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

211
(3) Taking the case of the Moon, the max correct-
ion amounts to (790 × 120) / 270 = 5' — 51''.
Verses 66, 67. The computation of Tithi, Nakṣatra
and Yoga.
The elongation of the Moon ie. the excess of the
Moon's longitude over that of the Sun in degrees. (In
case the former is smaller, add 360° to it and subtract
Sun's longitude) being divided by 12 and 6, the quotients
represent the elapsed tithis and Karaṇas. If the elapsed
Karanas are K, count K — 1 beginning from Bava to
get the current Karaṇa and count from Śakuni to get
the current one beginning from the mid-moment of
Kṛṣṇachaturdasī. Take again the planetary position or
that of the Moon in particular as well as the sum of
the longitudes of the Sun and the Moon both expressed
in minutes of arc and divided by 800. The first quotient
gives the elapsed stars ie. the Stars covered by the planet
or the Moon; whereas the second quotient gives the
elapsed Karaṇas. Then take the remainders in seconds
and divide by the respective daily motions in minutes i.e.
in the case of the planets or the Moon divide by their
respective motions and in the case of yogas divide by the
sum of the motions of the Sun and the Moon. Then the
results give the times in nādīs as to how much the next
nakṣatra or yoga have elapsed. If it be required to find
as to how long the next nakṣatra or yoga will last,
substract the remainder from 800, and divide by the daily
motions in minutes as mentioned above. The results
give as to how many nādīs beginning from the morning
concerned, the next nakṣatra or yoga will last.
Comm. (1) A lunation i.e. the time from the moment
of New Moon to the next New Moon is divided into
30 parts called Tithis. Their names are pratipat, Dwitiyā
etc. upto the 15th purnima or full Moon and again

212
pratipat, Dwitīyā etc. upto the 30th i.e. Amāvāsyā or New
Moon. Thus pratipat starts when ☾ = ☉ i.e. when the
longitude of the Moon is equal to that of the Sun i.e.
from the moment of New Moon when the Moon is in
conjunction with the Sun. अमा सह = वसतः सूर्य्याचन्द्रमसावस्या
मित्यमावस्या i.e. Amāvāsyā is that point of time when
the Sun and the Moon are together. Pratipat lasts
till ☾ = ☉ + 12°; then Dwitīyā begins and lasts till
☾ = ☉ + 24; Thus purnima begins from the moment
when ☾ = ☉ + 168 and lasts till ☾ = ☉ + 180° and
Amāvasyā begins when ☾ = ☉ + 348 and lasts till ☾ is
again equal to ☉ Thus a tithi is the time that is taken
by the Moon to overcome the Sun by 12° beginning from
the moment of New Moon. In other words the tithi is a
measure of the phase of the Moon with a particular
convention.
(2) In modern astronomy the phase of the Moon or
that of a planet is measured by the formula (1 + cos EPS) / 2
where P is the planet or the Moon. Since ES is almost
Fig. 29
parallel to PS (Fig. 29) ÊPS may be taken to be nearly
equal to P̂ES which is the elogation of the planet or the
Moon so that phase = (1 - cos EPS) / 2 approximately. So

213
in modern astronomy also the phase is measured through
the elongation. But the maximum phase in modern
astronomy is taken to be unity as could be seen from the
formula, for, when EPŜ = 180°, the phase = 1. Thus
the phase multiplied by 15 gives approximately the tithi.
We shall have occasion to deal with this topic later.
(3) Suppose ☾ − ☉ = E°. The number of the
tithis elapsed is equal to the quotient in E / 12. Take the
remainder r°. It means during the current tithi which
has a duration of 12°, r° have elapsed. Let the daily
motions of the Moon and the Sun be m and s so that the
Moon overtakes the Sun by m − s (expressed in minutes
for convenience) during 60 nādīs. The elapsed time of
the current tithi in nādīs is given by (r × 60 × 60) / (m − s). But
r × 60 × 60 = Seconds of arc of the remainder so that
it is said “गतैष्यविलिप्तिकाः”. If it be required to find how
many nādīs the next tithi lasts, (12−r)° or (12−r) × 60′
is to be gained by the Moon over the Sun. So the time in
Nādīs = ((12 − r) × 60 × 60) / (m − s) = एष्यविलिप्तिकाः / (m − s) .
(4) A lunation is again divided into 60 Karaṇas
and so 6° of increase in elongation correspond to a Karaṇa.
These Karaṇas are eleven in number, out of which 4 are
fixed to occur during the latter half of Kṛṣṇa Chautùrdasī,
during the two halves of Amāvāsyā and the first half of
the Sukla pratipat. Their names are Sakuni, Chatuṣpāt,
Nāgava, and Kimstughna. So there remain 56 halves of
tithis during the lunation during which the remaining
seven Karaṇas, Known as Bava, Bālava etc. rotate eight
times. Thus the first half of the duration of any tithi is
covered by one Karaṇa and the latter by another. The
computations of Karaṇas proceeds as with respect to tithis

214
but dividing the elongation by 6, we are asked to count
K — 1, (where K is the quotient) beginning with Bava
because the first half of Sukla pratipat is covered by the
fixed Karaṇa Kimstughna. The computation of the elapsed
as well as the remaining nādīkas of a particular Karaṇa,
it proceeds on the same lines as that of a tithi.
(5) The Nakṣatra in which a planet or the Moon
is situated is then found. The zodiac is divided into 27
equal divisions beginning with its zero point and each
division is named after the brilliant star of that division.
Those stars are Aświnī, Bharaṇī etc. The star occupied
by the Moon has a special significance in the Hindu
calendar, it being spoken of that every day is presided over
by a nakṣatra. This happens so because the Moon’s
sidereal period is approximately 27 days.
(6) To compute the Nakṣatra in which the planet
or Moon is, take its longitude in minutes of arc λ; divide
by 800, since each star division consists of 360° / 27 = (360 × 60) / 27
minutes = 800'. The quotient gives the elapsed nakṣa-
tras. To get the elapsed nādikās of the current nakṣatra
or the remaining, let the remainder be r'. Then the
proportion is ‘If during the day of 60 nādīs, the planet
or the Moon goes p minutes what time does it take to
cover r' or 800 — r'’. The answer is
(r' × 60) / p' or ((800 — r') × 60) / p' = गतैष्यविलिप्तिकाः / p' .
(7) Regarding yogas, let the longitudes of the Sun
and the Moon be ☉ and λ in minutes of arc; let their daily
motions be s and m in minutes. If the sum of the longi-
tudes is 800', we say the first yoga named Viṣkambha is
over, if the sum is 1600', the second Yoga, named prīti
has elapsed. Thus going on if the sum is 360°, we say the
27 yogas have elapsed and the first again begins. Since
the sum of the daily motions of the Sun and the Moon

215
are on the average 59'—8''+790'—35''=850', to cover
one Yoga, they take roughly one day since the duration
of a yoga is of 800'. To get the number of elapsed yogas,
divide (⊙+M)' where M is the longitude of the Moon by
(s+m). To get the elapsed nādīs which are given by the
remainder r' or the remaining nādīs of the current yoga
which are given by 800; the proportion is ‘If by s+m
gain in the sum of the longitudes we have 60 nādīs, what
time is indicated by r' or 800—r'?’. The answer is
(r' × 60)/(m + s) or ((800 — r)' × 60)/(m + s) = गतैष्यविलिप्तिकाः/(m + s) .
By the word गतैष्यविलिप्तिकाः is meant therefore the number
of seconds of arc as many as the remainder r or (800—r)
is in minutes or what is the same the remainder or 800—r
converted into seconds.
(8) The tithi, karaṇa, nakṣatra of the Moon and
yoga constitute four of the Angas of the panchānga or the
Hindu Calendar the fifth being the week-day. All these
five are supposed to have their effects good or bad on living
beings.
Verses 68, 69. The correction what is called Nata-
karma.
The Zenith distances of the Sun and the Moon at the
end of Purṇimā or Amāvāsyā at the time of lunar or solar
eclipse, being expressed in nādīs, are multiplied by 6 to get
the degrees. Let their H sine be got from the short table
of H sines. Multiply it by the equations of centre of the
Sun and the Moon. Divide by 4920, and 4361 respectively.
If the Sun be in the Eastern hemisphere, let the result
pertaining to the Sun be subtracted from his position ; if
in the Western, let it be added to his position. If the
Moon be in the Eastern hemisphere and if his equation
of centre be negative let the result be added to his position;
if the equation of centre be positive, let the result be sub-
tracted from his position in either of the hemispheres.

216
Again from these positions, let the tithi be computed and
again let the above process be carried out until a constant
time is arrived at for conjunction or opposition.
Comm. (1) The above procedure is accepted by
Bhaskara as Āgama enunciated by Brahmagupta and
reiterated by Chaturveda as giving results that accorded
with observation. We shall see that the correction known
in modern astronomy as 'correction due to astronomical
refraction' is indicated here, though it was not stated
explicitly.
In fact what was stated by Brahmagupta was that
the periphery of the Manda epicycle given as 13⅔° for the
Sun and as 31-36 for the Moon hold good only on the
meridian but the periphery of the Sun is to be increased
or decreased by 20' according as the equation of centre is
negative and the Sun is the Eastern equatorial horizon,
or western equatorial horizon. If the equation of centre
be positive the reverse correction is to be effected in
the periphery i.e. for negative equation of centre.
Periphery on the Eastern equatorial horizon = 14°-0
On the meridian = 13°-40'
On the Western equatorial horizon = 13°-20'
For positive equation of centre
Periphery on the Eastern unmandala i.e.
equatorial horizon = 13°-20'
On the meridian = 13°-40'
On the Western unmandala = 14°-0
In the case of the Moon, for negative equation of
centre.
On the Eastern unmandala = 30 - 44
On the meridian = 31 - 36
On the Western unmandala = 32 - 28

."
"Include all Sanskrit shlokas, sutras, mathematical derivations, tables, numerical examples, Hindi translation, commentary, and footnotes. Prefix each line with line marker like , ."
Let's represent the fraction cleanly using standard text notation, e.g., 1/3 × 1/120 = H sin Z / 360 or similar clean text.
Wait, let's see how lines correspond to the printed page.
Let's trace every printed line:

Line 1: 217 (page number at the top)
Line 2: For positive Equation of centre.
Line 3: On the Eastern unmandala = 30 - 44
Line 4: On the meridian = 31 - 36
Line 5: On the Western unmandala = 30 - 44
Line 6: In between the meridian and the unmandala, pro-
Line 7: portion is to be used. If by H sine Z equal to R, there
Line 8: is a difference of 20' in the periphery of the Sun, what
Line 9: will it be for an arbitrary H sin Z? The result is
Line 10: `H sin Z × 1/3 × 1/120

218
(3) Consider for the Sun first for negative
equation of centre.
(a) On the east, the negative equation being
increased, elevation is effected.
(b) On the west, the negative equation of centre
being rendered less, elevation is effected.
(c) For positive equation of centre, on the east it
being lessened, again elevation is effected.
(d) For +ve equation of centre on the Western
side it being increased again it is elevated.
Thus with respect to the Sun, the stipulated correction
effects elevation in all the cases which accords with the
effect of the modern refraction.
Then let us consider the case of the Moon.
(e) In the case of negative equation of centre, it
being lessened in the East, the effect is
depression.
(f) And being increased in the west, the effect
is depression.
(g) In the case of positive equation of centre, it
being reduced in the east, elevation is
effected.
(h) And in the west, it being reduced, elevation
is effected.
Thus in the case of the Moon, the phenomenon is
recorded as depression in the case of negative equation of
centre. Out of these two cases again the equation of
centre being lessened is truly desirable as the Hindu
equation of centre is in excess of the true value. So it
need not be interpreted as depression. Regarding the
other case, the error might have been due to the fact that
a lesser parallax being taken, whose effect is to depress

219
the celestial body, depression might have been noticed
during the course of an eclipse wherein conjunction or
opposition had to be belated; or again the greater equation
of centre as was postulated for the Sun, than what it
should be when his equation of centre was negative might
have depressed the Sun, so that the Moon had to be
depressed to arrive at the correct moment of conjunction
or opposition. Thus this correction of Natakarma which
was accepted by Bhāskara on the reported Āgama of
Brahmagupta and also on the right endorsement of
Chaturveda, must have been in fact no other than the
effect of the phenomenon of astronomical refraction, and
what further strengthens this observation is the prescri-
ption of H sin Z which is proportional tan Z. Also the
modern formula being A tan Z where A = 58″ approxi-
mately, when Z is sufficiently large, the magnitudes given
by Brahmagupta are of the same order as that of the
moderns. This correction of Natakarma really reflects
much credit on the ancient Hindu observations.
Verse 70. Computing the planetary position for a
given moment.
The daily motion of the planet being multiplied by
the time that has elapsed or that is to elapse at which
the planetary position is to be found, and divided by 60,
and the result being substracted from or added to the
planetary position found, will render the position hold
good for the moment in question. The Sun and the Moon
will become by this process of what is called तात्कालिकीकरण
equal up to minutes for the moment of conjunction or
opposition. For opposition only the Rāśis differ whereas
the degrees, minutes and seconds in their positions will
be equal whereas for opposition, the positions are equal
in all respects ie. Rāśis, degrees, minutes and seconds
too.
Comm. The meaning is clear.

220
Verses 71 to 75. Obtaining what are called Sukṣma-
nakṣatras.
The computation of the nakṣatras done as pres-
cribed before, is only approximate. Now I shall give
the method of obtaining what are called Sukṣma-
nakṣatras as prescribed by the Ṛiṣis that are required to
note auspicious occasions regarding marriages, journeys
etc. people who knew about it, told that the six stars
Viśākha, Punarvasu, Rohiṇī, and the three Uttaras or
Uttaraphālgunī, Uttarāṣādhā, Uttarabhadrā have the
duration of one and half stars ie. 3/2 × 790′ — 35″ =
1185′ — 52″. The six stars Āslesha, Ārdrā, Swāti,
Bharaṇī, Jyeshṭhā and Śatabhishak have half the duration
of a star ie. 395′ — 17″. The remaining 15 alone have
one nakṣatra duration ie. 790′ — 35″. A star's duration
is the mean daily motion of the Moon ie. 790′ — 35″. The
sum total of all the above 27 stars being subtracted from
360°, give the duration of the star what is called Abhijit
which occurs after Uttarāṣādha and before Śravaṇa. To
obtain the star in which a planet is situated, convert its
longitude in minutes of arc and substract the durations
of the stars from Aświni as many as could be substracted .
The number of stars whose durations are thus substracted
are deemed to have elapsed. The remainder is called the
gata or elapsed portion of the current star and the
difference of this gata and the duration of the current
nakṣatra is called the Ēṣya, ie. unelapsed portion. To
obtain the elapsed time or the unelapsed time of the
current star the gata or the ēṣya is to be multiplied by
60, and divided by the daily motion of the planet
concerned the result being in nādīs.
Comm. One line in the verse 72 is evidently missing
which should name Rohiṇī and the three Uttaras. We
are able to know them from Bhāskara's commentary
as well as from Brahmagupta and Śrīpati. In the course
of the commentary Bhāskara reiterates what was stated

221
by Brahmagupta, that Ṛiṣis like Puliśa, Vasiṣṭha and
Garga spoke about these Sukṣmanakṣatras. The
duration of Abhijit calculated as directed is 254' — 18''.
The computation as directed is easy for understanding.
The reason for the durations indicated is not clear but
is to be taken as based on Astrology.
Verses 76, 77. The duration of the planets’ transit
into new Rāśis and the duration of the interval between
successive stars, tithis, Karaṇas and yogas.
The disc of the planet multiplied by 60, and divided
by its daily motion gives the nāḍīs of transit of the
planet from Rāśi to Rāśi. This duration is considered
to be holy for performing Vedic rites. It is the holiest
with respect to the Sun’s transit in particular. A planet
in its transit gives partly holy results not so much as
the Sun, depending upon the nature of the previous and
succeeding Rāśis. The duration of Sandhi for tithis is
got by dividing the disc of the Moon expressed in seconds
by the difference of the daily motions of the Moon and the
Sun; so also with respect to Karaṇas. The Sandhi
between two Nakṣatras is obtained by the same measure
of the disc of the Moon expressed in seconds of arc being
divided by the Moon’s daily motion. The Sandhi between
two yogas is got by dividing the same numerator by the
sum of the daily motions of the Sun and the Moon.
Comm. (1) The Sandhi is the period that elapses
during the transit of the disc concerned between the Rāśis
and nakṣatra divisions. With respect to tithi, Karaṇa
and yoga, the divisions are imaginary not being seen in
the Sky and the disc concerned is that of the Moon, and
not that of the Sun though the Sun’s motion is also taken
into account. We say a transit from a division to another
is current so long as the disc lies partly in the previous
and partly in the latter. So the Sandhi begins when the
disc touches the next division and ends when its hind part

222
touches the previous division. In other words it is the
interval between the first contact of the disc with the
succeeding division and the last contact with the previous
division.
(2) In the case of the Sun the duration of the Sans-
krānti is equal to (32' ½) / (59 — 2) of a day taking the maximum
magnitude of the disc and the minimum daily motion
approximately = 65 / (2 × 57) × 60 nāḍīs = 650 / 19 = 34 — 12
nāḍīs approximately or more accurately (32 ²¹/₄₀) / (56 ¹¹/₁₂)
= 1301 / 40 × 12 / 683 = 3903 / 6830 of a day
= 3903 / 683 × 6 nāḍīs = 23418 / 683 = 34 — 17 nāḍikās.
Taking liberal boundaries, the वृद्धकारिका or the elders
saying is विंशतिः पूर्वे, विंशतिः परे ie. 40 nāḍikās on the whole.
Here ends the Spaṣṭāshikāra.

ॐ
THE TRIPRAŚNĀDHIKĀRA
Verse 1. The purport of this chapter.
Pandits say that this is the science of time in as
much as, herein there is described the method of knowing
the direction and the point of space (where a celestial body
is situated) given the time. Hence I expound that
chapter, which gives that knowledge and which abounds in
very important statements, which forms the quintessence
of the science of astronomy.
Comm. This chapter is called Tripraśnādhikāra,
since this deals with the three questions pertaining to the
direction and the point of space of a celestial body for
a given time ie. dealing with Deśa, Dik and Kāla. In
this chapter we come across the Hindu methods of
spherical trigonometry, and gnomonics or Śaṅkuvedha
or observations with the help of a gnomon. Also we find
herein a usage of what is called ‘Golayantra’ or the
armillary sphere, which helped the Hindu astronomers to
solve all diurnal problems. We find herein Bhāskara
excelling himself. This chapter abounds in a good number
of technical terms and without a knowledge of this
chapter, no one could call himself a Hindu astronomer.
Verses 2-4. To compute what is called lagna given
the time.
The lagna or the ascendant as it is called or the point
of intersection of the ecliptic with the horizon at a given
point of time is obtained as follows. Obtain the Sāyana
longitude of the Sun at the point of time at which it is
required to find the lagna. Supposing the Sun is in the
rth degree of a particular Rāśi, the number of asus which

224
give the rising time of the arc of (30 — r)° of that Rāśi
are called the Bhogyāsus, or the asus which are taken by
the remainder of the Rāśi to rise at the place. They are
equal to. ((30 — r) × T) / 30 where T gives the asus of the
rising time of that Rāśi. The Bhuktāsus on the other
hand are the asus which pertain to the rising time of the
arc of the first r° of that Rāśis, which are therefore
equal to (r × 𝒯) / 30 ; from the given time substract the
Bhogyāsus formulated above; then substract also the
rising times of as many subsequent Rāśis as could be
substracted. Let 'R' be the remainder of the time
given. If t be the rising time in asus of the next Rāśi,
(R × 30°) / t gives the number of degrees by which the lagna
has advanced in the next Rāśi. These degrees added to
the previous Rāśis beginning from ♈ the equinoctial point
give the Sāyana or the modern longitude of the lagna.
From this Sāyana longitude if we substract the Ayanamsa,
we have the Nirayana or the Hindu longitude measured
from the Hindu Zero-point of the ecliptic. If, however,
the given time after Sun-rise expressed in asus say 'a'
falls short of the Bhogyāsus defined above, then, (a × 30) / 𝒯
where 𝒯 is the rising time in asus of the Rāśi in which
the Sun is situated, added to the longitude of the Sun,
gives the Sāyana longitude of the lagna.
Comm. The substance of these verses, though appears
to be simple, yet is complicated which can be better under-
stood with the help of a figure (Ref. fig. 30).
Let SEN be the horizon, ♈ER the celestial equator
and ♈AL the ecliptic where L is the point of lagna.
Required to find ♈L the Sāyana longitude of L from
which if Ayanamsa be substracted we get the Hindu

225
longitude. Let rA, AB, BC, CD, DF be the successive
Sāyana Rāśis called Sāyana Meṣa, Sāyana Vṛṣabha etc.
(The Nirayana Rāśis also starting
from the Hindu zero-point are called
Nirayana Meṣa, Nirayana Vṛṣabha
etc. and if in Hindu Astronomy
we use the words simply as Meṣa,
Vṛṣabha etc. especially in panch-
āngas ie. the Hindu almanacs, we
have to construe them as belong-
ing to the Nirayana or the Hindu
system whose zero-point is called
Fig. 30
the first point of the constellation Aswini and not r).
rL = rD + DL = an integral number of Rāśis, say, ‘n’
of them ie. n × 30° + DL, where DL is the arc of the
Rāśi carrying the Lagna L. The question then resolves
itself into knowing how many Rāśis precede ‘D’ from r
and what the measure of DL is. The data are (1) the
Nirayana longitude of the Sun as computed by the
methods of Hindu astronomy (2) The Ayanamsa of the
year ie. ‘ro’ where o is the Hindu zero-point of the ecliptic.
(3) The time after Sun-rise at the place, given in Sāvana
units at which we are required to find the Lagna.
Finding the Lagna at a given point of time at a
given place is not only necessary in astronomy but its
importance is more in what is called horary astrology
or Muhūrta Sāstra, whose purpose is to fix an auspicious
moment for the performance of marriages etc. as well as
in astrology in casting a horoscope.
The finding of the rising times of the various Rāśis
at the equator, as well as at a given place was dealt with
in the previous chapter. Those rising times are found in
sidereal units, a sidereal day being divided into 60 nādis,
or 60 × 60 Vinādis or 60 × 60 × 6 asus. These units
are of constant magnitude since a sidereal day, which is
29

226
the period of diurnal rotation of the earth is of constant
magnitude. In the data given above, the time is normally
given in Sāvana units, ie. mean solar units. A Sāvana
day is of 60 × 60 × 6 + 59 asus, which is greater than a
sidereal day by 59 asus because the mean Sun advance by
59' — 8'' per day among stars and as such the Sun-rise the
next day is belated by 59 asus approximately. The given
time in Sāvana units could be converted into sidereal
units as per the approximate ratio 21600 : 21660 which
means for every Sāvana nādi we have to add one asu or
for every hour four seconds. Then the procedure of finding
the Lagna will be a little different from what it would
be if we proceed with the Sāvana units. The complex-
ity mentioned before, arises out of the Sāvana units,
and this has been explained by Bhāskara in Golādhyāya
under the title तात्कालिकीकरणवासना in the beginning of the
chapter called त्रिप्रश्नवासना. Now the procedure will be
explained. Let (Fig. 30) S' be the position of the Sun
at the Sun-rise and S his position at the time at which
the Lagna is to be found so thst the Sun has advanced
by S'S ie. which measured in minutes of arc is called gati-
kalās. In a mean solar day these gatis-kalas would be 59,
and in the given time after Sun-rise they will be
proportional. The time given after Sun-rise pertains
to the rising time of the arc S'L which we have in sidereal
units. We shall first find the Lagna using sidereal units
by measuring S'L, so that we may latter understand
Bhāskara's reasoning for his stipulation of तात्कालिकीकरण
on which basis he finds the Lagna with the Sāvana units
taking the position of the Sun at S instead of S'.
We know the position of the sun at sun-rise ie. S', so
that the rising time of As' ie. the previous arc in the Rāsi
in which the Sun is situated is given by Bhuktasus defined
above and the rising time of S'B the remaining arc
of the Rāsi is given by Bhogyāsus. Subtract from the
given time converted into sidereal units if they are not

, CD.->BC, CD.- no periods between B and C! In line 10:s'B, BC, CD and T the rising time of the Rāsi ... Wait, what is that symbol? Wait, look at&c.again: &has a loop at the top, a loop at the bottom, and a stroke to the right. Thenc. Look at the image: it is literally&c.! Wait, let's look at: the Rāsi &c. Wait, does it saythe Rāsi &c.? Wait, "T the rising time of the Rāsi &c." -> why "&c."? Because T is the rising time of the Rāśi in which DL lies! (or the Rāśi containing DL, etc.?) Wait, look at formula on line 3: formula is [S'B (indegrees) × T] / 30 where T is the rising time of that Rāsi expressed in asus. In formula on line 8: the magnitude of DL in degrees by the formula (R × 30) / T where ‘R’ is the remainder in time after subtracting the rising times of s'B, BC, CD and T the rising time of the Rāsi ...`
Wait, in line 3: "T is the rising time of that Rāsi

228
the argument of an imaginary opponent namely “ Is the
time given measured after Sun-rise Sāvana (mean solar)
or Nākṣatra (ie. sidereal)? If it be the former, how is it
you are subtracting the rising times of sA, AB etc. which
are sidereal from your Sāvana units? Further, should you
not take the position of the Sun at Sun-rise namely S',
because the time given is what has elapsed after Sun-rise ?
Also, why should you complicate matters by accepting
Sāvana units when the question is simple if dealt with
sidereal units ?
To this Bhāskara answers as follows —
“ True it is, what you say. Generally in day to day
life, time is given only in Sāvana units and not side-
real. Further you cannot avoid Sāvana measure, for,
in the case of an arc moved by a planet, in its diurnal
circle time is measured in the Sāvana units pertaining
to the planet. (The Sāvana units of a planet are
different from what they are for the Sun depending upon
the arc moved by the planet in question during a day).
These Sāvana units are what are termed Kṣetra-Vibhā-
gātmika or what depend upon the arc covered in the
diurnal path in contradistinction to the Kāla-Vibhāgat-
mika units or sidereal units. (In other words pure time
is what is measured in sidereal units which is a standard
measure, whereas time which has the bias of the motion
of the planet also ie. which we seek to measure by the arc
moved by a planet in its diurnal path, is Kṣetra-Vibhāgāt-
mika). Thus having had to accept the Sāvana measure
also, we seek to proceed on that basis, though we could
convert the Sāvana measure into the sidereal and
proceed without complication ”. The argument which
Bhāskara gives for तात्कालिकीकरण is further as follows.
The measure of the arc SL using the rising times in
asus ie. sidereal units, is the Sāvana measure of the arc
S'L done in sidereal units. Instead of subtracting the
rising time of S'A from the given time converted into

229
sidereal, we subtract the rising time of SA from the given
Sāvana time and the result will be the same, for,
— SA = — (S′A — S′S) = S′S — S′A. which means
subtracting SA from the time tantamounts to increasing
by S′S and subtracting S′A. This increase by S′S is add-
ing what have been defined as gati-kalas so that automati-
cally the Sāvana units got converted into sidereal units,
by taking the position S and subtracting SA instead of
taking the position S′ and subtracting S′A. The result is
the same. So, it is said ‘तात्कालिकार्कस्य’ in the beginning
of the verse 2.
In the second case mentioned in verse 4, if the time
given, falls short of the Bhogyāsus, then simply (x × 30) / T
where x is the time given in asus and T the rising time of
the Rāsi in which both the Sun and the lagna are then
situated gives the arc in degrees which if added to the
longitude of the Sun gives that of the lagna. Here also
the position ‘S’ counts.
Verses 5 to 6½. To find conversely the time that has
elapsed after Sun-rise given the lagna.
The Bhogyasus of the Sun and the Bhuktāsus of the
Lagna together with the rising times of intermediate
Rasis gives the time required.
If the Sun and the Lagna both be in the same Rāsi,
then the arc in between them, multiplied by T and divide
by 30, gives the time required.
If, however, the longitude of the Lagna falls short of
that of the Sun, ie. if the Sun be below the horizon, (in
this case SL > 180°) then finding the time of rising of SL
and subtracting from a day, we have the time of the Lagna
before Sun-rise.

230
However, here. there is one complication if we
consider the तात्कालिकार्क ie. s the Sun at the given time.
This position of s cannot be had unless we know the time,
which is itself required. So we get in the first place the
time pertaining to S'L, which is the time measured in
sidereal units the position S' being that of the Sun at Sun-
rise. If we don't take recourse to convert this time in
sidereal units to Sāvana units using the proportion
between them, then the alternative is to obtain the position
S using the time obtained and then calculate the time
again in Sāvana units.
If the time given to find the Lagna, be sidereal, it
goes without saying that we find it from S'. Also S'
being given and if the time after sun-rise is required in
sidereal units for a given Lagna, the method of succes-
sive approximation is unnecessary.
Verse 7. To find the Lagna before Sun-rise called
Vilōmalagna.
Suppose it be required to find the lagna before Sun-
rise, given the time before Sun-rise. Obtain the then posi-
tion of the Sun and find his Bhuktāsus; subtract them from
the given time; from the remainder, subtract the rising
times of as many Rāsis as could be, rāsis behind the Sun's
position. If R be the remainder in the time after these
subtractions, then R × 30/T where T is the rising time of
is the next preceding Rāśi, together with n × 30, where n
the number of integral Rāśis subtracted and the arc
of the next Rāśi by which the Sun has advanced in his
Rāśi at the time of the Lagna (known from the position
of S found) the sum total of these three items being
subtracted from the position of S gives the longitude of L.
Comm. Easy.
Verse 8. To obtain the East-West line.