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

244
Hṛti = O₁A = OO₁ + OA
= Kujyā + H cos δ or
Dhujyā VI since OO₁ =
DO₄ = Kujyā. The line in
the Equatorial plane cor-
responding to Kujyā is
Charajyā which is equal to
R tan φ tan δ (13) so that
we can write Kujyā =
R sin δ tan φ = H sin δ tan φ
(13'). Also from VI
[Fig. 40]
Antyā = R + Charajyā
= R + R tan φ tan δ (14)
which gives us the duration
of half the day where R
gives 6 hrs and Charajyā
the increment in day due
to φ and δ.
[Fig. 40]
Verse 18. To obtain the
magnitudes of the various
chords or the elements of
the latitudinal triangles.
The elements or the sides of all these latitudinal
triangles are mutually derivable from similarity.
Comm. Easy.
Second half of Verse 18. The radius multiplied by
the Kotis or Bhujas and divided by the Karnas gives
H cos φ and H sin φ.
Comm. The seven triangles except the second, of the
eight latitudinal triangles are compared here with the
second. Remembering Bhujas are the sides of the
triangles opposite to φ and Kotis opposite to 90 − φ,

245
Bhuja / Karṇa in any triangle = Bhuja / Karṇa in the second latitudinal
triangle = (H sin φ) / R ∴ (R × Bhuja) / Karṇa = H sin φ Akshajyā
or Palajyā (15) Similarly Koti / Karṇa in any triangle is equal
to the Koti / Karṇa in the second latitudinal triangle = (H cos φ) / R
so that (R × Koti) / Karṇa = H cos φ = lambajyā (16).
Verse. The arcs of H sin φ and H cos φ are respec-
tively the Akshamsas and lambamsas as they are called ie.
latitude and colatitude. H sin φ and H cos φ are also
obtainable thus √(R² - H sin² φ) = H cos φ and H sin φ =
√(R² - H cos² φ) Or again (H sin φ × Koti) / Bhuja = cos φ and
(H cos φ × Bhuja) / Koti = H sin φ where the Bhuja and Koti
may belong to any latitudinal triangle.
Verse 20. Agrajyā can be had by multiplying Krāntijyā
by Karṇa of any lat. triangle and divided by its Koti.
Also Sama-Sanku = Karṇa / Bhuja × Krāntijyā and
(Sama-Sanku × Karṇa) / Koti = Taddhṛti.
Comm. The first of these statements pertains to the
similarity of the third triangles to the others. The second
of the statements pertains to the similarity of the fifth
latitudinal triangle to others whereas the third pertains to
that between the fourth and the others.

246
Thus Sama-Śaṅku or S.S.
= (H sin δ × R) / (H sin φ) or (R sin δ) / (sin φ) (17).
Taddhṛti = R × (H sin δ) / (H sin φ) × R / (H cos φ) = (R sin δ) / (sin φ cos φ) (18).
Verse 21. Taddhṛti = (Karṇa × Agrajyā) / Bhuja
Comm. This pertains to the similarity between the
fourth latitudinal triangle and the others.
Latter half of Verse 21 and first half of Verse 22.
Sama-Śaṅku = (Taddhṛti × Koṭi) / Karṇa = (Agrajyā × Koṭi) / Bhuja
(Sama-Śaṅku × Bhuja) / Koṭi = Agrajyā.
Comm. The first statement is made out of the
similarity between the fourth lat. triangle and the others
whereas the second statement and the third as well are
made out of the similarity between the third and others.
II half of Verse 22. Sama - Śaṅku = Upper
segment of Taddhṛti × Karṇa / Koṭi
Comm. The similarity is between the fifth triangle
and others.
Verse 23. Kujyā = Krāntijyā × Bhuja/Koṭi
Upper segment of Taddhṛti = (Krāntijyā × Koṭi) / Bhuja and
Kujyā + Upper segment of Taddhṛti.
Comm. The first statement is through the similarity
of the third triangle with others, whereas the second is
through the similarity of the fifth with the others. The
third statement is clear from Fig. 21.

247
Verse 24. (Kujyā × Bhuja) / Karṇa = second segment of
Agrajyā (Krāntijyā × Koṭi) / Karṇa = first segment of Agrajyā
and Agrā = Sum of the two segments.
Comm. The first statement is based on the similarity
between the seventh triangle and others and the second on
the similarity between the 6th and the others.
Verse 25. (First segment of Agra × Bhuja) / Koṭi
= Un-mandala-Śaṅku and (Krāntijyā × Bhuja) / Karṇa
= Un-mandala-Śaṅku. (19)
Comm. Both the statements are based upon the
similarity of the sixth triangle and others. Thus un-
mandala śaṅku = U.S. = (H sin δ × H sin φ) / R =
= R sin δ sin φ.
Verse 26. (First segment of Agrā × Koṭi) / Bhūja
= Un-mandala-Śaṅku = Kujyā × Koṭi / Karṇa.
Comm. The first statement is based on the similarity
of the sixth latitudinal triangle and others and second
that between the seventh and others.
Sama-śaṅku − unmandala śaṅku = upper segment of
Sama-śaṅku.
Verse 27. (Agrā × Bhuja) / Karṇa = Kujyā
Taddhṛti − Kujyā = upper segment of Taddhṛti.

248
Comm. The first statement is based on the similarity
between the third triangle and the others and the second
statement is evident.
Second half of verse 27. Other elements could be
derived from what is already known and from what has
been obtained. Also by alternando and invertendo we
could pass from one element to the other and vice-
versa.
Verse 28. Karṇa = √(Bhuja² + Koti²)
Bhuja = √(Karṇa² − Koti²)
Koti = √(Karṇa² − Bhuja²)
Thus the third could be had from the other two in all
the triangles.
Verse. There are sixty-three ways of obtaining H sin ϕ
and H sin ϕ. On account of hundreds of ways of
obtaining Agrajyā etc., there are an infinite number of
ways of obtaining H cos ϕ etc.
Comm. Under verse 23 Bhāskara says that there are
98 ways of obtaining Taddhṛti. Taking the third latitude
triangle, H sin δ could be obtained in seven ways, from
this H sin δ, Kujyā could be obtained in seven ways;
hence, according to the principle of association namely
that when one thing could be done in m ways and another
in n ways, both the operations could be together performed
in mn ways, so Kujyā could be obtained in 7 × 7 = 49
ways; similarly the upper segment of Taddhṛti could be
had in 49 ways; so that adding the two Taddhṛti could be
obtained in 98 ways.
Similarly suppose we have to find H sin δ. H cos ϕ
could be found in seven ways and from H cos ϕ, H sin δ
could be found in seven ways. Thus H sin ϕ could be

249
found in 7 × 7 ways = 49 ways. From R, H sin φ could be
found in seven ways by using similarity with the other
seven latitudinal triangles except the second. Also
obtaining H cos φ in seven ways and using the formula
H sin φ = √(R² — H cos² φ) we have seven more ways.
Thus in all there are 7 × 7 + 7 + 7 = 63 ways. Similarly
H cos φ could be found in 63 ways. Extending this to
Agrajyā etc. which could be in as many or more ways
themselves finding H sin therefrom means again finding
it in 69 × 69 ways and so on. Since there is no end in
counting all these ways, it is said that there are infinite
ways to find it. The word 'infinite' here connotes only a
very large number of ways not exactly what we mean by
the word 'infinity'.
Verse 30. To find what is known as Koṇa-Śaṅku.
As a first approximation take
Koṇa-Śaṅku = √(R² — 2A²) where A = Agrajyā and Koṇa-
Śaṅku means H cos z when the azimuth is equal to 45°.
Then take Agrajyā ± the above Koṇa-Śaṅku × s/12 = Śaṅku-
bhuja = b (say) then again Koṇa-Śaṅku = √(R² — 2b²).
Then again take Agrajyā ± the above Koṇa-Śaṅku × s/12
as the new bhuja and proceeding thus by the method of
successive approximations, we arrive at a constant value
which gives the Koṇa-Śaṅku.
Comm. Bhāskara gives later the method of obtaining
H cos z ie. the Śaṅku pertaining to any zenith-distance.
So, he need not have given a separate treatment for this
Koṇa-Śaṅku. But in as much as Brahmagupta and other
previous writers gave it he has also given the same. He
gives here the method of finding the Koṇa-Śaṅku by the
method of successive approximations as given by Śrīpati.
32

250
We saw before that Agrā = Śaṅku-tala + Śaṅku-
bhuja. So, in the first place as a first approximation,
Agrā is taken as Śaṅku-bhuja. Since, when z = 45° the
perpendiculars from the celestial body on the planes of
the prime-vertical as well as meridian are equal, and since
the perpendicular on the plane of the prime-vertical is
called Śaṅku-bhuja = b (say) 2b² = H sin² z. This is so
because H sin² z = Sum of the squares of the perpendiculars
on the planes of the meridian and prime-vertical. But
H sin² z = R² − H ocs² z. ∴ R² − 2b'² = H cos² z. So, taking
Agrā as the bhuja b as a first approximation, √R² − 2b²
gives us H cos z. From this using formula III under
latitudinal triangles namely H cos z × s/12 = Śaṅku-tala,
obtain the approximate Śaṅku-tala, from the approximate
H cos z got above. Now using the formula Agrā =
Śaṅku-tala + Śaṅku-bhuja obtain Śaṅku-bhuja as Agrā ⩲
Śaṅku-tala, where the +ve sign is taken when the Sun
has a southern declination, and the difference sign when
the declination is north. Taking this Śaṅku-bhuja, b',
Koṇa-Śaṅku is now √R² − 2b'². In the first place we
took the Agrā itself as the bhuja; but here we have a
better approximation for the Śaṅku-bhuja. From this
Koṇa-Śaṅku again, obtain a still better approximation for
Śaṅku-bhuja and proceeding thus till a constant value is
obtained, we have the required Koṇa-Śaṅku. This is a
beautiful example where the method of successive approxi-
mation was used by the Hindu Astronomers to a good
advantage. It will be noted here that the Śaṅku-tala is
always treated as extending south ie. the Śaṅku-tala will be
south of the Śaṅku since India's latitudes are all north.
Also it is said that when the Sun has southern declination,
A + S = B and when northern A − S = B when A =
Śaṅku-Agrā or simply Agrā (in contradistinction to
Karṇāgrā reduced to a circle of radius K the chayākarṇa)
S = Śaṅku-tala and B = Śaṅku-bhuja. This convention

251
of signs is to be correlated with the modern. We have
from the formula derived out of PZS, A = S + B.
According to modern convention when δ is north, it will be
taken to be positive and when a is to the north of the East
point it also will be taken to be positive so that, (1) when
δ is north and a north, A = S+B ie. B = A-S; this accords
with the Hindu convention namely ‘सौम्येऽन्तरम्’ (2) when
δ is south and a south, - A = S - B so that B = A + S;
this also accords with the Hindu convention, namely
याम्ये योग: (3) But, however, when δ is north and a
south ie. when the Sun having northern declination comes
to the south of the prime-Vertical, A = S - B so that
B = S - A. This accords with the Hindu convention
if only we take B = | A-S | when δ is north.
Bhāskara makes two statements at the end of the
commentary under this verse namely that when δ is south
and A > 2431, there will be no Koṇa-Śaṅku and that
when δ is north and s > 17'' - 5''' there will be four
Koṇa-Śaṅkus. We have to verify these statements. The
first statement is evident because H sin (Agrā) > H sin 45°

252
as may be seen by taking δ = 20°, φ > 61° - 6'. Thus
Bhāskara gave the minimum latitude which could enjoy
four Koṇa-Śaṅkus.
Fig. 41
Fig. 42

253
From fig. 41, it is clear that there are two Koṇa-
Śaṅkus at S₁ and S₂ during the forenoon and similarly
two at S₁' and S₂' in the afternoon where S₁' and S₂'
are the symmetrical points of S₁ and S₂. From fig. 42, it
is clear that if Agrā < H sin 45° when δ is south, there
will be one Koṇa-Śaṅku in the forenoon at S₁ and one in
the afternoon at the symmetrical S₁'.
Verses 31 and 32. H cos z at noon known as
Dinārdha-Śaṅku.
By 'northern hemisphere' it is meant that the Sun
is in the northern hemisphere ie. his Sāyana longitude
ie. modern longitude lies between 0° and 180°, and 'the
southern hemisphere' means that the Sun's longitude lies
between 180° and 360°. The direction of δ may be got
from the above convention. The latitude and colatitude
are always deemed as south and north respectively.
The latitude and colatitude being 'added to subtracted
from or being decreased by' as the case may be, the
declination, we have the zenith-distance and the altitude
of the celestial body at Noon. The zenith-distance and
the altitude are mutually complements.
Comm. In Hindu Astronomy the words “उत्तरगोले”
“दक्षिणगोले” are very often used to connote that the Sun is
on the north or the south of the celestial equator respec-
tively, so that the declination could be automatically known
to be north or south respectively. Regarding the latitude,
the peculiarity in Hindu Astronomy is that what we call
north latitude in modern astronomy is construed as south
in as much as the celestial equator gets depressed south
in northern latitudes. The colatitude SQ in fig. 41 on the
other hand extends north from the south, so that, it is
construed as north.
The word 'Saṃskāra' is used in Hindu Astronomy
in the meaning given above in the translation. Fo

254
example in the equation A = S + B, we say that the
Bhuja is had by a Samskāra between A and S. The
meaning of Samskāra given by Bhāskara is “समदिशोर्योगः
भिन्नदिशोरन्तरम् संस्कारः” Latitude being regarded as southern,
if the Sun's declination is 12° north and the latitude 20°,
then as they are of opposite direction, effecting the
Samskāra as directed 20 - 12 = 8 = zenith-distance
(South) = Nata as it is called similarly 70 + 12 = 82 =
Altitude = Unnata; here we have added because, both
lamba and declination are north. Similarly when δ = 24°
north, and φ = 20° as before (south)
24 - 20 = 4° = zenith-distance (north) = Nata
70 + 24 = 94 = unnata (north). But, we take
180 - 94 = 86°.
In the above working in the first case we found
φ - δ, whereas in the second we found δ - φ. This
difference in treatment is not taken objection to, since,
the word Antara is used to take the positive value of the
difference alone and so in the first instance the nata is
pronounced as south, whereas in the second it is pro-
nounced north.
In modern astronomy, however, we have the formula
z + δ = φ, considering z as positive if south, δ and φ
positive if north. Here 8° + 12° = 20° (first case cited
above) and (- 4°) + 24° = 20° (2nd case, z being negative,
for, it is north. In the Hindu symbolism we have to
pronounce separately when z is south or north, whereas in
modern symbolism the sign alone informs its direction.
Similarly in the equation A = S + B, we have to pro-
nounce ‘north bhuja’ or ‘south bhuja’ as the case may be,
whereas having a convention that δ is + ve when north,
and also the Hindu azimuth (measured from the East
point) the sign of bhuja indicates its direction. In other
words we differentiate the two cases A - S and S - A
giving them signs and deducing the direction of the bhuja

255
from the sign itself without an appeal to a picture or
without ascertaining whether the northern Agrā prevails
over the Southern Saṅku-tala or the Southern Saṅku-tala
prevails over the northern Agrā. Thus the Dinārdha
Saṅku in symbolism = H cos (φ ± d) (20).
Verse 33. Here at noon, Dṛg-jyā is the H sine of
nata and the Saṅku is H sine of unnata.
Second half of 33 and first half of Verse 34.
The product of R and the unmandala-Saṅku divided
by Charajyā is called Yaṣṭi. The Yaṣṭi increased by
Un-mandala-Saṅku gives H cos z according as the Sun is
north or south of the equator.
Comm. Unmandala-Saṅku is H cos z when the Sun
is on the unmandala. From the sixth latitudinal triangle,
wherein Unmandala-Saṅku is Bhuja and Krāntijyā
Karṇa, so by comparing with the second latitudinal
triangle (or rather operating with the second triangle to
signify the Hindu method).
(Krāntijyā × Bhuja) / Karṇa = Unmandala-Saṅku
= (H sin δ × H sin φ) / R (already derived under (19)).
We saw before Charajyā = R tan φ tan δ. Hence as
directed in the verse (R × H sin φ H sin δ) / (R × R tan φ tan δ) = Yaṣṭi
= (H cos φ H cos δ) / R (21).
∴ H cos z (at Noon) = (H cos φ H cos δ) / R ±
(H sin φ H sin δ) / R according as the Sun is on the north or
south of the equator.

256
Hence H cos z (at Noon) = Natajyā
= (H cos φ H cos δ ± H sin φ H sin δ) / R or in modern
symbolism cos (φ ∓ δ) already derived under (20).
We shall now show how the formulation is done by
the simple rule of three (Ref. fig. 39). If a parallel
through o₄d is drawn to cut Aa at a₁, then Aa₁ is called the
Yaṣṭi, which is vertical. The triangles Aoa₁ and Do₄d
are similar so that Aa₁ / Ao = Dd / Do₄ ∴ Aa₁ = (DD × Ao) / Do₄ .
But Ao / Do₄ = R / Charajyā for, all the lines of the diurnal
circle and the equator stand in the ratio
(H cos δ) / R = Ao / R = Do₄ / Charajyā = Kujyā / Charajyā ∴ Ao / Do₄ = R / Charajyā
∴ Aa₁ = (Unmandala Śaṅku × R) / Charajyā = Yaṣṭi as
formulated.
Now Dinardha Śaṅku = Aa = Aa₁ + a₁ a = Aa₁ + Dd
= Yaṣṭi + Unmandala Śaṅku. It is evident from fig. 21
why in the northern sphere the sum is to be taken
whereas in the southern, the difference is to be taken.
Latter half of verse 34. Definition of Hṛti and
Antyā.
The sum or difference of Dyujyā and Kujya will be
similarly Hṛti, whereas the sum or difference of Charajyā
and radius will be Antyā.
Comm. We defined formerly Hṛti and Antyā under
our commentary on the latitudinal triangles. From fig. 39
Hṛti = o₁A = o₁o + oA = o₄D + oA = Kujya + H cos δ
(Dyujyā) (22).

257
In the parallel great circle, the Equator we have
therefore Antyā = Charajyā + R (already derived).
Verse 35. Antyā = (Hṛti × R) / (H cos δ) = (Hṛti × Charajyā) / Kujyā and
∴ Hṛti = (Antyā × H cos δ) / R = (Antyā × Kujyā) / Charajyā by what is
called Guṇa-ccheda-Viparyaya ie. alternando.
Comm. Evident.
Verse 36. To obtain Dinārdha-Śaṅku from Antyā
and Hṛti
(Antyā × un-maṇḍala Śaṅku) / Charajyā = (Hṛti × 12) / K = Dinārdha-
Śaṅku.
Comm. The second formula is derived from the
similarity of △ Ao₄a (fig.) with the first latitudinal triangle.
From the similarity of Aao₄, Ddo₄ fig. 39,
Ao₄/Do₄ = Aa / Dd ∴ Aa = (Hṛti × unmaṇḍala-Śaṅku) / Kujyā .
But Aa = Dinārdha-Śaṅku
∴ Dinārdha Śaṅku = (Hṛti × U.S.) / Kujyā (U. S. = Un-
maṇḍala-Śaṅku.)
But Hṛti / Kujyā = Antyā / Charajyā
∴ D.S. (Dinārdha-Śaṅku) = Charajyā / Antyā × U.S.
= (Hṛti × 12) / K VIII.
Verse 37. Meridian Zenith distance.
Agrā ± (Hṛti × bhuja of a lat. triangle) / (Karṇa of a lat. triangle) = H sin z
where z is the meridian zenith distance.
33

258
Comm. This formula is a special case of the formula
A = S + B since the H sine of the meridian zenith
distance is the Śaṅku-Bhuja at noon. From fig. 39
(Hṛti × bhuja of a latitudinal triangle) / (Karṇa of a latitudinal triangle) = O₁a = Dinārdha
Śaṅkutala.
The operation of sign has been already explained.
Verse 38. An alternative method.
The meridian zenith distance of the Sun can be had
also by the formula (Hṛti ± Taddhṛti) B.L.T. / K.L.T. where
B.L.T. and K.L.T. are the bhuja and karṇa of any
latitudinal triangle.
Comm. (Ref. figures 43 and 21). Let E₁ be the
centre of the armillary sphere so that QE₁R is the
diameter of the celestial equator which is on the median
plane. Let S₁ F₁ S be the diameter of the diurnal circle
of the Sun, which is also on the meridian plane so that
S₁ F₁ is the Taddhṛti, S₁ S is the Hṛti and S the position
of the Sun on the meridian.
H sin z = SM = SF₁ sin F̂₁ = SF₁ sin ϕ =
(Hṛti − Taddhṛti) × s/K where s/K can be replaced by
B.L.T. / K.L.T. (s=equinoctical shadow and k the Viṣuvat-Karṇa).
In the Southern sphere, H sin z = S′N = S′F₁′ sin ϕ
= (S′S₂ + S₂F₁′) sin ϕ = (Hṛti + Taddhṛti) × s/k.
Verse 39. Still another way of obtaining the m. z. d.
(meridian-zenith-distance).
R − H versin (altitude) = H sin z

259
Fig. 43
This formula gives not only the meridian zenith
distance but H sine of the zenith-distances of the Sun
when he is on the Koṇa-Vṛtta or prime-vertical or
unmaṇḍala.
Comm. From fig. 43, H sin z = SM = LE₁ = E₁s −
sL = R − H versine (Ss) = R − H versine (altitude) as
given. Since R − H versine (altitude) = R − {R −
H cos (90 − z)} = R − (R − R sin z) = H sin z, so this
formula applies wherever the Sun be. This is almost
begging the question as H sine of z is being sought
through H versine of (90 − z).

260
First half of the verse 40. To obtain the shadow S
and K the Chāyakarṇa of any shadow
(H sin z × 12) / (H cos z) = S and (R × 12) / (H cos z) = K.
Comm. (Ref. fig. 44).
(12. H sin z) / (H cos z) = 12 tan z = S. Also 12 / K = cos z = (H cos z) / R
so that (12 R) / (H cos z) = K.
The Hindu method of looking at this through the
similarity of ΔS OM☉ and Ogn the gnomonic triangle.
is as follows. ☉M is called Mahā-Śaṅku ie. H cos z ; ☉L
is Dṛkjya or
H sin z = OM. 12 / (H cos z) = S / (H sin z) so that S =
12 H sin z/H cos z. Also,
On / O☉ = 12 / (H cos z) ie. K / R = 12 / (H cos z) ∴ K = (12 R) / (H cos z) .
It will be noted that fig. 44 pertains to any vertical plane.
Second half of Verse 40. The Dinārdha-Karṇa is
equal to (R × k) / Hṛti where k is the Viṣuvat-Karṇa.
Comm. The formula is derived through twice apply-
ing the rule of three or what is the same, through the
similarities of two sets of triangles
From fig. 39, O₁A / Aa = Hṛti / Dinārdha-Śaṅku = k / 12 (a)
and from fig. 44 On / O☉ = 12 / Dinārdha-Śaṅku = K / R (b)
where K is the required Chāyākarṇa.
Dividing (a) by (b)
Hṛti / 12 = k / 12 × R / K ∴ K = kR / Hṛti

261
Fig. 44
Verse 41. Alternate method of obtaining K
101530/H sin λ = para (say) where λ is the Sāyana
longitude of the Sun ; then, (Para × k) / s = K where K is
un-mandala-Karṇa
First half of Verse 42. To obtain K when the Sun is
on the prime-vertical— Para × s/k = Samavṛttakarṇa.
Comm. From fig. 19, from the similarity of triangles
BD ☉ and CMA, B ☉ / CA = ☉ D / AM ie. ☉ D = (B ☉ × AM) / CA ie.
H sin δ = (H sin λ H sin ω) / R (a)
Then consider the similarity of the first and the sixth
latitudinal triangles ; then Unmandala Sanku / Krāntijyā = s / k (b)
where s is the equinoctial shadow and k the Viṣuvat-
Karṇa. Again taking that ☉ the Sun lies on the unman-

262
dala in figure 44, (H cos z) / 12 = R / K = (Unmandala Śaṅku) / 12 (c)
Eliminating Krāntijyā and Unmandala Śaṅku from
(a), (b) and (c) (Unmandala Śaṅku) / (H sin λ H nis ω/R) = s / k
∴ (12R / K) / (H sin λ H sin ω / R) = s / k ie. 12R² / (KH sin λ H sin ω) = s / k
∴ K = (12R² × k) / (s H sin λ H sin ω). Here 12R² / (H sin ω) = (12 × 3438²) / 1397
= 101531 ; but Bhāskara has taken 101530 taking a
more correct value of R. Then 101530 / (H sin λ) is symbolized as
para so that para × k / s = K = Unmandala Karṇa. Regard-
ing the Samavṛttakarṇa, in the place of (b) above we have
(Sama-Śaṅku) / Krāntijyā = k / s (b') by the similarity between the
first and the fifth latitudinal triangles. Equation (c)
holds good with respect to any H cos z and the corres-
ponding K since 12 R = K × Śaṅku and 12 R is a cons-
tant. Noting therefore R / K' = (Sama-Śaṅku) / 12 (c')
eliminating Krāntijyā and Sama-Śaṅku among (a), (b'),
(c'), we shall have
K = (12 R² s) / (k H sin λ H sin ω) = para × s / k as stated.
Second half of Verse 42. To obtain the Dinārdha-
karṇa from the Unmandalakarṇa.
(Un-mandalakarṇa × Charajyā) / Antyā = Dinārdhakarṇa.
Comm. We have equation (c) above stating
12 R = K × Śaṅku. (c) But

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Iṣṭa Śaṅku / Iṣṭa Hṛti = cos φ = constant = Dinārdha Śaṅku / Hṛti
= Sama-Śaṅku / Taddhṛti = Unmaṇḍala Śaṅku / Kujyā (d) (23)
Again by virtue of the proportionality of
Iṣṭa Hṛti / Iṣṭāntyā = Hṛti / Antyā = Kujyā / Charajyā (e) (24)
We have Ishta-Śaṅku / Ishtāntyā = Dinārdha Śaṅku / Antyā
= Unmaṇḍala Śaṅku / Charajyā
∴ Iṣṭa Karṇa × Iṣṭāntyā = Dinārdha Karṇa × Antyā
= Unmaṇḍala Karṇa × Charajyā (f) (25)
∴ Dinārdha Karṇa = (Unmaṇḍala Karṇa × Charajyā) / Antyā
as stated in the verse.
Verse 43. (Unmaṇḍala Karṇa × Kṣitijyā) / Hṛti
= (Sama Vṛtta Karṇa × Taddhṛti) / Hṛti = Dinārdha Karṇa.
Comm. From (c) and (d) above Dinārdha Karṇa ×
Hṛti = Sama Karṇa × Taddhṛti = Unmaṇḍala Karṇa ×
Kujyā (g). Khitijyā is the same as Kujyā.
From this the statement follows :
Verse 44. The ancient Achāryas found the gnomonic
shadows when the Sun is on the meridian, prime-vertical
and the Kona-Vṛtta (ie, Vertical when the northern or
southern Hindu azimuths are 45°) by different methods.
I consider him to be the very Sun illuminating the lotus-
faces of aitronomers, if anybody could give a method to
find the shadow in any required direction, which holds
good in all cases universally.