science of Jyautisha being depicted
as the very eyes of the Person of the Veda, so it has been
acclaimed as the most important of the six Angas or limbs
of the Veda, in as much as, even if a person be endowed

24
with limbs like the ears, nose etc, if he be devoid of vision,
he could not do anything.
Comm. Not necessary.
Verse 12. (Since this Science of Astronomy has been
declared as the most important of the Vedangas) hence
this has to be studied by the Dwijas (ie Brahmins Kshatri-
yas and the Vaishyas who form the three higher castes),
also because it is sacred, secret and the best discipline. By
so doing they would acquire Dharma, Artha, Kāma as well
as fame (Life is depicted as having a four-fold purpose out
of which, Dharma, Artha and Kāma form the first trio,
Moksha being the ultimate goal of life).
Comm. Not neceasary.
Verses 13 and 14. The creator having created the
stellar circle along with the planets, placed the latter at the
beginning of the circle, put them in constant revolution, at
the same time putting the extreme two stars (on either
side) in a fixed position.
Comm. Bhaskara's own commentary Vāsanā Bhasha
under this verse mentions the following points, which are
to be noted. The twenty seven stars known as Aswini,
Bharani etc. occupy positions roughly at equal distances
along the Zodiac, arranged from west to east. The planets
were all placed in the beginning of the stellar circle in
such a way that they were in a straight line the moon,
Mercury, Venus, Sun, Mars, Jupiter and Saturn occupying
consecutive positions from the earth in increasing dist-
ances, not of uniform measure. The circle of stars known
as the Zodiac lies far behind all the planets. There is a
wind known as pravaha which keeps the entire Zodiac, as
well as the planets below going round and round in the
Westerly direction. At the same time, the planets while
participating in this westerly motion, have themselves
individual motion towards the east. Two stars, are placed

5
one at the north-pole and the other at the south, and they
are fixed¹. The diurnal motion due to pravaha is far
greater than the individual motion of the planets among
the stars in a direction from west to east ie the direction
opposite to that of diurnal motion.
In this context, it is worth-noting that Aryabhata I
mentioned the diurnal rotation of the earth in the verse
“अनुलोमगतिः...” Bhaskara must have been aware of this ;
yet, in spite of his rational outlook in all matters, mis-
guided himself in this respect (vide verse 3 Madhyagati-
vāsanā-Goladhyaya), apparently for fear of tradition.
Verse 15. The first Mahayuga, the first year, the first
day of the bright half of the first month named Madhu,²
all of them began simultaneously at the Sun-rise³ at Lankā
on Sun-day, at the beginning of the first Kalpa which
marked the beginning of creation.
Comm. Though, in the Commentary before, Bhaskara
gave expression to the fact that Time is eternal with no
beginning and no end, herein he mentions the point of
Time when creation commenced. So, at the back of his
mind, the concept of Time arose only at the beginning of
creation, whereas before creation, as well as after deluge
the आत्यन्तिकप्रलय, there was and would be neither the
concept of Time nor space. In other words, both space
and Time are manifest only after creation, and get extin-
guished after deluge.
Verses 16, 17 and 18. The unit of Time named Tat-
para is 1/30th of what is known as Nimeṣa or the time
taken during the fall of an eyelid; One-hundredth of a
Tatpara is known as a Tṛti ie the time taken to pierce a
lotus-leaf with the finest needle. Eighteen Nimeṣas are
equal to a Kāṣṭhā. Thirty Kāṣṭhas are equal to a Kalā,
and thirty Kalās are equal to one sidereal ghatī. Two

6
ghatis make a Kshana and thirty Kshanas make a sidereal
day. Thirty sidereal days are equal to a sidereal month
and twelve sidereal months make a sidereal year (not the
sidereal solar year). The Zodiac, divided into twelve rasis,
and 360 degrees, a degree divided into 60 minutes of arc
and a minute divided into 60 seconds of arc all correspond
to the year and its successive divisions.
Comm. In the Commentary under these verses, Bhas-
kara gives further details of division of time as follows.
The time taken to pronounce a guru, i.e. double the time
of pronouncing a short vowel, is one-tenth of a Prāna,
which is the time required by a healthy person to inhale
and exhale once. Six Prāṇas make one Vighaṭī and sixty
ghatis make one sidereal day. It may be clearly noted
here that this sidereal month consisting of thirty sidereal
days is not the sidereal month that is the time taken by
the Moon to go round the Zodiac, nor the sidereal year
defined above is the time that the Sun takes to go round
the Zodiac once in his aoparent annual motion. To distin-
guish these latter divisions of time, we shall use the
nomenclature sidereal lunar month and sidereal solar year.
There are further other divisions of time which will be
later elucidated.
Verses 19, 20. The time taken by the Sun to com-
plete one revolution with respect to the stars goes by the
name ‘The sidereal solar year’. This will be a day for
the gods and demons. The time that elapses between two
consecutive new moons or conjunctions of the Moon with
the Sun is called a Chāndra-māsa or a lunar month or
simply a lunation. This again is the day of the Pitṛs or
the Manes.
The time that elapses between two consecutive Sun-
rises at a place is termed the Sāvana day or civil day.
This is called Saura-Sāvana day and it is also the day of
the earth.

7
The sidereal day is the time taken by the stars to go
round the earth once. It is called Nākshatra-dina.
Comm. In Hindu mythology gods are supposed to
reside at the north-pole, where one civil day is the same
as one sidereal solar year for the other places; also the
demons are supposed to reside at the south pole so that
their civil day also is equal to a sidereal solar year. But,
what is day to the gods is night to the demons and vice-
versa.
If we call the earth, the moon of the moon so to say,
which it is so, when it is the moment of new moon for the
earth, it is the moment of Full Moon to the Moon. Thus
what is a Chāndra-māsa to the earth may very well be
called with respect to the Moon a Bhauma-māsa. In
Hindu mythology the manes are supposed to take residence
on the surface of the Moon. We know from modern
astronomy that the moon revolves about her axis once
roughly in a lunation. Thus for 'the man on the moon',
a day is roughly equal to our lunation. So if the manes
were to reside on the moon, their day is equal roughly
to a lunation of ours. We say 'roughly' because as we
see under the chapter of the lunar eclipse, the moon
almost shows the same face to the earth on account of
what are called 'librations in longitude'. On this count
the time of rotation of the moon does not exactly coincide
with the time of a lunation.
A civil day for a place is also the civil day for every
place of the earth so that it is called the earth's day. By
'day' here we do not mean the time when the Sun is
above the horizon, for that time differs from place to place
on the earth. A civil day is the sum-total of the duration
of day and the duration of night for any place and it will
be seen that this is the same for the entire earth except at
the places having perpetual day. The word Saura Sāvana
is used to signify that the time that elapses between two
consecutive rises of any other planet is termed the Sāvana

8
day pertaining to that planet. Thus the Chāndra-Sāvana
day is roughly equal to 24 hrs–52' and this is the longest
of the Sāvana days pertaining to the planets. The shortest
Sāvana days is that of the Saturn, since Saturn moves a
very little along the ecliptic and his Sāvana day is there-
fore just a little longer than the sidereal day. This nomen-
clature brings in the phenomenon that the Sāvana day of a
retrograde planet happens to be less than a sidereal day.
But in a given time, which is sufficiently long like a yuga,
the Sāvana days of a planet are equal to the number of
sidereal revolutions in that period minus the number of
the planetary revolutions in the same period. This will
be constant for a given planet in such a long period,
though individual days happen to be some shorter and
some longer than a sidereal day.
The modern sidereal day is the time between two
consecutive rises of the equinoctial points and as the equi-
noctial points have a slow retrograde motion, the modern
sidereal day is just a little shorter than the Hindu sidereal
day, which does not take cognizance of the revolution of
the equinoctial points round the earth in reckoning diurnal
motion. The Hindu astronomers speak of the revolution of
the stars only around the earth in the context of diurnal
motion. It will be noted that the Saura-Savana day or the
civil day will not be of the same duration, since the Sun has
unequal motion amongst the stars from day to day. When
the Sun is in perigee and has the max. daily velocity, his
Sāvana day at that moment is the longest and the Saura-
Sāvana day when the Sun is in apogee will be the shortest.
Though in Indian Chronology a day is divided into
sixty ghatis for convenience, these ghatis are evidently
longer than the Nākshatra-ghatis or the sidereal ghatis,
which are of a fixed duration. Thus a Sāvana ghati is a
little longer than a Nākshatra-ghati and what is more, a
Saura-Sāvana ghati is of a variable duration from day to
day, the variation being of course very small.

9
The concept of a month originally arose out of the
phenomenon of new-moons, for, this phenomenon alone
appeals to every lay man, when he could not sight the
Moon. The concept of an year arose originally out of
what is called a tropical year, which is the time between
two consecutive conjunctions of the Sun with an equi-
noctial point and which it is that makes the seasons recur.
Thus the primitive man must have had the concept of an
year when he once saw the mango trees blossoming and
again when he saw them blossom. This is why the Hindus
celebrate the new year's day with eating the neem flower
with unripe mangos, which goes by the phrase Nimba-
Kusuma bhakshanam or eating the neem flower. In the
Vedic times, however, the year began with the luni-Solar
month called Mārga-Śīrṣa, which brings in the new year
crops. In the Vedic sacrifices, there is thus what is called
the Āgrahāyaṇeṣṭi, where the word Āgra-hāyaṇa means
the Mārga-Śīrṣa month. The etymology of the word is
that अग्रे हायनं यस्य तत् आग्रहायणम् ie the year is ahead of this
month, which therefore is the beginning month of the year.
This is also why the ancient lexicon named Amarakośa
enumerates the months from Mārga-Śīrṣa. This is the
month when the full moon occurs when the Moon is in the
star Mṛgaśira. We have also an inkling from this that the
vernal equinoctial point was probably situated in the star
Mṛgaśira. The Veda however enumerates the stars from
the star Kritticā, and we have a statement in the Śata-
patha-Brāhmaṇa that "एता ह वै कृत्तिकाः प्राच्यै दिशो न श्च्यवन्ते
ie Behold ! these stars which go by the name the Krit-
ticas do not deflect from the east-point. As this group
of Krittikas is situated on the ecliptic, the statement that
they were rising in the east signifies that the Vernal equi-
noctial point was situated in the Kritticas". Arguing
about the situation of the vernal equinoctial point in the
so-called Vedic times, the late Lokamānya Bāla Gangā-
dhara Tilak concluded that Vedic literature must have had
its beginning about eight thousand years ago.
2

10
From the original concepts of the month and the year,
further concepts of the different kinds of month and the
year arose with the advance of astronomical knowledge.
We shall deal with these different kinds of the month and
the year in their respective contexts.
In these two verses, we have the definitions of Saura-
māna, Daiva-māna, chāndra-māna, Paitra-māna, Sāvana-
māna and Nākṣatra-māna, six of the nine mānas, ie mea-
sures of time.
Verses 21, 22, 23, 24, 25. The four yuga-pādas named
Kṛita, Tretā, Dwāpara and Kali consist of 4 × 432000,
3 × 432000, 2 × 432000 and 432000 mean solar years
respectively, the sum total of which consisting of
(4+3+2+1) × 43200 = 43,20000 mean solar years, is
calleda yuga. Each of the yuga-pādas above are inclusive
of what are called their respective Sandhyās and Sandh-
yāmsās which constitute one-twelfth of their own durations.
A Manu's duration consists of 71 yugas and 14 Manus
duration is reported to be the day-time of Brahma, whose
night is also of an equal duration.
The duration of a Manu, known as a Manvantara has
a Sandhyā-Kāla on either side, ie before and after, equal
to one Kṛita. If these are taken into account, the day-
time of Brahma amounts to one thousand yugas and it
goes by the name a Kalpa so that a complete day of
Brahma equals two Kalpas. The life-duration of Brahma
consists of one hundred years on this scale (where one
year = 360 days). This life-duration of Brahma goes by
the name Mahā-Kalpa, as reported by elders. In as much
as Time was without a beginning and will have no end
either, I do not know how many Brahmas have gone
before.
Comm. In these verses we are given what is known
as Brāhma-māna, the seventh of the nine mānas. Inciden-
tally we are also given the measures yuga-pādas, yugas,

11
Manvantaras and a Kaḷpa. Since in many ancient astro-
nomical works, the revolutions of the planets and the
planetary points like nodes, apogees or aphelia are given as
integers during the course of a yuga, the concept of a yuga
must have arisen as follows. The durations of the sidereal
revolutions of the planets and the apogee of the Moon and
its node having been ascertained by observation, a period
was calculated in which are contained integral multiples of
those durations. In other words a yuga of 4320000 mean
solar years is construed as the period in which the planets
the node and apogee of the Moon make an integral number
of revolutions with respect to the stars.
We have excluded here the aphelia and the nodes of
the planets, as we shall see later that their sidereal revolu-
tions were not based on observation but by an assumption
that those points also must be having an integral number
of revolutions during the course of a Kaḷpa, having started
at the beginning of the Hindu Zodiac ie the beginning
point of Aswini at the beginning of the Kaḷpa. On this
assumption cited, and using indeterminate analysis the
number of their sidereal revolutions were got as reported by
Bhāskarāchārya in his Commentary in the chapter Bhaga-
nādhyāya. He has given us a clue that the numbers of
sidereal revolutions of the planets including the node and
apogee of the Moon were originally determined by observa-
tions though he appeals to Āgama that those numbers were
given by Āgana, as accepted and transmitted by Brahma-
guptāchārya. Even in the Upapatthis or proofs that Bhās-
karāchārya gives regarding the numbers of sidereal revolu-
tions known as Bhagaṇas, we perceive that he consciously
commits the logical flaw known as इतरेतराश्रयदोष as we are
going to show in that context. The proofs he adduces are
indeed based upon Aryabhatāchārya's verse 48, Golapāda
namely
क्षितिरवियोगाद्दिनकृत्, रवीन्दुयोगात्प्रसाधितश्चेन्दुः
शशिताराग्रहयोगात् तथैव ताराग्रहाः सर्वे

12
and on Brahmaguptācharya's verse 12, ch. 20 namely
ज्ञातं कृत्वा मध्यं भूयोऽन्यदिने तदन्तरं भुक्तिः
त्रैराशिकेन भुक्त्या कल्पग्रहमण्डलानयनम्
Aryabhatācharya and his immediate followers Lallā-
charya and Vateswarācharya make the yuga-pādas of equal
duration. Brahmaguptācharya criticises Aryabhata for
having said so against the Canons of the Smṛtis as well as
Romaka for having ignored the concept of yugas, manvan-
taras and Kalpa, as this he deems as a heresy (Vide verses
9 & 13 ch. I).
Indeed there seem to be two schools among the ancient
Hindu Astronomers one of Brahmagupta who was followed
by Sripati, Bhaskara and a number of others and the other
of Aryabhata who was followed by Lalla, Vateswara, Bhas-
kara I, and a host of others mostly hailing from Kerala.
There is an Aryabhata who has been termed Aryabhata
II and who was the author of a book named Brihad-Arya-
bhatiyam or Mahā-Siddhanta as it is also called. M. M.
Sudhakara Dwivedi mentions in his Ganaka-Tarangani,
that this Aryabhata should have existed after the author of
Modern Surya Siddhanta. Aryabhata I, Aryabhata II, many
of the Kerala astronomers used a different nomenclature to
signify numbers, denoting them by letters. Thus one of
the distinguishing features of the Kerala school of astrono-
mers (not all of them) seems to be to use letters for numbers.
One Kalpa = 14 Manvantaras = 14 × 71 yugas + 15
Sandhis in between the Manvantaras each equal to a Krita
ie 4 Kalis = 994 yugas + 60 Kalis = 994 + 6 yugas =
1000 yugas = 4320000000 mean Solar years.
Bhaskara speaks of Sandhyas and Sandhyamsas each
of them equal to 1/12 of the yugapādas.
Thus one Kali = 432000 years = 1200 Divyābdas (gods'
years each year being equal to 360 Solar years) = 1000 +

18
100 + 100 Divyābdas since ¹/₁₂ × 1200 = 100 = measure
of the Sandhyā and Sandhyamsa times. At this rate Dwa-
para = 2000 + 200 + 200 Divyābdas and so on. However,
the measure of the Sandhyas with respect to a Manvantara
does not follow this one-twelfth rule, because a Krita is
not ¹/₁₂th of the Manvantara. Thus 1000 Divyābdas = one
Kali excluding Sandhyā and Sandhyāṁsa, whereas 1000
yugas = one Kalpa including the Sandhyas and Sandh-
yamsas.
When a yuga was conceived as a period wherein the
planets make an integral number of revolutions, it goes
without saying that they make integral numbers of revolu-
tions in a Manvantara or a Kalpa. When a Kalpa was con-
ceived as the period in which the slow-moving planetary
points aphelia and nodes also make an integral number of
revolutions, one wonders how a manvantara was conceived.
It is further peculiar why such an odd number 71 was
chosen, when it was said that 71 yugas make a Manvantara.
One also wonders why the modern Suryasiddhānta says
that after the Kalpa began, creation started only after 47400
Divyābdas whereas neither Brahmagupta nor Bhāskara
speaks of this. As a matter of fact Bhāskara mentions
later the number of years that had elapsed upto the begin-
ning of the Saka era, as equal to 1972947179, but does not
speak when the creation of planets and stars began actually.
Verse 26. Half the life-period of the present Brahma
has elapsed; some said that only eight and half years of
his life has elapsed — Let the Āgama or tradition be
whatsoever; we don't have any need of knowing it be-
cause the planetary positions have to be computed only
from the beginning of this Kalpa.
Comm. Vateswara it was that mentioned that only
eight and half years of the present Brahma had elapsed
(Vide verse 10 Madhyādhikāra ch. I Vateswara Siddhānta)
Vateswara prescribes that Ahargaṇa or the collection of

14
days has to be calculated from the birth-time of this
Brahma, but Bhāskara rightly points out that it is a waste
of labour, for, all the planets must have returned to the
Zero-point of the zodiac ie the beginning of the Star Aswini
at the beginning of this Kalpa and hence it is sufficient to
calculate only from the beginning of this Kalpa. Further
Bhāskara states that when the very planets did not exist
during the last elapsed night of Brahma, what is the fun
of calculating their positions.
There is also a tradition that there are nine Brahmas
and that the present one is the very first. This tradition
Bhāskara does not mention, because he exclaims that he
does not know how many Brahmas have gone by, Time
being without a beginning.
Verse 27. In as much as the creation started only
from the beginning of this Kalpa which is the present day-
time of Brahma, and because deluge takes place at the end
of the day-time, the question of Computing the planetary
positions arises only when the planets exist. If some (the
allusion is to Vateswara) propose to compute the planetary
positions even when the very planets did not exist, may we
salute those great people !
Comm. Not necessary.
Verse 28. Six Manus have elapsed in this Kalpa,
thereafter twentyseven yugas, as well as three yugapādas
namely Kṛita, Tretā and Dwāpara. Further 3179 years of
this fourth yugapāda namely Kali have elapsed by the end
of the Saka king (which moment was the beginning of the
Saka era). Hence in the present Kalpa ie the day-time
of this Brahma, 19729 47179 years had elapsed upto the
beginning of the Saka era.
Comm. The computation is as follows :
6 Manvantaras = 6 × 71 × 10 Kaliyugas since each
Manvantara Consists of 71 yugas and a yuga Consists of

18
ten Kaliyugas (one yuga = Krita + Tretā + Dwāpara +
Kali = 4 + 3 + 2 + 1 = 10 Kaliyugas). The Sandhis
that were there in between the Manus and in the beginning
of the first Manu are seven and each Sandhi being equal to
one Krita or four Kalis, the seven Sandhis = 7 × 4 = 28
Kaliyugas.
Further it is stated that 27 yugas had elapsed in the
present seventh Manvantara known as Vaivasvata which
are equal to 27 × 10 = 270 Kaliyugas. Further in the
present yuga, Krita, Tretā and Dwāpara had elapsed equal
to 4 + 3 + 2 = 9 Kaliyugas.
Thereafter in the present Kaliyuga 3179 years elapsed
upto the beginning of the Saka era. Thus totalling we
have 4260 + 28 + 270 + 9 kalis + 3179 years.
= 4567 kalis + 3179 years.
= 4567 × 432000 + 3179 years.
= 179 294 7179 year as mentioned.
Verse 29. The six Manus that went before the present
Vaivasvata were Swāyambhuva, Swārociṣa, Auttama
Tāmasa, Raivata and Cākshusha.
Comm. Clear—Upto this point we have seen the
Brāhmamāna, the seventh of the nine mānas.
Verse 30. The Sāmhitikas declare that a Samvatsara
is equal to the time of a mean sidereal revolution of Guru
the Jupiter (This is the Bārhaspatyamāna). The ninth
māna named the Manushyamāna is composite of the four
mānas (as detailed in verse 31).
Comm. The years of the Bārhaspatya māna are enu-
merated as Vijaya, Jaya etc which are sixty in number.
The same names are also adopted in the cāndramāna ie
the luni-Solar reckoning (In vogue in the Andhra Pradesh
and some other provinces too) only with the difference that
prabhava is taken as the starting year in which sequence
Vijaya happens to be the twenty-seventh year. Since the

10
mean sidereal revolution takes roughly 11.8 years, five
revolutions take 59 mean solar years. The sixtieth year
thereafter of the Jovian Cycle is considered as an Adhi-
Samvatsara thereof so that the luni-Solar reckoning as well
as the jovian reckoning get wedded together. Thus to get
the jovian year we have simply to add 27 to the luni, solar
year. The wedding of the two reckonings has its analogy
in the process of intercalation which weds the solar reckon-
ing with the luni-Solar only with the difference that
months of the latter reckoning are set apart as Adhika
or extra months. Just as the process of intercalation
brings in its train what is called a Kshayamāsa as per the
convention that, that month would be set apart as an
Adhikamāsa, which does not carry a Samkrānti ie entrance
of the Sun into the next Rasi, in which process there
appears a month in which there may occur two Samkrāntis
which is hence considered as a Kshayamāsa, just in a
similar way that luni-Solar year in which the Jupiter
enters the next Rasi, is supposed to be normal whereas that
year in which such an entrance does not take place is
deemed an Ādhika year and set apart while that year which
carries two entrances is deemed as a Kshya year. The pro-
cess of intercalation which weds together the solar and the
luni-solar reckonings will be elucidated further in its appro-
priate context. The word Sāmhitikas means the authors
of the works called Samhitas like the Varāha-Brihatsamhita
etc.
Verse 31. The manushya-māna or that which men
follow, adopts the year, the Ayana the Ṛtu or the season
(six in number during an year) and the yuga according to
the movement of the Sun ie according to Sauramāna the
months and the thithis according to the luni-solar reckon-
ing ie according to Chāndramāna the Vratas, upavāsas,
treatment of diseases, deliveries of ladies, the names of the
weeks all these according to the reckoning of civil days
ie according to Sāvanamāna and finally the ghatis Vighatis
etc. etc. according to the Nākshatramāna.

17
Comm. A lunation is divided into thirty tithis, which
go by the names pratipat etc. of which there are fifteen in
the brighter half of the lunation and fifteen in the dark
fortnight. The pratipat tithi is that duration of time be-
ginning from the moment of New Moon and ending when
the Moon has over taken the Sun by 12°; the second tithi
named dvitīyā begins at the end of pratipat and lasts upto
the point of time when the moon’s elongation is 24° and so
on. Thus the tithis are seen to be of unequal length in as
much as both the Sun and the Moon have unequal motion.
The mean duration of a tithi is seen to be a little less than
a civil day since a lunation has ronghly 29½ civil days.
There is a convention that the tithi which is current at a
Sun-rise will be considered to be the tithi of the whole day.
As per this convention it so happens that a tithi lasts just a
little after Sun-rise and the next tithi vanishes during the
same day so that the next but one will be taken for the next
day. This vanishing tithi is known as a Kshaya tithi which
is also called a Kshayāba. Again it so happens that a tithi
may be current at two consecutive Sun-rises beginning a
little before the Sun-rise of the first day and extending a
little after the next Sun-rise. Such a tithi is called Dina-
Traya or a tithi which touches three days. In this matter
if seems as though we have gained a tithi but ultinately as
a tithi must be less than a civil day, it so happens that on
the average there will be a Kshayāha rongbly in 64 tithis.
We shall see more about this matter subsequently.
Verse 32. Thus there are nine mānas Mānava, Divya
(or of gods), Bārhaspatya (Jovian) paitra, Nākshatra,
Saura, Chāndra, Sāvana and Brāhma. But the planetary
positions are to be computed by men by their own māna.
Comm. Not necessary.
Here ends the Adhyāya known as Kāla-māna in the
Madhyādhikāra.
2

MADHYĀDHIKĀRA — SECTION II
BHAGAṆĀDHYĀYA
Verses 1 to 6. The number of sidereal revolutions of
the Sun during a Kalpa is 4320000000. It is also the
number of those of Mercury and Venus, and those of the
Śīghrocclas of the planets Mars, Jupiter and Saturn.
The Moon makes 57753300000 sidereal revolutions
in a Kalpa, the Mars 2296828522, the Mercury's śīghroccha
17936998984, the Jupiter 364226155, the śīghroccha of
Venus 7022389492 and the Saturn 146567298. The sidereal
revolutions of the apogees of the Sun and the Moon and
those of the aphelia of Mars, Mercury, Jupiter, Venus and
Saturn in a Kalpa are respectively 480, 488105858, 292,
352, 855, 653, 41.
The retrograde sidereal revolutions of the nodes of the
orbits of Moon, Mars, Mercury, Jupiter, Venus and Saturn
are respectively 232311168, 267, 521, 63, 893, 584.
Comm. Since Mercury and Venus will be oscillating
about the Sun in their apparent motion as seen from the
earth in a long period of time, the number of sidereal re-
volutions made by the Sun is also equal to that made by
Mercury and Venus. Since, as we see in the Course of the
Spaṣṭādhikāra the Sun plays the part of what is called the
Śīghroccha of the three major planets named Mars, Jupiter
and Saturn, the number of the Sun's sidereal revolutions
is also the same as that of their śīghrocchas.
The reason why the sidereal revolutions of what are
called the śīghrocchas of Mercury and Venus are the same
as the heliocentric sidereal revolutions of Mercury and
Venus will be clarified in the spaṣṭādhikāra. The reason
also why the sidereal revolutions of the major planets are

19
the same as their heliocentric ones will also be clarified in
the same adhikāra.
The reason why we have termed the Mandocchas as
apogees in the case of the Sun and the Moon and as aphelia
with respect to the other planets is that the Sun moves
round the earth relatively while the Moon directly moves
round the earth whereas the remainning planets move
round the Sun while the Sun moves relatively round the
earth. In other words the Sun and Moon have apogees
whereas the remaining planets aphelia.
The nodes of the planets are the points of intersection
of their orbits with the ecliptic which is the apparent orbit
of the Sun. In other words the planetary orbits are
inclined to the ecliptic which means that their orbital
planes do not coincide with the ecliptic plane.
In Hindu Astronomy the Sun, the Moon and also the
two nodes of the lunar orbit which go by the names Rahu
and Ketu are also termed as grahas along with the other
five which are known as Tāra-grahas or planets resembling
stars. The etymology of the word ‘planet’ is that it
moves amongst stars; in this respect the nine Hindu grahas
also moving among the stars are eligible to have the same
appellation though it is not permitted in modern astronomy.
But the word graha has a different connotation etymologi-
cally namely गृह्णातीति वा गृह्यते अनेनेति वा ग्रह : ie that which
seizes the fates of men is known as a graha. In the course
of this work we use the word graha and planet synony-
mously so that we deem the Sun, the Moon and the nodes
of the lunar orbit also as planets.
When the ancient Hindu astronomers knew by obser-
vation that the nodes of the lunar orbit have a retrograde
motion, and could also measure their mean motion, they
extended the analogy to the nodes of the other planets also,
whose motion could not be measured during anybody's life-