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

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

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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

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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.

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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-

20
time. Hence the estimate of their mean motion by the
Hindu Astronomers naturally went wrong.
Noticing that the Mandoccha of the Moon has a
progressive motion which the Hindu astronomers could
measure correctly, they extended the analogy to the apogee
of the Sun and the aphelia of the other planets whose
motion also being very slow could not be measured during
the life-time of a man. So here also the estimate of their
mean motions of the Sun's apogee and the planetary aphelia
went wrong.
Bhaskarāchārya has given proofs as to how the sidereal
revolutions could be got but as we shall see later in the
Spaṣṭādhikāra, his proof occasionally suffers from what is
called ‘इतरेतराश्रयदोष’ ie ‘begging the question’. We shall
however construct our own proofs at that place deferring
them for the present, for, the proofs require an elucidation
which obtains in the Spaṣṭadhikāra alone.
Verse 7. The number of diurnal revolutions of the
stars in a Kalpa is 1582236450000.
Comm. In fact this number is that of the diurnal
rotations of the earth which is equal to the number of
apparent diurnal rotations of the stars, if the earth is
deemed as fixed. The only Hindu Astronomer who made
bold to say that the earth is rotating and not the stars
came under criticism by Brahmagupta and the latter Hindu
astronomers. Aryabhata said अनुलोमगतिः नौस्थः पश्यत्यचलं
विलोमगं यद्वत्, अचलानि भानि तद्वत् समपश्चिमगानि लङ्कायाम् ” ie
Even as a man stationed on a moving boat perceives that
the trees etc on the banks of the canal, river or lake to be
moving in the opposite direction supposing himself statio-
nary, so also men stationed on the surface of the earth
(which is like a moving boat) perceive the actually statio-
nary stars to be moving directly from east to west at
Lanka ie the equator ”.

Why none of the latter Hindu astronomers, though
many of them could intuit this simple phenomenon, boldly
came out asserting this, is rather mysterious. Even today
there is such an irrational orthodox type of scholars who
are not in touch with modern astronomy, holding the view
that the earth does not move.
In Hindu Astronomy it was postulated that there is
what is called the pravaha wind, which effects moving of
the entire stellar universe along with the planets from
east to West. It is a very simple matter to visualize
earths' rotation, instead of supposing that the entire stellar
Universe is being driven round the earth. The absurdity
in this latter supposition might not have been clear to the
orthodox type of the Hindu Astronomers, for, they could
not measure the dimensions of the giant stars and super-
giants, which are everyone of them mighty Suns. Though,
however, the dimensions of the Sun were known to them
to be far greater than those of the earth, it did not occur to
them why a mighty Sun should go round a pigmy earth.
Or even if it occured to astronomers like Bhaskara, they
dared not to go against the puranic tradition.
Since the stars do not move among themselves while
partaking this diurnal motion, the entire starry skies are
obliged to go round the earth as a rigid structure, if we
suppose that the earth is not rotating about, herself. This
kind of supposition is just like a revolving person, revol-
ving about himself and claiming that the entire Universe is
revolving round him and not he about himself.
Bhaskara gives the proof of getting this number of
diurnal rotations of the stars during an year in the verses
5-7 of Madhyagati Vāsanā, समं भसूर्यावुदितौ etc. as follows.
Suppose a star and the Sun rise together today. Tomorrow
the star will have arisen earlier than the Sun who will
have moved towards the east of the star by his own
(apparent) daily motion. So tomorrow's sun-rise will get

22
belated by the duration of time that the arc of the ecliptic
covered by the Sun's today's motion takes to rise. This
duration of time is variable on two counts; first by the
variable motion of the Sun and second by the obliquity of
the ecliptic on account of which even equal arcs of the
ecliptic will not rise in equal times. In other words the
duration of time between two consecutive Sun-rises will
not be the same. This duration of a particular day can
roughly be calculated by the rule of three as follows. Let
the Sun be in a particular Rasi, the rising time T of which
could be computed; let the Sun cover an arc of x° in that
Rasi on that day. Then the time taken by that arc to rise
is xT / 30, where a Rasi consists of 30°. This time computed
in Sidereal measure added to 60 Sideral ghatis is equal to
the length of the day. During the course of an year ie the
time taken by the Sun to move round the ecliptic starting
from the Zero-point of the zodiac and again returning to
the same point, the Sun will have made one revolution less
than the stars. Thus if ‘ R ’ the number of diurnal revolu-
tions of the Sun (where R will be not an integer) during an
year or what is the same the number of Sāvana or civil
days during an year, they are equal to R + 1 sideral days.
Hence the number of sideral days in a kalpa will be equal
to the number of civil days in a kalpa together with the
number 4320000.000 which is the number of revolutions
made by the Sun relative to the stars.
In this context we are to know the number of civil
days in a kalpa. The proof given by Bhaskara in Gaṇita-
dhyāya under verses 1-6 in Bhagaṇopapatthi is as follows.
Draw a circle on a horizontal plane and place a vertical
pole called gnomon at the centre of the circle. Observe the
point of intersection of the gnomon's shadow with the
circumference of the circle at Sun-rise on a day in the
Uttarāyaṇa ie during the course of the Sun’s north-word
journey, just at the time when his rising point is very near
the east point and also to the south thereof. Then from

that day go on counting the number of days, which will
be 365 when the Sun again rises very nearly at the same
point and just to the south of the east point. It will be
found that the Sun will rise the next day just a little to
the north of the east point, which means that the Sun has
taken 365 days and a fractional part of a day to complete
his revolution round the stars. Having noted the two
points of intersection of the gnomonic shadow with the
circle, on those two consecutive days when the Sun
happens to rise just a little to the south and then on the
next day just a little north of the east point, and having
measured the arcs in minutes between those points of
intersection and the western point of the horizontal circle
(western because the gnomonic shadow of the rising Sun
is cast towards west) then the following rule of three is
to be applied. If during 60 ghatis of the day, the sum of
the arcs in minutes is covered, what will be the time
taken by the shadow to traverse the arc between the west
point and the northern point of intersection. This added
to 365, gives the number of civil days in an year. Here
it will be noted that this year is tropical because rising in
the east signifies the Sun's position at an equinox.
Verse 8. The number of solar days in a kalpa is
equal to 1555200000000 and of the lunar days or tithis is
1602999000000.
Comm. The solar days here cited are counted at the
rate of 360 per year; and the lunar days at the rate of 30
per lunation. Tithi is defined as mentioned by us under
verse 31 of the previous section. Since there are thirty
tithis in a lunation, their enumeration is quite alright;
but there seems to be a little oddity in saying that there
are 360 solar days in an year. This kind of a solar day
is a little longer than a civil day and does not correspond
to any particular motion of the Sun, say for example the
time taken by the Sun to move a degree along the ecliptic.
The definition of solar days pertains only to a stipulation

24
that 360 solar days constitute a solar year and no defini-
tion is given for a single solar day. This definition of
solar days, though apparently artificial, has some signi-
ficance, namely that the difference of the solar and lunar
days defined above constitutes 159330000000 Adhikamāsas
in a kalpa at the rate of 30 tithis per month. In other
words the difference between the solar and lunar days, is
the number of tithis that the luni-solar reckoning gains
over the solar. This topic will be dealt with later.
Verse 9. The number of civil days in a kalpa is
equal to 1577916450000; the number of the diurnal revolu-
tions of the stars minus the number of sidereal revolutions
of any particular planet constitute the days of that parti-
cular planet with respect to the earth.
Comm. The civil days in a kalpa are evidently the
number of Sun-rises. These are as mentioned before the
difference of the number of diurnal revolutions of the stars
and the number of the sidereal revolutions of the Sun.
By analogy, the number of the days of a particular planet
with respect to the earth or what is the same the number
of risings of that planet in a kalpa as seen from the earth,
is the difference of the number of the diurnal revolutions
of the stars and the number of the sidereal revolutions of
that planet in a kalpa. Thus we have Saura-Ku-dināni,
Chāndra-Ku-dināni, Bhauma-Ku-dināni etc, where the
word ‘ Ku ’ means the earth. The Saura-Ku-dināni are
the civil days defined before. It will be noted that the
Chāndra-Ku-dināni are not the lunar days.
Verse 10. The number of Adhikamāsās or intercalary
months in a kalpa is equal to 1593300000 and the number
of Dina-Kshayas is 25082550000.
Comm. A luni-solar year as per Cāndramāna ie the
luni-solar reckoning consists of twelve lunations; as such
its length falls short of that of the solar year by 11 days
3 ghatis, 52 Vighatis and 30 Sukshmaghatis. If both the

25 .

solar and luni-solar years begin simultaneously this year,
by the end of the luni-solar year, it will have gained over
the solar year the above-mentioned days. This difference
which goes by the name Adhimasa-Sesha or Suddhi
accrues to the length of a lunation in about 32½ solar
months. Unless this accruing difference is set apart by some
device, and the beginnings of the two years again brought
together, the luni-Solar year looses its significance of an
year, for, it does not accord with seasons. It being a con-
vention that an year should begin with the spring, if the
luni-Solar year also is to begin with the spring, it is to be
wedded to the solar year by some device. The device
adopted in this behalf was to leave out a month in the luni-
Solar reckoning as soon as it could be seen that a month
has been gained by this reckoning over the solar. This
knowledge is had from the following fact. The zodiac is
divided into twelve equal portions called Rasi's beginning
from the first point of the Hindu Zodiac, ie. from the first
point of the asterism division called Aswini. The Sun
traverses each of these Rasis in one Solar month, and on
account of the unequal motion of the Sun, these solar
months are of unequal length. The entrance of the Sun
from one Rasi into another is called a Samkrānti, and the
moments of Samkrāntis are held to be holy for religious pur-
poses. The solar month being a little longer than a luna-
tion, normally a Samkrānti occurs in a lunation. But it
so happens that in a particular lunar month this Samkrānti
might not occur. The Suddhi which is the time between
the moment of New Moon and the subsequent Samkrānti
and which is therefore the time gained by the luni-Solar
reckoning over the Solar, having accrued to a lunation, it
is an indication that the luni-Solar reckoning has gained a
lunation over the Solar. That lunation not carrying a
Samkrānti is termed an Adhikamasa and is left out. That
it is left out is connoted by the word Adhika which means
extra, as well as by the convention that no auspicious
celebrations like a marriage etc. should not take place
4

26
during that month. The next lunar month is termed as
the Nija-māsa or the month which is the true. In this
convention however, there may occur two Saṁkrāntis in
one particular lunar month. For this to happen two cri-
teria are to be satisfied namely that, that particular lunar
month must be greater than the concurrent solar month
and secondly one of the two Saṁkrāntis must occur imme-
diately after a New-moon so as to permit the second Saṁk-
rānti to occur just before the lapse of the lunar month.
The Sun coming to his perigee roughly about the lunar
month Mārgasira, and as such having the quickest motion
at that point, he covers the length of the Rasi of 30° within
the shortest span of time making that Solar month shorter
than the corresponding lunar month. Since the months
Kārtica and pausha, one before and the other after Mārga-
sira, also being thus longer than the corresponding solar
months, two Sankrāntis could occur, if at all, in these three
lunar months. If they occur, that lunar month is termed
a Kshayamāsa. Since two lunar months are to correspond
to two Saṁkrāntis, convention has it that two lunar months
lapse simultaneously during that lunation, the previous
lunation running during the forenoon and the subsequent
lunation during the afternoon. Thus a Kshaya māsa is also
called a yugalībhūta māsa or a twin māsa, which therefore
makes one lunation virtually vanish. On account of this
vanishing’, it is called a Kshayamāsa. After what an inter-
val of time this Kshaya māsa occurs how a month precedes
it having no Saṁkrānti and how again a lunation follows
it without a Saṁkrānti will be seen in a subsequent
context. This convention pertaining to the institution
of an Adhikamāsa goes by the name ‘Inter-calation’.
It was mentioned before that on an average an
Adhikamāsa occurs once in 32½ solar months roughly–Also,
Adhikamāsas are the excess of lunations over the solar
months in a given period. There being 51840000000 solar
months in a Kalpa and 53433300000 lunations the number
of Adhikamāsas is therefore 1593300000 as mentioned,

27
Kshayāhas were explained before and their number in
a Kalpa is the difference of the civil days and the tithis.
Verses 11, 12. The numbr of solar months in a Kalpa
is 51840000000; the number of lunar months is
53433300000. The number of solar months being subtracted
from the number of lunations, we have the number of
Adhikamāsas—The number of solar days together with the
days of Adhika months are equal to the lunar days or the
tithis; or again the lunar days minus the Kshayāhas are
equal to the number of civil days or the reverse will be
had by a reverse process.
Comm. Already explained.
Verse 13. The excess of the sidereal revolutions of the
moon over the number of the Sun’s sidereal revolutions is
equal to the number of chāndramāsas or lunations.
Or again the excess of the sum of the sidereal revolu-
tions of the moon and the tithis over the sum of the luna-
tions and the diurnal revolutions of the stars is equal to the
number of Kshayāhas.
Comm. The first part is clear. Regarding the second,
let the number of lunations be x and the number of the
sidereal revolutions of the moon be y. Then y —x =z, the
number of the sidereal revolutions of the Sun because
y — z = x from the first part above. If now, the number
of the diurnal revolutions of the stars be t, then t — z = t —
(y — x) = t + x — y = civil days. Subtracting these civil
days from ‘U’ the number of Tithis, we have the number
of Kshayāhas namely U — (t + x — y) = (U + y) —
(t + x) which accords with the statement.
Verse 14. The number of Adhikamāsas is equal to the
excess of the number of sidereal revolutions of the Moon
over thirteen times the number of sidereal revolutions of
the Sun.