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

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.

231
The East-West line is roughly the join of the
extremities of the morning shadow as well as that in
the after-noon of a gnomon placed at the centre of a
circle drawn on a plane with any arbitrary radius,
when those shadows equal the radius of the circle. But
this line is to be deflected keeping its western point ie.
the extremity of the morning shadow fixed through a
distance K (sin δ₁ ~ sin δ₂) / cos ϕ at the eastern end perpendi-
cular to it, where the above distance is measured in units,
which measure K.
Comm. The east and west points are where the
celestial equator cuts the horizon. The east point is
thus the point where the Sun-rises when he is exactly at
the vernal equinox. The question is how to draw the
east-west line on a plane. For this we are asked to
draw a circle with any radius on that plane. The plane
is described here as अम्भःसुसमीकृतक्षिति that kind of sur-
face as is determined by the surface of water there. Such
a kind of surface forms approximately a horizontal plane
not of course exactly because such a surface is really
spherical, the earth being a sphere. But because the
radius of the earth is sufficiently large, we can take such
a surface to be a horizontal plane for all practical purposes.
Having drawn a circle place the gnomon vertical at the
centre. In the morning note the extremity of the shadow
when it equals the radius. In the afternoon also mark
the point when the shadow equals the radius. Join those
two points. It represents roughly the east-west line,
roughly because the Sun's declination changes in between
the two moments however small the change might be.
Ignoring the change in the declination, this line will
be east-west because of the following reason. The length
of the shadow is 12 tan z where 12 units are the measure
of the gnomon. But since the shadow on both the occasions
is equal to the radius of the circle z, the zenith-distance

232
will be the same on both the occasions. Then the spherical
triangles PZS₁ and PZS₂ where P is the celestial pole, Z the
Zenith and S₁ and S₂ are the positions of the Sun on the
two occasion, are congruent their three sides being respe-
ctively equal, provided we take PS₁ = PS₂ ie. 90 — δ₁ =
90 — δ₂ ie. δ₁ = δ₂ on both the occasions. When the
two triangles are thus congruent, PẐS₁ = PẐS₂ ie. S₁ and
S₂ are equidistant from the plane of the Prime-Vertical.
Hence the extremities of the shadows will be equidistant
from the East-West line ; or this may be seen in another
way ; S₁ S₂ will be perpendicular to the meridian plane
and as such parallel to the plane of the Prime-Vertical.
The correction mentioned in the verse is known as
the Agrāntara correction which was originally given by
Chaturvedā chārya and then accepted by Sripati. Why
it is called Agrāntara is because it is a change in what
are called Karnavrittāgras of the two occasions where we
shall see in due course that the formula for Karnavrittāgra
is (K sin δ) / (cos φ) where K is hypotense of the gnomonic triangle
formed by the gnomon and its shadow S at any place and
time. This correction is a very minute correction and
as a matter of fact could be ignored. But the fact that the
correction was cognized and correctly formulated testifies
to the knowledge of the sphere which the above
acharyas had. Assuming the formula of the Karnāgra
here (it will be proved by us later in this chapter) if
δ₁ and δ₂ be the declinations on the two occasions
respectively the Karnāgras will be (K sin δ₁) / (cos φ) and (K sin δ₂) / (cos φ) ,
K being equal on the two occasions because the shadows
are equal. Hence the correction being the difference of
the Agrās, it is [K (sin δ₁ ~ sin δ₂)] / (cos φ) as stated by Bhās-
kara,

233
We shall now prove it in modern terms from the
spherical triangle PZS. We have the formula sin δ =
sin ϕ cos z + cos ϕ sin z sin a where PZS = 90 — a, a
being the Hindu azimuth measured from the East point.
Multiply the above equation by K and divide throughout
by cos ϕ where K is called the Chāyākarṇa equal to
√(12² + S²), S being the gnomonic shadow at the moment.
Hence
K sin δ / cos ϕ = K cos z tan ϕ + K sin z sin a (1)
Fig. 31 Fig. 32
But from Fig. 31, K cos z = 12, K sin z = S, (2)
and from Fig. 32, S sin a = b where b is called the
Chāyābhuja ie. Chāyābhuja = K sin z sin a (3). Thus
K sin δ / cos ϕ = 12 tan ϕ + b.
But again from Fig. 33, when ☉ the Sun at vernal
equinox is on the meridian and as such has a meridian
zenith-distance equal to ϕ, 12 tan ϕ = s where s is called
the Vishuvat-chāyā or equinoctial shadow. Thus we have
K sin δ / cos ϕ = s + b (4).
Again if the Sun be on the horizon, from Fig. 34,
E ☉ is called the Agrā, A, so that from the triangle P ☉ N,
30

234
Fig. 33 Fig. 34
cos (90 — δ) = cos ϕ cos (90 — A) ie. sin δ / cos ϕ = sin A or
in the Hindu form H sin A = R (sin δ / cos ϕ) = Agrajyā (5).
This Agrajyā is in a circle of radius R, and if it be re-
duced to a circle whose radius is K, is will be (K / R) × (R sin δ / cos ϕ)
= (K sin δ / cos ϕ) which is called Karṇāgrā. Hence we have
Karṇāgrā = s + b which we shall write as a = b + s (6).
This is an important formula which is going to be
formulated later in verses 72, 73. In the above formula,
s being constant, by differentiating δa = δb which means
that the variation in the bhuja is on account of the
variation in the Karṇāgrā. If in Fig. 35, Cω', CE″ be
morning and evening shadows when they are equal as per
the verse under comment, Mω' = b the morning bhuja,
NE″ = b', the evening bhuja dE' is the variation in the
bhuja ie. b — b' = δb which is formulated and equal to
δa. But δa = δ(K sin δ / cos ϕ) = K δ (sin δ) / cos ϕ, ϕ being con-
stant and K also being constant because the shadow
S is constant and K = √(S² + 12²) = constant on both
the occasions. ∴ δb = δK = (K / cos ϕ) (sin δ₁ — sin δ₂)
as stated by Bhāskara.

235
Fig. 35
Verse 9. Having determined the East-West line as
mentioned, by drawing with a compass with the same radius
on either sides of Eω (Eω is the East-West line through the
centre of the circle drawn parallel to the rectified East-
West line namely E′ ω′ where E″ ω′ is the approximate
East-West line obtained by the join of the extremities
of the shadows) two arcs which intersecting each other
form a fish-like enclosure as shown in the figure and
as such called Matsya meaning a fish, and joining the
ends of the fish namely f, g and producing fg, we have
the south and the north points s and n. Or again,
the north-south line could be had from Fig. 36, where
EC is a rod held in the direction of the north-pole
as seen by the eye at e and EA and CB the lines indicated
by the plumb-line called अवलम्बकसूत्र A and B being on
the plane and AB joined passing through N, the north-
point. Or again the directions could be determined as
follows from a single shadow S namely Cω′ in Fig. 35,

236
calculating the bhuja ω'M and Koti
CM; Bhuja and Koti being known,
and the shadow Cω' being drawn,
holding two rods whose lengths are
equal to the bhuja and Koti perpendi-
cular to each other, one extremity of
the bhuja-rod being held at ω' and
one extremity of the Koti-rod being
held at C, and the rods making a
right angle at M, then the bhuja-rod
determines the north-south direction
and the Koti-rod the East-West dire-
ction.
Fig. 36
Verse 10. The Bhuja is defined as the distance of the
extremity of the shadow from the east-west line where
the S'anku or gnomon is placed at the intersection of
Eω and NS. Koti = √(S² − b²) ∴ Koti will be in the
East-West direction. So Chayā-koti = K sin z cos a (7).
Comm. Easy.
Verse 11. The Chayākarṇa, K is equal to √(S² + 12²),
so that √(K² − 12²) = S or √[(K + 12) (K − 12)] = S.
Comm. Easy.
Verse 12. The S'anku is also called Nara or Nā.
The zenith-distance of the Sun at Noon when the Sun
is in ♈ is the latitude of the place, called pala or
Aksha; the altitude then is called lamba or colatitude.
Comm. The word S'anku we have previously used
for the gnomon. It is also used for the H cosine of
the zenith-distance and to differentiate it from the
previous S'anku called Dwādasāṅgul'a-S'anku or
twelve-unit-length S'anku, it is termed Mahā S'anku
and occasionally Iṣṭa-S'anku. Mahā S'anku or Iṣṭa
S'anku = H cos z (8). Thus in figure (37) ☉M =
H cos z. In the fig. where gn = gnomon, go = S

237
the shadow, ☉ = the position of the Sun whose zenith-
distance is ☉ z. If z' be taken as the zenith ☉ z' meas-
ures the zenith-distance whereas if z be taken as
the zenith ☉ z is the zenith-distance. The apparent
inconsistency that both ☉ z' and ☉ z are taken
as the zenith-distance is not there if we consider the
^ ^
zenith-distance as the angle ☉ nz' = ☉ oz. O ☉ = R,
☉ z = z, so that ☉ L = H sin z, and LO = ☉ M =
H cos z = Śanku or Nara or Nā. it is called Nara or
Nā which means man, because
a man may consider himself
as a gnomon, which is called
Śanku. So the word is also
applied to the parallel H cos z
parallel to the gnomon and
called Mahāśanku. H sin z is
called Drigjyā (9) zenith-dis-
t

238
Fig. 38
plane of the meridian - where E₁ is the centre of
the sphere.
(4) The fourth latitudinal triangle is E₁ S₁ F₁ the
projection of ESF (Fig. 21) on the meridian plane
where E₁ F₁ is the Sama-Sanku or the Sanku of the
celestial body when it is on the prime vertical. E₁ S₁
is the Agrajyā as mentioned, S₁ F₁ is what is called
Taddṛti.
(5) The fifth latitudinal triangle is E₁ B₁ F₁ where
E₁ B₁ is Krāntijyā or H sin δ, E₁ F₁ is the Sama-
Sanku defined above and B₁ F₁ is what is called the
higher segment of the Taddṛti which is equal to
Taddṛti minus Kujyā or Kshitijyā.
(6) The sixth latitudinal triangle is E₁ D₁ B₁
where E₁ D₁ is called the first segment of Agrajyā, D₁ B₁ is
what is called un-mandala Sanku or H cos z of the
celestial body when it is on the unmandala or the
Equatorial horizon and E₁ B₁ Krāntijyā.

239
(7) The seventh latitudinal triangle is D₁ S₁ B₁
where D₁ S₁ is the second segment of the Agrajyā, S₁ B₁ is
the Kujyā, and D₁ B₁ is the unmandala Sanku.
(8) The eighth latitudinal triangle is B₁ L₁ F₁
where B₁ L₁ is equal to the first segment of Agrajyā,
L₁ F₁ is the higher segment of Agrajyā, L₁ F₁ is the
higher segment of the Sama-Vritta-S'anku, and F₁B₁ is
the upper segment of Taddṛti mentioned before.
Comm. It was already mentioned that a latitudinal
triangle is such a right-angled tri-angle constituted by
the chords of the celestial sphere where the angles in the
triangle are ϕ, 90 − ϕ, 90°. The side opposite to ϕ is
called the Bhuja, that opposite to (90 − ϕ) is called
Koti and the third Karna. Such triangles are not only
eight as have been mentioned, but many more will be
there as mentioned by Bhāskara. They are all formed
as mentioned by him by the intersections of the diurnal
paths and the celestial equator with the circles of the
sphere namely horizon, prime vertical, meridian,
Equatorial horizon and declination circles. These
circles clearly intersect at ϕ or 90 − ϕ. The eight
triangles mentioned are those whose elements will be
entering computations. There is another important
latitudinal triangle with which we have to deal later
namely that formed by what is called Hriti, S'anku, and
S'ankutala (Fig. 39).
O₃K = Agrā ; O₃C = S'anku-tala ; CN = S'anku-bhuja
= CK ∴ Agrā = Sankutala + Sanku-bhuja
Agrā = R (H sin δ / H cos ϕ) ; S'ankutala = H cos Z tan ϕ
S'anku-bhuja = (H sin z H sin a) / R (Ref. fig. 40')
∴ R (H sin δ / H cos ϕ) = H cos z tan ϕ + (H sin z H sin a) / R I