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

162
Comm. Ref. verse 45, Spaṣṭādhikāra, Śiṣya Dhī-
vṛddhida तद्रहिताऽऽशुभुक्तिः, त्रिज्ज्याहता स्वचलकर्णहताऽऽशुचापभोग्य
ज्यया विगुणिता विहृताऽऽद्यमौर्व्या, लब्धं त्यजेत्, स्वचलतुङ्गगतेः सदैव शेषं
स्फुटा भवति च ग्रहभुक्तिरेवम्" i.e.
( शीघ्रगति-मन्दस्फुटगति ) × R Śīghraphala Bhōgyakanda
--------------------------- × -----------------------
K 225
= Sīghragatiphala.
Here the quantity within the brackets is δm. What
Lallācharya had in his mind is as follows. ‘If for the Ādya
Khanda 225 we have H cos E = R, what shall we have for
the Bhōgya Khanda of the Sīghraphala ?’ The result
would be [ H sin (E + 225) - H sin E ] × R / 225 =
( H sin E H cos 225 + H cos E H sin 225 / R - H sin E ) × R / 225
Taking H cos 225 = R and H sin 225 = 225 it would be
H cos E × 225 / R × R / 225 = H cos E. So Lallācharya’s
formula would become (δm / K) × H cos E as given by Bhās-
kara. The charge levelled at Lallācharya is due to the fact
that by the word ‘Āsu-chāpa’ by which Lallācharya meant
‘आशुफलचाप’ Bhāskara meant ‘आशुकेन्द्रचाप’. When the
Kendra = 90 or 270, the Bhōgyakhanda being Zero, the
Sīghragatiphala would be Zero. Also where it ought to be
zero namely (90 + H sin⁻¹r) & (270 - H sin⁻¹r) it would not
be zero. If Lallācharya had really meant what Bhāskara
allributed to him, the formula would be (H cos mδm / K) and
it is very unlikely that Lallācharya would have meant this
wrong formula, for, even if K were taken by him to be
steady, δ (r H sin m / K) = (r H cos mδm / K) and Lallāchar-
ya’s formula does not contain ‘r’. The only non-rigorous
part in Lallācharya’s formula is at the point where he took

163
H̄ cos 225 = R and H sin 225 = 225 which is rather crude.
Bhāskara of course improved on this crudeness by taking
δm to be an increment in E.
In the commentary under this verse, Bhāskara,
having misinterpreted Lallācharya's phrase आशुचाप as
meaning आशुकेन्द्रचाप and not as आशुफलचाप which was
in the mind of Lallācharya, goes on recounting examples
where the wrong formula attributed to him would give
wrong results. So, we need not enter into those details.
Now we shall correlate Bhāskara's formula with its
modern counterpart. Assuming coplanar heliocentric cir-
cular orbits for planets, let us see at what points two planets
appear mutually stationary, that is, have a Zero relative
angular velocity before they appear mutually retrograde.
Let S = Sun, E = Earth, J = Jupiter, u = Earth's
linear velocity, v = Jupi-
ter's linear velocity r and
R the orbital radii of the
Earth and Jupiter respe-
ctively. Let EE' and JJ'
be perpendiculars to EJ so
that when the relative velo-
city of Jupiter with respect
to the Earth perpendicular
to EJ is Zero, Jupiter will
appear stationary as seen
from the Earth. This
Fig. 17
means that u cos θ + v cos ξ = 0 I
∴ u / v = - cos ξ / cos θ II
But from triangle ESJ,
R cos ϕ + K cos θ = r III and
r cos ϕ + K cos ξ = R IV

164
From III & IV cos ξ / cos θ = (r cos ϕ − R) / (R cos ϕ − r) V
Equating cos ξ / cos θ from II and V
− u / v = (r cos ϕ − R) / (R cos ϕ − r) so that (ru + Rv) / (rv + Ru) = cos ϕ
If m be the Śīghra anomaly m = 180 − ϕ
so that cos m = − (ru + Rv) / (rv + Ru) VI
Now we shall show that Bhāskara's formula accords with
this Spaṣṭagati = Śīghragati − (H cos E δm) / K . As per
Bhāskara
Spaṣṭagati = 0 if Śīghragati = (H cos E δm) / K VII
ie. Jupiter appears stationary as seen from Earth, if Śīghra-
gati = (H cos E δm) / K
The angular velocities of Earth and Jupiter are respect-
ively u / r and v / R so that the Sun's apparent velocity is also
u / r and δm = Kēndra gati = Sun's apparent velocity
minus Jupiter's heliocentric velocity = u / r − v / R
∴ Substituting in VII
Śīghragati = u / r = (H cos E / K) (u / r − v / R)
∴ u / r ( (H cos E / K) − 1 ) = (H cos E / K) × v / R
ie. u / r ( (H cos E − K) / K ) = (H cos E / K) × v / R

M₂ + E₂ = S I where M₂ = Mandasphuta-
graha and E₂ = the Sīghraphala where from we have
δM₂ + δE₂ = δS II i.e. Mandasphutagati + Sīghragati
phala = Spastagati (a) Let E₂ be maximum so that
δE₂ = 0, then δM₂ = δS. This means that in the helio-
centric figures 11 and 12, at the points a and b,: the
Sīghraphala being maximum, the Mandasphuta gati will be
itself the Spashtagati. This line ab, it will be seen corres-
ponds with the so called “कक्षामध्यगततिर्यग्रेखा (प्रतिवृत्तसम्पातरेखा)”
i.e. the line cutting the eccentric, drawn through the centre
of the deferent.
(b) The planet begins to retrograde only after the
Spashtagati vanishes i.e. after δM₂ + δE₂ becomes Zero.
Taking δM₂ to be almost a constant since the Mand

166
negative when the planet courses the arc of the smaller
segment ab, because E₂ decreases from a maximum value
to zero. As a matter of fact δE₂ negatively increases along
ac and again increases from a negative minum at c to Zero
at b as we course along cb (vide fig. 11 & 12). Thus the
planet will assume zero velocity at two points symmetrical
about c and in between ab. In other words the planet will
be retrograde along the arc d₁ cd₂ not entirely along ab as
some have misconstrued. Regarding E₂, it is clear that it
is zero at S'. then gradually increases to a maximum as the
planet traces S'a, the maximum being assumed at a, then
it decreases from the maximum to zero as the planet
courses the arc ac. and then increases from zero to a maxi-
mum at b and finally decreases from that maximum to
zero at S' again. Keeping the Earth constant it will be
noted that an Inferior planet always goes anticlockwise
whereas a Superior planet always goes clockwise. Also
it will be seen that the Sīghraphala is positive as the planet
courses the arc Sac, whereas it is negative as it courses
cbS'.
(c) The values of the spashtagati at S' and C will be
respectively putting H cos E = R in the formula.
Spashtagati = Sīghragati — (H cos Eδm) / K . Here for
Sīghragati we may put U and U — V for δm, and putting
K = R + r at S' and R — r at C, the values of the spash-
tagati would be
[U — R (U — V)] / (R + r) and [U — R (U — V)] / (R — r) ie. (RV + ru) / (R + r),
(RV — rU) / (R — r) respectively.
(RV + rU) / (R + r) > (RV — rU) / (R — r) if R²V — rRV + RrU — r²U >
R²V — RrU + rRV — r²U

167
ie. if rR (U - V) > Rr (V - U). U - V is + ve and
equal to V - U.
So the positive velocity at S' of the planet will be
equal to its negative or retrograde velocity at C. A quan-
tity assuming values K, O, - K,O,K must have a value
numerically less than K in between. Thus the velocity
direct or retrograde at any point of the orbit is less than
the numerical value of the velocity at S' and C.
Verse 41. Retrograde motion.
The planets Mars, Mercury, Jupiter, Venus and Saturn
will be retrograde when the Sīghra anomaly assumes values
163, 145, 125, 165 and 113 respectively and the direct
motion again ensues at (360 - 163), (360 - 145), (360 - 125),
360 - (165) and (360 - 113) respectively.
Comm. Since we have had the formula VI
cos m = - ( (rU + Rv) / (rV + RU) ) for stationary points,
noting that cos (180 - θ) = - cos θ = cos (180 + θ) the
stationary points are symmetrically situated with respect
to S' of figures 11 and 12. As mentioned before substitut-
ing the values of U, V, r, R for all the planets we can
prove the veracity of Bhāskara's statement.
Verse 42. Heliacal rising and setting of planets.
Mars rises heliacally in the East by 28°, Jupiter by
14°, Saturn by 17° of Sīghra anomaly and set heliacally in
the west by degrees which are the differences of the above
and 360° respectively.
Comm. The Sun's velocity being greater than that of
the Superior planets, the Sun overtakes them so that they
set in west and rise in the East. When these planets are

168
situated within particular limits from the Sun, they will be
invisible in the rays of the Sun. As these superior planets
will be near the Sun, near the moment of conjunction,
they will not be seen at conjunction and within particular
limits from the position of the Sun. The limits cited
above mostly depend upon their respective brilliance and
to some extent upon their distances from the Sun too.
Their brilliance again depends upon the extent of their
gibbosity. The formula given for the phase in modern
astronomy is phase = (1 + cos EpS) / 2 and noting that Êps =
E₂ = the Śīghraphala, phase = (1 + cos E₂) / 2. Hence the Supe-
rior planets are always gibbous ie. the disc illuminated will
be always greater than ½. Though at conjunction E₂ = 0
and the entire discs of the major planets will be illumina-
ted, we cannot see them as they are immersed in the rays
of the Sun. As they emerge out of conjunction gradually
E₂ will be increasing so that the discs will be illuminated
lesser and lesser gradually. But K decreasing, the brilli-
ance will not be so much effected. Along the arc acb (figs.
11, 12) the planets gradually gain in illumination and will
be brightest when they are in opposition both as cos E in-
creases and K decreases. In other words the Superior
planets appear more and more brilliant when they are
retrograding, being most brilliant at ‘C’. The spherical
radii of Jupiter Saturn and Mars being in decreasing
order, their brilliance will be in decreasing order so that
they will be rising at distances from the Sun which are in
increasing order. The inverse square law of courses works
here but combining the two factors (1) the spherical radius
of the planet and (2) the inverse square law, we may take
it that Bhāskara’s numbers cited above namely 28°, 17°
and 14° for Mars, Saturn and Jupiter respectively accord
with truth, for, these numbers are given by Bhāskara or
his authority Brahmagupta only after observing the planets
rising heliacally.

169
That the positions of the planets while setting or rising
heliacally will be situated symmetrically with respect to
the Sun, goes without saying.
But one thing. In the chapter called Udayāstamayā-
dhyāya, the degrees known as Kālāṁśas which are given
as the arcs between the planets and the Sun for heliacal
rising or setting, are different from the numbers given
above, for, the latter are the values of the Śīghra anomaly.
In the case of Mars when the Śīghra anomaly is 28°, the
Śīghraphala will be 11°, so that the apparent distance of
the planet will be 28° - 11° = 17° which are the Kālāṁśas
for Mars. Similarly in the case of Jupiter, when the
Śīghra anomaly is 14°, the Śīghraphala will be 3°, so that
the distance between the planet and the Sun as seen from
the earth will be 11°, which are given as Jupiter's Kālā-
ṁśas. Also in the case of Saturn, the Śīghraphala for 17°
of anomaly will be 2°, so that the Kālāṁśas would be 15°.
Verse 43. Mercury and Venus rise in the West by
50° and 24° of Śīghra anomaly respectively, and set in
the West by 155°, and 177° respectively. They rise in the
East by 205° and 183° of Śīghra anomaly and set there by
310° and 336° respectively.
Comm. Regarding the Inferior planets, they rise
heliacally in the East after Inferior conjunction and then
they are retrograde. They attain gradually the maximum
elongation in the East and after they revert to direct
motion, their elongation gradually decreases. They then
set in the East and heliacally rise thereafter in the West.
There again their elongation attains a maximum value;
then they begin to retrograde and gradually set in the
West only to rise in the East. This is all clear from the
heliocentric figure 11. When the Śīghra anomalies of
Mercury and Venus happen to be respectively 50° and 24°,
their Śīghraphalas would be 13° and 11° respectively, so
that they are themselves the Kālāṁśas, in as much as in
22

170
the case of the Inferior planets, the Sighraphala will be
itself their elongation eastern or western. Then they
rise in the West, being near Superior conjunction. When
again their Sighra anomalies equal respectively 155° and
177°, the same Sighraphalas will arise so that they set
heliacally in the West. Then as the Sighra anomalies
attain the symmetrical values on the other side ie. (360-
155) and (360-177) ie. 205° and 183° the Sighraphalas being
the same, they rise in the East. Again when they attain
the values (360-50) and (360-24) ie. 310° and 336°, they
set in the East on account of the same Sighraphalas or
Kālāmsas.
Verse 44. When the Sighra anomalies have parti-
cular values, to decide when the planets rise or set helia-
cally, we have to take the difference of those particular
values and the numbers given above for the respective
Sighra anomalies, convert them into minutes of arc and
divide the results by the daily motion in the Sighra ano-
malies in minutes of arc. Then we have the number of
days in which the rising or setting takes place thereafter.
Comm. Suppose it is required when Mercury rises in
the West. Suppose we want to compute this on a parti-
cular day when Mercury's Sighra anomaly is x°. Then
because we know that Mercury rises in the West when his
Sighra anomaly is 50, we have to calculate by how many
days the difference | x-50 | of the Sighra anomaly would
be covered. Let the daily motion of the Sighra anomaly
be y' per day, Then by rule of three (| x-50 | × 60) / y will
be the number of days before or after as the case may be
for Mercury to rise in the West.
Verse 45. To obtain the Mean planet knowing the
True.
Assume the True planet to be the Mean; compute the
Manda and Sighraphalas and applying them inversely,

171
we have an approximation of the Mean planets. Treating
these as the Mean planets, again obtaining the Manda and
Śīghraphalas and again applying them inversely and
repeating the process till constant values are obtained, we
have by this method of successive approximation the Mean
planets required.
Comm. The method of successive approximation is
clear.
Verse 46. To obtain the equinoctial shadow.
Convert the Ayanāṁśas into minutes of arc, and
divide by the mean daily motion of the Sun; then we
have the number of days before the Meṣa or Tulā Saṁ-
krānti day, or before Makara and Karkaṭaka Saṁkrānts
days, when the Sun will be in equinoxes or Solstices
respectively. The mid-day shadow of the Sun cast by the
gnomon on such an equinoctial day, will give us the
equinoctial shadow required.
Comm. The palabhā or equinoctial
shadow as it is called is the length of
the shadow cast by a gnomon taken
to be of 12 units in length, (measuring
the shadow also in the same units), at
noon of an equinoctial day. In other
words, if this shadow be of s units,
clearly s / 12 = tan ϕ (Vide fig. 18).
The Ayanāṁśas are the degrees of
the arc of the ecliptic in between the
Hindu Zero point of the Zodiac and the
first point of Aries which is now behind the former due to
the phenomenon known as the precession of the equinoxes.
They are called Ayanāṁśas because the solstices are also
behind the Makara and Karkaṭaka Saṁkrānti points of
the Hindu Zodiac or points which have Hindu longitudes
270° and 90° resply, by the same arc. The Hindu astro-

172
nomers came to know that the solstices are preceding by
observations made with the gnomonic shadow at mid-day
around the solstitial days. The day on which the maxi-
mum mid-day shadow is cast by the gnomon is the Winter
solstitial day whereas the day on which the mid-day
gnomonic shadow is maximum in the southern direction
(assuming the place to be of northern latitude and ϕ < ω)
or minimum in the northern direction (ϕ > ω) is the
summer solstitial day. Calculating the Sun's longitude
on that day at noon, we know how far the solstitial points
have preceded behind the points of the Hindu Zodiac which
have Hindu longitudes 90° and 270°. The word अयनांशाः
has therefore the meaning अयनविलोमगत्यंशाः where the
Samāsa may be viewed as a मध्यमपदलोपी समासः
Verses 47, 48. Calculating the five fundamental
H sines of a point of the Zodiac pertaining to a point of
the ecliptic.
The declination has to be computed from the sum of
the Hindu longitude of that point and the Ayanamsas.
Similarly if it be required to find the time before or after
the rise of a point of the ecliptic, we have to compute them
from the sum of the longitude of that point and the
Ayanamsas.
H sin 24° × H sin λ
─────────────────── = H sin δ I
R
√(R² — H sin² δ) = H cos δ = Dyujyā II
δ = H sin⁻¹ (H sin δ) III
s
── × H sin δ = Kujyā IV
12
Kujyā × R
───────── = Charajyā V
H cos δ
H sin⁻¹ (Charajyā) = Charam VI
Comm. (1) Let rA☉ be the ecliptic where r =
Vernal Equinox, A = first point of the Hindu Zodiac,

in between r and ☉,
not shown in the figure
☉ = the Sun. Let rA =
a° = Ayanamsas defined
before so that r☉ = (a + λ).
From the spherical triangle
r☉M, by Napier's rule,
sin δ = (sin λ + a) sin ω
which in Hindu trigono-
metry becomes H sin δ =
[H sin (λ + a) × sin ω] / R
In Hindu Astronomy the
obliquity of the ecliptic was
taken to be 24°. The value
of the obliquity is now 23°–27' approximately and it has
been know that this has been decreasing. At the time
when the Hindu Astronomers observed this, it should
have been greater than 23°–27' so that if it was taken to be
24°, their observations were not far from truth. This
means that the antiquity of Hindu Astronomy might be far
more than what the Moderns estimate it to be.
(2) How the formula I was derived in Hindu tri-
gonometry was as follows. Let ♋ be the summer solstice
♋C, ☉B, the perpendiculars dropped from ♋ and ☉ on
the line of intersection of the Equatorial and Ecliptic
planes namely rBC. Let perpendiculars be dropped from
♋ and ☉ on the plane of the Equator. Let them be ♋N,
☉D, Join NC and DB. Then ☉B̂D = ♋ĈN = the
dihedral angle between the two planes = ω; ♋C = R
since in Hindu trigonometry H sin 90° = R. Also ♋N =
H sin ♋E (in Hindu trigonometry) = H sin ω; ☉D =
H sin ☉M = H sin δ; ☉B = H sin r☉ = H sin (λ + a)
where λ is the Hindu longitude of the Sun and a =
Āyanamśas. It will be seen that ☉D is a segment of the

line of intersection of the planes PCM a plane perpendi-
cular to the plane of the Equator, and ☉BD a plane per-
pendicular to the Ecliptic plane. Since ⚍C 11 ☉ B and
⚍N 11 ☉ D and BD☉ = CN⚍ = 90°, so the two tri-
angles are congruent
∴ ☉B / ⚍C = ☉D / ⚍N ie. H sin (λ + a) = (R × H sin δ) / (H sin ω)
∴ H sin δ = [H sin (λ + a) H sin ω] / R
NC = ⚍L = √(⚍C² — ⚍N²) = √(R² — H sin² δ) =
H cos δ. ⚍L is called Dyujyā as explained below in
note (3).
Another way of looking at the similarity of the tri-
angles C⚍N and B☉D is from the fact that they are
formed by the intersection of parallel planes, both of which
are perpendicular to the plane of the Equator. This idea will
be elaborated when we explain fig. 21, wherein the so-called
latitudinal triangles will be shown to be formed by the
intersection of the plane of the Equator and planes of
diurnal circles with the planes of the horizon and the
prime vertical. In fact, the plane of a great circle and
the parallel planes of the corresponding small circles form
the same dihedral angle with the planes of the celestial
sphere namely the planes of the meridian, horizon and
prime-vertical so that right-angled triangles formed by
their intersection will be all similar.
Formula II is derived from the formula H sin² δ +
H cos² δ = R² derived from fig. 6 from which formula,
formula III follows.

176
(3) Why H cos δ is called Dyujyā in formula II
is clear from fig. 20 wherein p is the celestial pole, (C) is the
Fig. 20
celestial Equator and (C) is the diurnal circle of a celestial
body say the ☉ ie. the Sun. Let ☉N be the perpendicular
dropped from ☉ on the plane of the Equator so that
☉N = H sin ☉A = H sin δ CN = H cos δ = C☉ =
radius of the diurnal circle called Dyujyā (Dyu = day)
द्युवृत्तत्रिज्या = द्युज्या (Madhyamapadalōpi Samāsa) formulae
IV, V and VI will be dealt with in the next chapter
Triprasnādhyāya more elaborately but one has to under-
stand what Charajyā is to follow the subsequent verses of
this chapter so that we shall give its location and defini-
tion in the light of fig. 21. Let fig. 21 represent the
celestial sphere in which SEN = Horizon, Z = Zenith,
N = Nadir Ez = prime vertical EQR = celestial equator.

176
Fig. 21
PP′ = Polar axis, SBS′ = the diurnal circle of a celestial
body S; EW = the East-West line, SS′ = Udayāstasūtra
or the join of the rising and setting points S, S′. This
SS′ is evidently a diameter of the diurnal circle which is
bisected by the plane of the horizon; PEP′ is called the
unmandala or the Equatorial horizon. PSA is the decli-
nation circle of S cutting the Equator in A. EA is called
the charam whose H sine is called Charajyā. The H sine
of SB in the diurnal circle is called Kujyā so that as
corresponding lines in the diurnal circle and the Equator
stand in the ratio H cos δ : R,
Kujyā / Charajyā = (H cos δ) / R I

177
In the beginning of the Triprasnādhyāya, Bhāskara says
“साक्षे देशे खगोलवलयानां, तिरश्चीनभगोलवलयानां च संपातात्त्र्यस्राणि
क्षेत्राण्युत्पद्यन्ते, तान्यक्षक्षेत्रसंज्ञानि” ie. In a place having a
latitude the diurnal paths of stars and planets will be
inclined to the fundamental circles of the celestial sphere
namely horizon, meridian and prime vertical and so their
intersection gives rise to what are called latitudinal tri-
angles, in which the angles would be ϕ, 90-ϕ and 90
where ϕ is the latitude. Thus in fig. 21 the projections
of the triangles ESB, EDB, DSB, EDF, EBF, ESF and
EQG on the meridian plane will be all triangles in which
the angles will be ϕ, 90-ϕ and 90° so that they are all
latitudinal triangles. These are all similar to the funda-
mental gnomonic triangle gmn of fig. 18 wherein also the
angles are ϕ, 90-ϕ and 90°. Here, there is one important
point to be observed. We have said “their projections
are all similar”. In fact the corresponding spherical
triangles enumerated above are all apparently similar,
though they are not viewed in modern astronomy as
regular spherical triangles, because all the three sides of
the triangles are not arcs of great circles. It is not possi-
ble to apply Napier’s rules to these triangles for the reason
mentioned above. None the less, the property of simil-
arity of the projected right angled triangles is made use
of in Hindu Astronomy to obtain the magnitudes of the
sides of the projected triangles.
EW is called the prāk-pratīchī sūtra, SS′ the Uda-
yāsta sūtra; similarly if lines through F, B, D, A parallel
to EW be drawn, these sūtras will intersect the meridian
planes in points which constitute the projected triangles
mentioned above. Let us study these triangles which we
connote by the same letters with lowered indices. Thus
for example E₁ S₁ B₁ is the projected triangle of ESB
where of course E₁ will be the centre of the celestial
sphere. The lines drawn parallel to SS′ or EW through
B, D etc. will be denoted as BB′, DD′, FF′ etc.
23

178
In the triangle E₁ S₁ B₁, E₁ S₁ is called Agrajyā, S₁ B₁
Kujyā, and E₁ B₁ Krāntijyā. Of the sides of ESB. ES
and EB are arcs of great circles whereas SB is the arc of
a small circle. In the projected triangle E₁ S₁ B₁, E₁ S₁
will be equal to the perpendicular from S on Eω, which
will be H sin (ES) and is called Agrajyā ; E₁ B₁ will be
equal to the perpendicular from B on Eω which will be
H sin BE and as such called Krāntijyā. But S₁ B₁ which
is equal to the perpendicular from S₁ on BB' the diameter
of the diurnal circle is the H sine of SB in the diurnal
circle and is called Kujyā. It will be noted that the per-
pendiculars from S on Eω, and BB' and the perpendicular
from B on Eω do not form a triangle by themselves but
by the theorem of three perpendiculars, if SL be the
perpendicular from S on the plane of the unmandala ie.
the great circle EBP, and if LM be perpendicular from
L on Eω, SM will be perpendicular on Eω. Here the
perpendicular SL will be the Kujyā, and SM the Agrajyā,
whereas LM is not actually the H sine of BE but is equal
and parallel to it.
Similarly take the spherical triangle SAE. This is a
regular spherical triangle because the three sides are arcs
of great circles. Napier's rules can be applied to this
triangle and we have the formula sin SA = sin SE sin
SÊA or in Hindu form H sin SA = (H sin SE × H sin Ê) / R
or RH sin δ = Agrajya × H cos ϕ ie. Agrajyā = (RH sin δ) / (H cos ϕ) II
Again sin EA = tan SA × tan Ê where H sin EA is
called Charajyā and tan E = cot ϕ so that Charajyā =
tan δ tan ϕ in modern form, and the Hindu form is
<u>R tan δ tan ϕ</u>

179
From I H cos δ / R = Kujyā / Charajyā so that
Kujyā = H cos δ / R × R tanδ tanϕ = H sinδ tanϕ III
In Triprasnādhyāya, the elements of all the eight
latitudinal triangles are found by using their similarity
with the fundamental gnomonic triangle. The important
elements that will enter into computation are (a) Charajyā
(b) Kujyā (c) Agrajyā (d) Taddhriti ie. S₁ F₁ the proje-
ction of SF on the plane of the meridian (e) Sama-Sanku
= E₁ F₁ = H sin EF (f) Krāntijyā = E₁ B₁ = H sin EB
= H sin δ (g) Lambajyā = H sin QS = H cos ZQ =
H cos ϕ (h) Akshajyā = H sin ZQ = H sin ϕ (i) B₁ D,
= Ud-Vritha-Sanku = H cos ZB (j) Dinārdha-Sanku =
H cos Zq. The following points will be noted.
(i) In the triangle E₁ Q G₁, the projected triangle
of EQG on the meridian plane, noting that E₁
is the centre of the celestial sphere E₁ Q = R,
QG₁ = H cos ϕ and E₁ G₁ = H sin ZQ =
H sinϕ.
(ii) S₁f₁ = H sin SB + H sin fB (both the H sines
pertaining to the diurnal circle.
(iii) Sama Sanku is the H cosine of the Zenith-
distance when the Sun or celestial body is on
the prime-vertical.
(iv) BD is an arc of the great circle ZB, so that the
Unmandala Sanku is the H cosine of ZB.
(v) Dinārdha — Sanku = H cosine ZQ.
(vi) These Sankus are the H sines of altitudes or
H cosines of Zenith-distances and they are in
the planes of the respective great circles.

180
(vii) Lambāṁsa-chāpa is the arc of the colatitude
QS where S is the South point so that Lam-
bajyā is the H sine of QS or H cosine of ZQ.
This Lambajyā will be seen to be the diameter
of a small circle parallel to the Equator and
passing through Z.
Verses 49-51. A different method of obtaining
chara.
This chara can be had by the so-called chara-segments
of the locality using a process similar to that of finding
the H sines of the smaller table of nine H sines using
λ/3 where λ is the Sāyana longitude of the Sun (i.e. The
Hindu longitude plus the arc of ayanāṁśas is called the
Sāyana longitude or the modern longitude measured from
v along the ecliptic to the Sun). Find the charas of 30°,
60° and 90° of the ecliptic measured from v. Subtract the
first from the second, the second from the third. Thus we
have C30°, C60°-C30°, C90°-C60°, (where Cθ° signifies the
chara of θ°) which are called chara-Khandas or chara-seg-
ments. The equinoctial shadow multiplied by 10, 8, 3⅓
gives the approximate values of the chara-segments in
Vinādīs (a sidereal day is divided into 60 nādīs and each
nādī consists of 60 Vinādīs). The chara-segments thus
measured in Vinādīs are rather approximate. If further
exactitude is required, better take the arc in units each
of which rises in ⅙th of a Vinādī (This ⅙th part of a
Vinādī is known as a prāṇa i.e. the duration of the
interval between two inhales of a healthy person reckoned
as 4″ of time).
Comm. The charas of 30°, 60° and 90° of modern
longitude are the values of the arc EA, (Ref. Fig. 21)
when S has longitudes 30°, 60° and 90°. We have the
folmula
Charajyā = R tan δ tan ϕ

181
Taking the equinoctial shadow equal to 1 Angula
means tan ϕ = 1/12. As charajyā is proportional to tan ϕ,
for any equinoctial shadow of s angulas, tan ϕ being
equal to s/12 the charajya got above is to be multiplied by
S only to obtain the charajyā in any place where the equi-
noctial shadow is s angulas. Putting the modern longitu-
des equal to 30° & 60°, if the corresponding declinations
be δ₁, δ₂ sin δ₁ = sin 30 sin ω, sin δ₂ = sin 60 sin ω. Tak-
ing ω = 24° and applying logarithmic tables
log sin δ₁ = 9.6990 + 9.6093 = 9.3083 so that δ₁=11°–44′
log sin δ₂ = 9.9375 + 9.6093 = 9.5468 so that δ₂=20°–38′
Now from the formula for charajya cited above viz.
H sine (chara) = R tan δ tan ϕ or sine (chara) =
tan ϕ tan δ, putting tan ϕ = 1/12 and applying tables,
using the values of δ got above,
charajya for 30° = (tan 11°–44′) / 12
and charajyā for 60° = (tan 20°–38′) / 12 so that
log (sine chara) = 9.3175 – 1.0792 for 30° and for 60°
log (sine chara) = 9.5758 – 1.0792
∴ Chara for 30° or C (30)° = 59′ and C (60°) = 1° – 48′
Converting these arcs into their rising times at the rate of
6′ per Vinadi, we have C (30°) = 10, and C (60°) = 18
Noting δ₃ = ω, sin (chara) for 90° = (tan 24°) / 12 so that
log sin (C 90°) = 9.6486 – 1.0792 = 8.5694 so that
C (90°) = 2°–8′ = 21⅓ Vinādis.
Thus (C 30̄) = 10, C (60) – C (30) = 18 – 10 = 8
C (90°) – C (60°) = 21⅓ – 18 = 3⅓ so that the chara
Segments are respectively 10, 8, 3⅓ as given by Bhāskara.