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

187
ecliptic successively each of 30°, we have what are called
Sāyana Rasis or Rasis taking the Ayanāṁsa into consi-
deration or in other words measuring from r. On the
other hand successive arcs each of 30° measured from the
zero point of the Hindu Zodiac, constitute what are called
the Nirayana Rāśis. The words Meṣa, Vṛṣabha etc.
signifying the shapes of the constellations apply strictly
to the Nirayana Rāśis. But nonetheless, by the conven-
tion what is called Upachāra, we name the Sāyana Rāśis
also by the same names. In as much as the zero-points
of the Hindu Zodiacs is ahead of the modern zero-point by
an arc which is the arc of Ayanamsa or accumulated pre-
cession, the words Sāyana and Nirayana came into vogue.
Once upon a time approximately in Varāha’s time or
rather 499 A. D, the two zero-points coincided and then
the Sāyana and Nirayana Rāśis were the same. Gradually
on account of the phenomenon of precession r preceded,
and today the distance between r and the Hindu zero-
point is about 20°–30′. Of late, there has been a big
controversy as to what exactly the arc is and the Calendar
reform committee has adopted a value far more than what
could be justified. The present author opines, that we
have no right to set aside a statement of no less an astro-
nomer than Varāhamihira who stated explicity ‘आश्लेषार्धात्
दक्षिणमुत्तरमयनं रखेर्धनिष्ठाद्यम् , नूनं कदाचिदासीत् येनोक्तंपूर्वशास्त्रेषु,
साम्प्रतमयनं सवितुः कर्कटकाद्यं मृगादितश्चाऽन्यत् उक्ताभावो विकृतिः
प्रत्यक्षपरीक्षणव्यक्तिः” i.e. True it is that once upon a time,
the Sun began his southern journey when he was mid-way
the constellation of Asleṣa and his northern when he was
at the beginning of Dhaniṣtha, because it was stated in
ancient texts. (The allusion is to Vedāṅga Jyotiṣa
where it was stated as such); but now the southern journey
of the Sun begins from the point marking ¾ of the con-
stellation of punarvasu which is the beginning point of
the Karkaṭa Rāśi and his norther from the beginning of
Makara i.e. from the point marking ¼th of the constellation
of Uttarāṣādha; there has been a change from what was

188
stated in the ancient texts ; let people verify this by actual
observation". Accepting an Ayanāṁsa which goes
against the statement of this great astronomer, who said
that he observed and called upon others to observe, is
really unwarranted, especially when the adopted Ayanāṁsa
of the Calendar reform committee goes on the basis of
surmises and consensus. We shall deal with this topic
in further detail in an appendix to this work, because it
is a really important issue and has been wrongly solved.
The importance of knowing the exact value of the arc is
clear when we observe that from the correctly computed
planetary positions of modern astronomy this ayanāṁsa
is being subtracted to give the positions measured from
the Zero-point of the Hindu Zodiac. An error in the
Ayanāṁsa therefore vitiates the positions obtained by the
above method.
Coming to the point, the problem on hand is to
obtain the rising times of the Sāyana Rāśis at a place of
zero latitude i.e. what are called Laṅkōdaya times of
Sāyana Rāśis (Ref. fig. 21). Let rS = 30° so that rA the
equatorial arc gives the rising time of rS. We could have
the magnitude of this arc by the formula derived from
Napier's rules namely cos ω = tan α tan λ I.
But as in Hindu trigonometry the tangent functions of
angles are not used, Bhāskara gave the following formula
Sin α = √(sin² λ − sin² δ) / cos δ II. This formula could be
derived easily from formula I and the formulae cos λ =
cos α cos δ III and sin δ = sin λ sin ω IV ; for multi-
plying the right-hand sides of I and III we have cos ω
cos λ = (sin α cos δ) / tan λ i.e. sin λ cos ω = cos δ sin α
∴ sin α = (sin λ cos ω) / cos δ V. But from IV sin ω =

--- ------------------ = ------------------- sin λ cos δ cos δAnd then:H·cos δ as stated.? Wait, where does H·cos δ as stated.come from? Wait! Look at line 5:H sin α = R √(H sin² λ - H sin² δ) / (H cos δ) ...Wait, is line 5:H sin α = R √(H sin² λ - H sin² δ) / (H cos δ)Wait, why is thereH·cos δ as stated.? Could it be: ... / H·cos δ as stated.? Wait! Look at the vertical alignment: Line 5: H sin α = R[fraction numerator:√(H sin² λ - H sin² δ)] Wait, what is the denominator of line 5? H cos δ! Then on line 6: sin λ × 1/sin λ ...Wait, why isH·cos δ as stated.BELOW line 6? Wait, look at the line aboveH·cos δ as stated.: There is a horizontal line above H·cos δ: H̄·cos δ as stated.? Wait! Is that H·cos δ as stated.or is that a fraction line? Wait, look at the letter beforecos δ`:
It has a bar above it:

190
From M drop a perpendicular ML on r ♎ so that SL will
be perpendicular from S on r ♎ by the theorem of three
perpendiculars. SL = H sin λ. Also ML will be equal
to the perpendicular from S on the diameter of the diurnal
circle of S parallel to r ♎ so that ML = SN = H sine of
the arc in the diurnal circle corresponding to Kr
∴ SW² = SL² - LN² = H sin² λ - H sin² δ
∴ SN = √(H sin² λ - H sin² δ)
∴ The length of the perpendicular from K on r ♎ =
R × √(H sin² λ - H sin² δ) / (H cos δ) since corresponding lines of
the diurnal circle and the equator stand in the ratio of
H cos δ : R (Vide fig. 20). But this ⊥ᵃʳ is H sin α =
R √(H sin² γ - H sin² δ) / (H cos δ) .
If α₁, α₂, α₃ be the Right ascensions of the points on
the ecliptic whose modern longitudes are 30°, 60° and 90°,
expressed in asus, then α₁, α₂ - α₁, α₃ - α₂ will give the
rising times of the arcs of the ecliptic which stand for
Sāyana Meṣa, Sāyana Vriṣabha and Sāyana Mithuna.
The rising times of the next three Rasis will be the same
in reverse order since Karkata is symmetric with Mithuna
with respect to the Equator and similarly Simha and
Kanya symmetric with Vriṣabha and Meṣa. The next
three are again symmetric with Meṣa, Vriṣabha and
Mithuna and the last three with Mithuna, Vriṣabha and
Meṣa.
Verse 56. The H cosines of the ends of the Rasis
Karkata etc., being multiplied by the radius, and divided
by the H cosines of their respective declinations, and the
arcs of those H cosines being taken, subtract as before the
preceding from the succeeding. Then we have the rising
times of the Rasis beginning with Karkata.

191
Comm. This is clear from fig. 24. If S be the end of
Vṛiṣabha, S' in the figure denotes the end of Karkaṭa
them from the right-angled triangle SAP,
^
sin S'A = sin SA = sin APS × sin PS = sin A'K × cos SK
sin S'A
∴ sin A'K = ————— . But sin S'A = cos S'♈
cos SK
and cos SK = cos δ where δ is the declination at S.
A'K converted into time gives the rising time of AS'
ie. Karkaṭa.
cos S'
∴ the rising time of Karkaṭa = ————
cos δ
In tbe Hindu form, it will be
Karkaṭa-anta-Koṭijyā
Karkaṭa-Rāsi-Udayakāla = ————————————— × R
Karkaṭa-anta-Dyujyā
as stated.
cos ♈S
In modern terms sin A'K = cos ♈K = ————
cos δ
from the formula cos λ = cos α cos δ. Thus, virtually
the formula is a statement of the formula cos λ = cos α
^
cos δ. Aslo the formula sin AS' = sin P sin PS is
parallel to the formula sin δ = sin λ sin ω which we
proved already from Hindu methods.
Verse 57. Still an alternative method.
The H sines of Meṣa etc. being multiplied by
H cos ω divided by their respective H cos δ's and the
arcs thereof being subtracted as before the preceding from
the succeeding we have the rising times of Meṣa etc.
Comm. From figure 24,
^ SN
SN = SL cos LSN and ————— = perpendicular from
cos SK
K on ♈

192
∴ (SL cos ω) / (cos δ) = H sin rK. SL = H sin rS
∴ H sin rK = (H sin rS × H cos ω) / (H cos δ)
We proved the above in a modern way. The Hindu
concept is derived from the similarity of SML and ACM′
where C is the centre of the sphere and M′ is the foot of
the perpendicular from A on the plane of the equator
∴ LM / CM′ = SL / CA ∴ LM = (H cos ω × H sin rS) / R
Since LM = SN.
LM divided by H cos δ and multiplied by R gives H sin rK
∴ H sin rK = [(H cos ω × H sin rS) / R] × [R / (H cos δ)] =
= (H cos rS × H cos ω) / (H cos δ)
i.e. H sin α = (H sin λ × H cos ω) / (H cos δ)
Here H cos ω is called Trigṛha-dyu-maurvī because
it is the H cosine of the declination of λ when λ = 90°.
Verses 58, 59. The magnitudes of the rising times.
Those rising times are 1670, 1793, 1937 ; these in the
same and reverse orders diminished or increased by their
respective Chara segments which are also in the same and
reverse orders give the rising times of the Sāyana Rasis
beginning from Meṣa for the locality. The Rasis from
Tulā are in a reverse direction i.e. as the Meṣa is proje-
cting upwards above the horizon, Tulā will be projecting
below the horizon so that, the time taken by Meṣa to
rise is exactly the time taken by Tulā to set.
Comm. We shall compute the rising times of
Sāyana Rasis for Lanka first i.e. for zero latitude using
modern methods from the formula tan α = cos ω tan λ

198
log tan α = log cos ω + log tan λ; Put λ₁ = 30, and λ₂ = 60
and take ω = 24°; Let the corresponding α's be α₁, α₂
log tan α₁ = log cos 24 + log tan 30° (1)
log tan α₂ = log cos 24 + log tan 60° (2)
log tan α₁ = 9.9607 + 9.7614 = 9.7221
∴ α₁ = 27° - 48'
log tan α₂ = 9.9607 + 10.2386 = 10.1993
∴ α₂ = 57° - 42'
At the rate of 1 asu for 1', α₁ = 1668 asus α₂ = 3462;
α₂ - α₁ = 1794 and since α₃ = the right ascension of 90°
Longitude = 90°, α₃ = 5400 so that α₃ - α₂ = 1938.
These are given by Bhāskara as 1670, 1793, 1937, the first
exceeding by 2 asus, the second and third each less by
one asu and the total according with the total. The rising
times of Karkaṭa etc. will be 1937, 1793, 1670, 1670, 1793,
1937, 1937, 1793, 1670 respectively.
Let us then find the rising times of these Sāyana
Rasis at a locality say of latitude 13°. Refer to fig. 21.
Let rS represent Meṣa so that the rising times of rE is
equal to that of rS. But rE = rA - AE. We have seen
rA = 1670 using Bhāskara's value. Sin EA = tan 13°
tan δ₁ where δ₁ is the declination of S where rS = 30°.
Sin δ₁ = sin 30° sin 24°.
log sin δ₁ = 9.6990 + 9.6093 = 9.3083
∴ δ₁ = 11° - 44'
∴ log sin EA = log tan 13° + log tan 11° - 44 =
9·3634 + 9.3175 = 8.6809
∴ EA = 2° - 45' ∴ rE = 1670 - 165 = 1505 asus.
Similarly for λ = 60°, putting α₂, δ₂ in the place of
α₁, δ₁ and proceeding as before rE = rA₁ - A₁E; rA₁ =
3463; sin EA₁ = tan 13° tan δ₂ sin δ₂ = sin 60° sin 24°
25

194
∴ log sin δ₂ = 9.6093 + 9.9375 = 9.5468
∴ δ₂ = 20° – 30′
∴ log sin EA₁ = 9.3634 + 9.5758 = 8.9392
∴ EA₁ = 4° – 59′ = 299′
∴ rE = 3463 – 299 = 3164 asus
∴ Rising time of Sāyana Vṛṣabha for the locality
= 3164 – 1505 = 1659 asus.
rA₂ = 5400, sin EA₂ = tan 13 tan δ₂ = tan 13 tan 24°.
∴ log sin EA₂ = 9.3634 + 9.6486 = 9.0120
∴ EA₂ = 5° – 54′ = 354 asus
∴ rE = 5400 – 354 = 5046
∴ Rising time of Mithuna is 5046 – 3164 = 1882 asus.
Before we proceed to find the rising times of Karka-
taka, Simha and Kanyā, we shall cast our previous proce-
dure into the Hindu form. In fig. 21, let rS be the Sāyana
Mesha. The rising time of rS is measured by rE, because
when r is at E, Meṣa is just about to rise and when r is
in the position indicated, the extremity of Meṣa namely
S is rising. So it means that as rS of the ecliptic has
risen, a portion rE of the Equator has risen. As time is
measured by the arc of the equator which rises with a
uniform speed, we measure the rising time of rS by the
arc rE; but rE = rA – AE. rA is the Equatorial rising
time of rS, because when A is at E, S will be at B i.e.
A and S will then be on the equatorial horizon EB simul-
taneously. Hence rA = 1670 as proved before and stated
by Bhāskara. EA is the chara for 30°. The chara for one
angula or inch (inch is here used technically, and does not
mean what it means in ordinary parlour) as has been stated
by Bhāskara and proved by us is 10 Vinadis. (Vide page 181)

195
But we have taken 13° as our latitude, so that tan ϕ = .2309.
Since s/12 = tan ϕ = .2309 ∴ s = 2.7708; let us take
this as 2.8″ so that the charas 10, 8, 3⅓ found for one inch
are to be multiplied by 2.8 to give the local charas. They
are in Vinādis, 28, 22.4, 9⅓ or in asus 168, 134·4, 56; for
convenience let us take 134.4 as 134, so that the charas
are 168, 134, 56. Thus the rising time of Sāyana Meṣa
at this locality is 1670 − 168 = 1502 asus i.e. 250 Vinādis
= 4-10 Nādis. Then let rS now represent 60°, instead of
subtracting EA from rA to get the combined rising time
of Meṣa and Vṛiṣabha, the Hindu practice is to subtract
the chara pertaining to Vṛiṣabha from the equatorial
rising of Vrishabha i.e. rE = 1793 − 134 = 1659 as got
before. Here it must be noted that EA° is the chara not
pertaining to Vṛiṣabha alone but to Meṣa and Vṛiṣabha
put together. That is why for ease, the chara to Meṣa,
the increase in chara for Vṛiṣabha, and the increase in
chara for Mithuna as well as their individual equatorial
rising times are given. The increments in the charas are
called chara-khandas just as the increments in the H sines
are called Jyā-khandas (khands means segments). Simil-
arly the rising time of Sāyana Mithuna is equal to 1937 −
56 = 1881 asus = 313.5 Vinadis = 5-14 Nadis. Now with
respect to Karkataka, its equatorial rising time is 1793,
for, from fig. 25, the equator at the equatorial place being
prime Vertical, if rM be Meṣa, its time of rising is given
by rE where E is the foot of the declination circle of M.
Similarly if MV represents Vṛiṣabha, when V comes to
the horizon, N the foot of the declination circle comes to
the horizon. Thus the rising time of any arc of the
ecliptic at an equatorial place is given by the correspond-
ing arc of the equator, which is intercepted between the
declination circles of the ends of the arc. So from fig. 26
if r ed ≃ be the equator, rED. ≃ the ecliptio, A the
Ayana or Summer solotice, P the pole rE, ED, DA etc.
the Sāyana Rasis Meṣa etc. a, b, c etc. the feet of the

196
Fig. 25
Fig. 26
declination circles of A, B, C etc., since PAE is secondary
both to the ecliptic and equator (i.e. perpendicular circle)
spherical triangles PAD, PAB are congruent; PBC, PDE
are congruent and PC ♎ is congruent with PE ♈. Hence
ab = ad i.e. rising times of Karkataka and Mithuna are
equal; bc = de i.e. those of Simha and Vriṣabha are

197
equal and similarly those of Meṣa and Kanyā. It will
be noted that the equatorial risings alone are equal in the
above cases but not at any other place, for in a place with
some latitude when a point like B (fig. 26) comes to the
horizon, the foot of its declination circle namely b will
not be on the horizon and there arises the chara in bet-
ween, which has to be taken into account, and be subtract-
ed from or added to the equatorial rising time as the case
may be.
Now regarding the chara-khanda of Karkaṭa it is
again 56; why it should be so is not proved by Bhāskara
but merely stated, nor any commentator took the pains to
prove. It can be proved as follows. Let in fig. 26,
δ₁, δ₂, δ₃ be the declinations at the ends of Vṛṣabha,
Mithuna and Karkaṭaka respectively. We know δ₁ = δ₃.
The chara-segments for Mithuna and Karkaṭaka i.e. when
Mithuna and Karkaṭaka are rising are to be proved to be
equal, here 56 asus. Their expressions are tan ϕ tan δ₂ −
tan ϕ tan δ₁ and tan ϕ tan δ₃ − tan ϕ tan δ₂ i.e. tan ϕ
(tan δ₂ − tan δ₁) and tan ϕ (tan δ₃ − tan δ₂). Since
δ₁ = δ₃ we perceive that they are equal but of opposite
signs. So Bhāskara says rightly “अपचीयमानत्वात् धनम्”
i.e. because of negative sign, the chara-segment of Karka-
taka while being subtracted will be rendered positive’.
Hence the rising times of Karkaṭaka, Simha and Kanya
will be respectively 1937+56, 1793+134, 1670+168 asus
or 1993, 1927, 1838 asus or 332, 321, 306 Vinadis or
5-32, 5-21 and 5-6 nadis. Thus, in as much as the rising
times of Meṣa to Kanyā are 1670−168, 1793−134,
1937−56, 1937+56, 1793+134, 1670+168 their total
is 30 nadis as should be expected because the equator
bisects the ecliptic between ♈ and ♎ and the equatorial
interval between ♈ and ♎ is 30 nadis. It will be noted
that while the equatorial rising times of Meṣa to Kanyā
are symmetrical as 1670, 1793, 1937, 1937, 1793, 1670,
their rising times at any other place are not like that but

198
constitute a different kind of symmetry as 1670—168,
1793—134, 1937—55, 1937＋55, 1793＋134, 1670＋168.
We have now to comment upon the statement
“तुलादितोऽमी च विलोमसंस्थाः ” which means that the rising
times from Tulā to Mīna are in the reverse order i.e. the
rising time of Tulā equals that of Kanyā; that of Vris-
chika equals that of Simha and so on the rising time of
Mīna equalling that of Meṣa. Thus the rising times of
Tulā to Mīna being in the reverse order are 1670＋168,
1793＋134, 1937＋55, 1937—55, 1793—134, 1670—168
for the aforesaid locality. Why it should be so can be
easily seen from the fact that Kanyā and Tulā are sym-
metric with respect to the line ♈ ♎ which bisects the
ecliptic (Ref. fig. 27). Or again we can see this in another
Fig. 27
way; the chara-segments are successively (tan ϕ tan δ₁ —
tan ϕ tan 0), (tan ϕ tan δ₂—tan ϕ tan δ₁), (tan ϕ tan ω—
tan ϕ tan δ₂), (tan ϕ tan δ₂—tan ϕ tan ω), tan ϕ tan δ₁—
tan ϕ tan δ₂ (tan ϕ tan 0—tan ϕ tan δ₁), (tan ϕ tan δ₁—
tan ϕ tan 0), (tan ϕ tan δ₂—tan ϕ tan δ₁), (tan ϕ tan ω—
tan ϕ tan δ₂), (tan ϕ tan δ₂—tan ϕ tan ω) (tan ϕ tan δ₁—
tan ϕ tan δ₂), (tan ϕ tan 0—tan ϕ tan δ₁) where δ₁ =
declination of 30°, and δ₂ that of 60°. These are there
as found before 56, 134, 168, —168, —134, —56 upto Kanyā.
But the remaining, though apparently are 56, 134, 168,
—168, —134, —56 must be taken with a reverse sign because
the rising point of the ecliptic will be to the south of the
east point and δ will be negative from 180° to 360° longi-
tude. Hence the chara Segments are 56, 134, 168, —168,
—134, —56, —56, —134, --168, 168, 134, 56 so that from
Tulā onwards they are in the reverse order as Bhāskara

100
has stated. Bhāskara's statement yet has another mean-
ing. The ecliptic from 180° to 360 being in the reverse
as can be seen in fig. 27, the time by which a particular
Rāsi rises, is equal to the time by which its seventh Rāsi
or diametrically opposite Rāsi sets. It is important to
note that the rising time of a particular Rāsi is not equal
to its setting time as can be gauged from the rising times,
for, then the rising times of Meṣa and Tulā must be equal
which is not the case. The word विलोमसंस्थाः has this
meaning as well, rising and setting being reverse directions.
Incidentally there is what is called a वृद्धकारिका
handed down i.e. a traditional statement which men-
tions "मीनमेषौ चतुर्नड्यः सार्धं चत्वारिगोघटौ etc." This
clearly pertains to the local rising times of the Sāyana
Rāsis at a place of latitude 17°-45' i.e. in between
Rajamundry and Vizianagaram, as in this latitude the
chara Segments in asus will be 230, 184, 73 which give
rise to such rising times.
We next find what are called Nirayana Swōdayas i.e.
the rising times for the locality of the Nirayana Rāsis i.e.
the Rāsis from the zero-point of the Hindu zodiac. We
shall obtain these magnitudes for Rajahmundry whose
latitude is 17°-2'. We shall however take it as 17°; also
we shall take the Ayanāṁsas to be 21° at present following
Varāhamihira. This is one of the contexts where the
knowledge of Ayanāṁsas is essential. The method is the
same as followed before; only, we have to find the decli-
nations, right ascensions, charas and therefrom the rising
times of arcs of magnitude 21°, 51°, 81°......351°. Sub-
tracting the preceding rising times from the following we
have successively the required rising times. We shall
give them under the following table.

200

201
In modern terms, the rising times of Rasis could be
found by solving the spherical triangle γEA (fig. 28) using
the Inner side inner angle formula namely .
[Fig. 28]
Fig. 28
cos 𝑥 ✕ cos 𝜔 = sin 𝑥 cot 𝜆 + sin 𝜔 tan 𝜙 I
sin 𝑥 cot 𝜆 − cos 𝑥 cos 𝜔 = − sin 𝜔 tan 𝜙
putting sin 𝑥 = t, this reduces to
t cot 𝜆 − √(1 − t²) cos 𝜔 = − sin 𝜔 tan 𝜙
or (t cot 𝜆 + sin 𝜔 tan 𝜙)² = (1 − t²) cos² 𝜔
∴ t² (cos² 𝜔 + cot² 𝜆) + 2t ✕ sin 𝜔 tan 𝜙 cot 𝜆 −
cos² 𝜔 = 0
∴ t =
−sin 𝜔 tan 𝜙 cos 𝜆 ± √[sin² 𝜔 tan² 𝜙 cot² 𝜆 + cos² 𝜔 (cos² 𝜔 + cos² 𝜆)]
──────────────────────────────────────────────────────────────────────────
cos² 𝜔 + cot² 𝜆
Putting successively 𝜆 = 30°, 60°...360° and ignoring
the negative sign of the radical we have the sines of rising
times of arcs of the ecliptic of 30°, 60° etc. Subtracting
26

202
the preceding from the succeeding, we have successively
the rising times. Thus the rising time of Mesha is
√3 sin² ω tan² ϕ + cos² ω (cos² ω + 3 − √3 sin ω tan ϕ II
────────────────────────────────────────────────────────
3 + cos² ω
That of Meṣa and Vṛṣabha put together, the rising
time is .
√sin² ω tan² ϕ + 3 cos² ω (cos² ω + ⅓) − sin ω tan ϕ III
───────────────────────────────────────────────────
√3 (cos² ω + ⅓)
The combined rising time of the 3 Rasis Meṣa, Vṛṣa-
bha and Mithuna is from I using tables cos⁻¹ (tan ω tan ϕ)
= 5046 asus which exactly accords with what we have
found previously namely 1505+1659+1882 = 5046. Also
putting in I λ = 180°, we have sin x = 0 or x = 180° =
10800 asus ; subtracting 5046, we have the combined times
of rising of Karkataka, Simha ahd Kavya to be 5754 asus
as we have had. Putting again λ = 270, we have cos°⁻¹
(tan ω tan ϕ) = 2π − 5046 whioh signifies that the sum
of the rising times of the last three rasis is the same as
that of the first three which again means that the sum of
the rising times of Tula to Dhanus is equal to the sum of
the rising times of Karkaṭaka, Simha and Kanya establish-
ing Bhaskara's statement “विलोमसंस्थाः”.
Verse 60. Computations of Lagna, Udayāntara and
the like from the rising times of big arcs of the ecliptic
like Rasis will be approximate, whereas one desirous of
greater approximation has to find the same from the
rising times of smaller arcs likes Horas and Dṛkkāṇas, so
as to be more correct.
Comm. Rasis divided into halves are called horas
and if divided into one-third parts are called
Dṛkkāṇas. The meaning of the verse is that if after
having found the rising time of a particular Rāśi say
Meṣa, we say that one-third of that rising time is that of
one-third of that Rāśi, we will be making only an appro-

203
ximate statement just like saying that ‘Since 12 Rasis
rise in the course of a sidereal day, so each rasi rises in
1800 asus’ which is far from truth being based on a crude
rule of three. So, Bhāskara says, Acharyas like Aryabhata
insisted on finding the rising times of Dṛkkāṇas, in as
much, as while computing the Lagna or the point of inter-
section of the Ecliptic with the horizon, we will be nearer
the truth by using the rising times of smaller arcs like
Dṛkkāṇas than broadly using those of Rasis.
Verse 61. Bhujāntara correction.
The equation of centre of the Sun multiplied by the
equatorial rising time of the Rāsi occupied by the Sun,
and divided by 1800, and then again multiplied by the
true daily motion of a planet and divided by the number
of asus in a day, is the correction in the planet positive
or negative according as the equation of centre of the Sun
is positive or negative.
Comm. This Bhujāntara correction arising out of
the Sun’s equation of centre is prescribed even for the Sun,
as well as for the other planets. The planets are originally
computed for the rising time of the Mean Sun, whereas
we are interested to know the positions at the True Sun-
rise. The position of the Sun also is originally computed
for his mean rise and is therefore to be rectified to get his
position at his true rise. So even the Sun is not exempt
from the correction. It will be noted here that as the
equation of centre pertains to the eccentricity of the Sun’s
orbit, this correction of Bhujāntara is a correction for the
so-called modern ‘Equation of time due to eccentricity’.
In other words, the Sun’s equation of centre converted
into time is exactly what is called the Equation of time
due to eccentricity.
The formula prescribed for the correction is as follows.
Suppose the Sun is in a particular Rāsi which rises at the
equator in x asus. Let the equation of centre of the Sun be

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E minutes of arc. Then the equatorial rising time of E is
(E × x) / 1800 asus because there are 1800' in a Rasi. We have to
provide a correction in the planetary position including that
of the Sun for this time. If the planet goes Y' during
21659 asus of a day, what arc is covered by the planet or
the Sun in (E × x) / 1800 asus? The result is (E × x) / 1800 × y / 21659
minutes of arc. This correction is positive if the equation
of centre of the Sun is positive, for, we want the planetary
position for a latter time than Mean Sunrise, since the
positive equation of centre advances the True Sun over
the Mean. This correction will be appreciable only in the
case of the Moon having a rapid motion.
Verses 62, 63. The correction known as Udayāntara.
The difference in minutes of arc in the longitude of
the Sāyana mean Sun and the asus in his Right ascension,
multiplied by the daily motion of the planet and divided
by 21659 is the result to be added to or subtracted from
the planet's longitude according as the asus in the Sun's
Right ascension are greater or less than the minutes of arc
of his longitude. This is what is called Udayāntara
correction in the planetary position.
Comm. (1) We have seen that the Bhujāntara is a
correction in the mean planetary position due to the
Equation of time in Eccentricity.
(2) This Udayāntara is a correction in the same
due to the Equation of time in obliquity.
(3) Some have misconstrued that this Udayāntara
correction is the Equation of time in the obliquity itself,
whereas it is a correction to be effected in the planetary
position due to the Equation of time due to obliquity.

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(4) The maximum equation of centre in the Sun
has a magnitude = 13 2°/3 × 1/2π = 41/3 × 7/44 = 287/132 = 2° — 10'
Converting this into time at the rate of 15° per hour
(since in one hour diurnal rotation of the earth is equal to
15°) we have 13/6 × 4 minutes = 26/3 = 8'—40''. The maxi-
mum equation of centre according to modern astronomy
is 2e expressed in radiaus where e = 1/60 (.0167339) = 2 × 1/60
radiaus = 2 × 1/60 × (180 × 7)/22 degrees = 21°/11 = 1°—54' — 33'.
As such the max. Equation of time due to eccentricity is
21/11 × 4' = 84'/11 = 7'—38''. The small difference in the
two values arises out of the difference in the max. equ-
ations of centre. Any way it is clear that, in as much as
the Bhujāntara correction is necessitated on account of
the equation of centre in the Sun, to obtain the planetary
positions computed for the mean Sunrise at the time of
True Sunrise, the Bhujāntara correction is a correction in
the planetary position, on account of the equation of time
due to eccentricity. The max. correction to be effected in
even the quick moving Moon amounts to (8 × 790)/(24 × 60) = 79/18 =
4'—23.3''.
(5) We have said in the translation of the verse.
‘The asus in the Right ascention of the mean Sun’,
where what exactly is stated by Bhāskara is, the time of
rising of the small arc covered by the Sun in the particular
Sāyana Rasi in which the Mean Sun is, (भुक्तासवः) together
with the rising times of the previous Sāyana Rasis covered
by the Sun. The meaning is therefore the rising time of
an arc of the ecliptic equal to the Sāyana longitude of the
ecliptic which is measured by his Right ascension at the

rate of 15° per hour or 6° per nadi or 10 Vinādis per degree
or 60 asus per 60 minutes of arc or as many asus as there
are minutes of arc in the right ascension of the mean Sun.
So, what is stated by Bhāskara is the difference of the
minutes of arc in the mean longitude of the Sun and the
minutes of arc in his right ascension ie. (l—a) expressed
in minutes. We know that the total equation of time
arising out of unequal motion in the true longitude of the
Sun ie. ☉ in comparision with the equal motion in his
right ascension ie. a is measured by ☉—a which may be
expressed as (☉—l) + (l—a) where l is his mean longi-
tude, ☉—l = Equation of centre and so the time
expressed by ☉—l is the equation of time due to obliquity.
The difference l—a arises out of the obliquity of the
ecliptic and so the time expressed by l—a is the equation
of time due to obliquity. We have the modern formula
Cos ω = tan a / tan l so that (1 — cos ω) / (1 + cos ω) = (tan a — tan l) / (tan a + tan l) =
(sin a — l) / (sin a + l)
∴ Sin (a — l) = tan² ω/2 sin (a + l). As a is very
nearly equal to l, we could write, when expressed in
radiaus a—l = tan² ω/2 sin 2l. Thus the maximum
difference between a and l arises when 2l = 90° ie. l = 45
degrees ie. at the mid-point of the first quadrant; the mini-
mum difference is when 2l = 270 ie. l = 135 ie. at the middle
point of the 2nd quadrant. Also the numerical magnitudes
of the max. as well as the minimum value are each tan² ω/2
ie. they are equal. Since this is expressed in radiaus,
converting into time the numerical value of the max. and
minimum equation of time due to obliquity is 9.87′. Thus
we can write l—a = 9.87′ sin 2l. Again where l = 225,
2l = 450 so that l—a will have a positive max.; and again
when l = 315, 2l = 630 so that l—a will have a negative
maximum value. Also when l has values 0, 90, 180, 270 it
is zero. Thus the equation of time due to obliquity is zero
at ♈, ie. the vernal equinox; +ve in the first quadrant