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

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∴ b = KA / R - s. But (H sin a) / b = R / S where S is the
shadow so that
S = bR / (H sin a) = (KA / R - s) R / (H sin a) = √(K² - 12²)
∴ (KA - sR) = H sin a √(K² - 12²)
ie. K² A² + s² R² - 2 AsRK = H sin² a (K² - 12²)
∴ K² (A² - H sin² a) - 2 AsRK = - s² R² - 12² H sin² a
∴ K² (H sin² a - A²) + 2 AsRK = 12² H sin² a + s² R² I
This quadratic is of the form ax² + 2bx = c
ie. x² + 2 (b/a) x = c/a II Here 'c'/a is called Adya and b/a Anya.
The solution of the above equation is given by x = - b/a ±
√(b²/a² + c/a)
ie. - Anya ± √(Anya² + Adya). III
When b = KA / R + s, putting - s in the place of s in I,
K = Anya ± √(Anya² + Adya) IV
Out of the four solutions given by III and IV we have
taking the positive solutions
K = √(Anya² + Adya) ± Anya as stated.
Verse 51. If H sin a < A, then in the northern
hemisphere ie. where δ is north,
± √(Anya² - Adya) + Anya = K.
Comm. We have initially put Anya = H sin² a - A².
If H sin a < A, them to avoid a negative value for the

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Anya, we could put Anya = A² - H sin² a. As a matter
of fact in verse 50, we are asked to take H sin² ~ A²,
as Bhāskara wanted that the second case also be included.
Thus putting Anya = A² - H sin² a, equation I becomes
K² (A² - H sin² a) - 2 AsRK = - (Adya) so that
K = Anya ± √(Anya² - Adya) as given.
Verse 52. The Bhuja is to be obtained through
Karṇāgra from the equation a = b + s (already proved)
and Rb / S = H sin a {ie. S = bR / H sin a as already proved}.
This H sin a will be the same in the case of obtaining
two values of K ie. two shadows one in the morning and
the other in the afternoon, (the only difference being that
they will be on alternate sides of the East-West line).
Verses 53 and 54. Obtaining the shadow when time
is given.
In the two previous examples the magnitude of the
shadow was obtained in a given direction ; now we shall
obtain the same when time is given. The word unnata
stands for the time that has elapsed after Sun-rise or that
which is the balance of the day time. The unnata sub-
tracted from half-the-day gives what is called Nata.
The H sine of the unnata minus Chara or increased
by the Chara according as the Sun's declination is north
or south, is called Sūtra ; this multiplied by the H cos δ
and divided by the radius, is called Kalā.
Comm. (Ref fig. 51) The time measured by the arc
MN, that is the time in between the moment when the Sun
S is on the horizon and the moment when he is at L is
called the unnata ie. the time measured after rising and
the time measured by the arc NQ ie. the time in between

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Fig. 51
the moments when the Sun is at L and when he reaches
the meridian. is called Nata. In modern astronomy this
Nata is known as the hour angle h and unnata = H - h
where H is the rising hour angle. The time measured by
the arc ME is called Chara. Thus unnata-chara = EN
and H sin EN H sin (90 - h) = H cos h is called Sutra
which is BN shown in fig. 52 representing the Equator.
The corresponding line bn in fig. 53 which represents the
diurnal circle, is known as Kalā. Thus Sūtra = H cos h (26)
and Kalā = (H cos h × H cos δ) / R (27).
When the Sun is in the Southern hemisphere, unnata
is measured by the arc SL or MN ie. (H - h) where H is

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Fig. 52
Fig. 53

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the rising hour angle and ℎ = L P̂ Q and Chara by the arc
EM and Sūtra = H sin (MN + EM) = H sin (unnata +
chara) = H sin EN = H sin (90 - ℎ) = H cos ℎ.
Verse 55. Sūtra multiplied by Kujyā and divided by
Charajyā will be also Kalā and Kalā multiplied by the Koti
and divided by the Karṇa of any latitudinal triangle will
be Iṣṭa yaṣṭi.
Comm. In as much as Kalā is the corresponding line
in the diurnal circle to the Sūtra in the plane of the
celestial equator
Sūtra / Kalā = R / (H cos δ) = Charajyā / Kujyā ∴ (Sūtra × Kujyā) / Charajyā = Kalā
In verse 33, we saw that
Dinārdha Sanku — Unmandala Sanku = Yaṣṭi.
This Iṣṭa-yaṣṭi will be therefore the perpendicular dropped
from 𝑛, the Sun's position in the diurnal circle on the
plane parallel to the horizon and passing through the
head of the Unmandala Sanku ie. passing through B and
parallel to the horizon in fig. 21. Since the angle between
the diurnal plane and the vertical plane of the yaṣṭi is
equal to φ the latitude, the Iṣṭayaṣṭi forms a latitudinal
triangle with the Kalā, it being the Koti or side opposite
to the angle 90 - φ and the Kalā being the hypotenuse,
∴ Yaṣṭi / Kalā = cos φ = Koti of a latitudinal triangle / Karṇa of a latitudinal triangle
∴ Iṣṭayaṣṭi = Kalā × (Koti of a latitudinal triangle) / Karṇa
When the Sun is on the meridian, Iṣṭayaṣṭi becomes yaṣṭi
of verse 33.
The formula for Iṣṭayaṣṭi is therefore
(Kalā × H cos φ) / R = (H cos δ · H cos ℎ) / R × (H cos φ) / R
from formula (27)
= (H cos φ · H cos δ · H cos ℎ) / R² 27'

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First half of verse 56. The Unmandala S'anku
multiplied by the Sūtra and divided by Charajyā is also
Iṣṭayaṣṭi.
Comm. The Unmandala S'anku and Iṣṭayaṣṭi are
the lines in vertical planes corresponding to Charajyā and
Sūtra in the Equatorial plane. Hence the proportion. It
will be noted that the Unmandala S'anku and Iṣṭayaṣṭi
are not in the same vertical plane but parallel vertical
planes. None the less the proportionality holds good.
Latter half of verse 56 and first half of verse 57.
The Sūtra increased or decreased by the Charajyā
according as the Sun is in the northern or southern hemi-
sphere is what is known as Iṣṭāntyā ; similarly the Kalā
increased or decreased by Kujyā is what is known as Iṣṭa-
Hṛti.
Comm. In fig. 52, Iṣṭāntyā = AN = AB + BN =
ME + BN = Charajyā + Sūtra. Similarly in fig. 53,
Iṣṭa Hṛti = an = ab + bn = sg + bn = Kujyā + Kalā.
∴ Iṣṭāntyā = H cos h + R tan δ tan φ =
R (sin φ sin δ + cos φ cos δ cos h) / (cos φ cos δ) in modern terms (28)
Latter half of verse 57. Similarly Iṣṭayaṣṭi
increased or decreased by the Unmandala S'anku is Iṣṭa
S'anku or H cos z.
Comm. Let in fig. 54 which represents the plane of
the prime-vertical AA', EW, BB', FF', qq' represent the
lines of intersection of this plane with planes parallel to
the horizon and passing through A, B, F, q of fig. 21. Then
Oa = Unmanda-S'anku, Oβ = Sama S'anku, Or =
Dinārdha S'anku. If xx' be the line of intersection of this
plane of the prime-vertical with a plane passing through
an arbitrary position of the Sun in the diurnal circle and
36

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Fig. 54
parallel to the horizon. then Ox = Iṣṭa-S'anku = Oα +
ax = Un-mandala S'anku + Iṣṭa yaṣṭi. ar = yaṣṭi.
In the Southern hemisphere Oα, the Unmandala
S'anku will be below the horizon so that Iṣṭayaṣṭi
decreased by the Unmandala S'anku will be Iṣṭa-S'anku.
Thus we have the method of obtaining the Iṣhta-
S'anku from the Unnata Kāla as detailed above. We shall
see what this process means in modern terms.
Unnatakāla-Charakāla = ⊙P̂A − AP̂E (Fig. 21) =
⊙P̂E where ⊙ is the foot of the declination circle of the
Sun e in any arbitrary position in his diurnal path. But
⊙P̂E = QP̂E − QP̂⊙ = 90−h where h is the hour angle
of the Sun. Thus Sutra = H sin (90−h) = H cos h I

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∴ Kalā = (H cos h × H cos δ) / R II
∴ Iṣṭa-yaṣṭi = [(H cos h × H cos δ) / R] × [(H cos φ) / R] III
Now Unmaṇḍala-Śaṅku is derivable from the sixth
latitudinal triangle in which Krāntijyā is the Karṇa and
Unmaṇḍala-Śaṅku is the Bhuja. Comparing it with the
second latitudinal triangle Krāntijyā / R = U. S. / (H sin φ) where
U. S. is Unmaṇḍala-Śaṅku.
∴ U. S. = (Krāntijyā × H sin φ) / R = (H sin φ H sin δ) / R IV
∴ Iṣṭa-Śaṅku as per the above formulation is given by
Iṣṭa-Śaṅku (I. S.) =
(H sin δ H sin δ) / R + (H cos φ H cos δ H cos h) / R² = H cos z V
[चित्र: गोलीय त्रिभुज PZS (Spherical triangle PZS) — शीर्ष: Z (zenith), P (celestial pole), S (celestial body / Sun); भुजाएँ: ZP = 90 - φ, ZS = z, PS = 90 - δ; कोण: ∠PZS = 90 - A, ∠ZPS = h, ∠PSZ = η]
(Ref. fig. 55) Fig. 55 Formulae for △PZS
Z = zenith ; P = celestial pole. S = celestial body, say,
the Sun.
z = zenith-distance of the celestial body S.

284
PZ = colatitude ; PS = north-polar-distance or co-decli-
nation.
η is called the parallactic angle ; h = hour-angle of S.
(90 — a) = The complement of the Hindu
azimuth a being measured from the East point.
cos (90 — δ) = cos (90 — φ) cos z + sin (90 — φ) sin z cos (90 — a)
sin δ = sin φ + cos z + cos φ sin z sin a (1)
cos z = cos (90 — φ) cos (90 — δ) + sin (90 — φ) ×
sin (90 — δ) cos h
= sin φ sin δ + cos φ cos δ cos h (2)
cos (90 — φ) cos (90 — a) = sin (90 — φ) cot z — sin (90 — a) cot h
ie. sin φ sin a = cos φ cot z — cos a cot h (3)
cos (90 — φ) cos h = sin (90 — φ) cot (90 — δ) — sin h ×
cot (90 — a)
ie. sin φ cos h = cos φ tan δ — sin h tan a (4)
sin z sin (90 — δ)
─────── = ───────────── ie. sin z cos a = sin h cos δ (5)
sin h sin (90 — a)
This means in modern terms
cos z = sin φ sin δ + cos φ cos δ cos h V which we
derive from the triangle PZS.
Verse 58. To get H cos z from h the hour-angle or
nata Kāla.
The H. vers (Nata) is called Sara (CQ of fig. 52) (29)
Antyā — Śara = Iṣṭāntyā ie. FQ — CQ = FC = AN
(fig. 51)
H cos δ Śara × Kujyā
Śara × ───────── = ───────────── = phala (CQ) (fig. 53)
R Charajyā
(30)

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Hṛti — phala = Iṣṭa-Hṛti ie. fq — cq = fc = an
(fig. 53)
Comm. (Ref. fig. 52). Nata = arc QN
∴ H. vers (Nata) = QC. (Called Śara) = H. vers (h)
Antyā — Śara = FQ — CQ = FC = AN = Iṣṭāntya II
Śara × (H cos δ / R) = phala = cq (fig. 53) = (Hvers h × H cos δ) / R
Hṛti — phala = fq — cq = fc = an = Iṣṭa-hṛti IV
Verse 59. (Phala × Koti of a latitudinal triangle) / Karṇa
= Ūrdhwa V
= βr (of fig. 54)
Comm. (Ref. fig. 54) βr is the corresponding line in
the plane of the meridian corresponding to phala in the
plane of the diurnal circle. As the angle between these
two planes is the latitude itself, by the principle of ortho-
gonal projection namely.
Magnitude of a projected segment = cosine of the
dihedral angle × the magnitude of the segment projected,
since Ūrdhwa is the orthogonally projected segment of
phala, so, Ūrdhwa = phala × cos φ
= (phala × H cos φ) / R VI (31)
= (phala × Koti of a latitudinal triangle) / (Karna of the latitude triangle) as stated
Thus Ūrdhwa = βr (of fig. 54).
Verse 60. Ūrdhwa is also given by
Ūrdhwa = [U.S. (Unmandala Śanku) × Śara] / Charajyā
Dinārdha-Śanku (D.S.)—Ūrdhwa = Iṣṭa-Śanku (I.S.) =
H cos z.

286
Comm. Since U.S. is the projected segment of
Charajyā on a vertical plane and since the diahedral angle
between the planes is φ the latitude
U.S. / Charajyā = Cos φ and so Śara × cos φ = (Śara × H cos φ) / R
= Ūrdhwa.
From fig. 54, Dinārdha-Śanku — Ūrdhwa = o'q — βr
(fig. 54) = o'β' = yX = Iṣṭa-Śanku.
In modern times, this means,
Śara = H. vers (h) = (R — cos h) = 1 — cos h in modern terms
phala = (Śara × H cos δ) / R = (1 — cos h) × cos δ ,, ,,
phala × (H cos φ) / R = Ūrdhwa = (1 — cos h) cos δ × cos φ ,, ,,
Dinārdha Śanku — Ūrdhwa = Iṣṭa Śanku H cos z
= cos z (in modern terms)
= H cos (φ — δ) (taking northern declination and following
Hindu convention with respect to signs)
= Dinārdha·Śanku
= cos (φ — δ) in modern terms.
∴ cos (φ — δ) — (1 — cos h) cos φ cos δ = cos z
ie. cos φ cos δ + sin φ sin δ — cos φ cos δ + cos φ
cos δ cos h = cos z
ie. cos z = sin φ sin δ + cos φ cos δ cos h as before.
Verse 61. Computation of H cos z (Mahā Śanku)
through Antyā and Hṛti.
Let the Dinārdha Śanku (D.S.) be computed through
Iṣṭāntyā and Iṣṭa Hṛti and therefrom Iṣṭa Śanku. From
the Śanku, Drik·jyā ie. H sin z and the shadow (KH sin z) / R
could be computed —H sin z should not be computed from
Hṛti.

287
Comm. We computed Dinārdha Śaṅku by the formula.
D.S. = (Antyā × U.S.) / Charajyā = (Hṛti × Koṭi of a L.T.) / (Karṇa of L.T.)
under verse 36 ; similarly
Iṣṭa Śaṅku (I.S.) will be given by
Iṣṭa Śaṅku = (Iṣṭāntya × U.S.) / Charajyā = (Iṣṭa Hṛti × Koṭi of a L.T.) / (Karṇa of the L.T.)
But Iṣṭāntyā = [R (sin φ sin δ + cos φ cos δ cos h)] / [cos φ cos δ]
(as under verse 56)
∴ Iṣṭa Śaṅku = [R (sin φ sin δ + cos φ cos δ cos h)] / [cos φ cos δ]
× (R sin δ sin φ) / (R tan φ tan δ) from formulae (13) and (19)
= R (sin φ sin δ + cos φ cos δ cos h) = R cos z = H cos z
Or again Iṣṭa Hṛti = (R cos z) / (cos φ) from formula (11)
∴ Iṣṭa Śaṅku = (R cos z / cos φ) × (H cos φ / R) = R cos z = H cos z
Having got H cos z, using the formula H sin² z = R² —
H cos² z, H sin z ie. Dṛk-jyā can be computed. Also K =
12R / (H cos z) and S = (KH sin z) / R give the Chāyā Karṇa and Chāyā.
Bhāskara cautions us that H sin z cannot be computed
from Hṛti as mentioned in verse 37. because there in that
verse, the H sin z computed is that at noon alone.
Verse 62. Alternate method of obtaining K.
The Chāyākarṇa when the Sun is on the unmaṇḍala
multiplied by Kujyā or that when the Sun on the prime-
vertical multiplied by Taddhṛti, or again that when the
Sun is on the meridian multiplied by Hṛti, divided by
Iṣṭa Hṛti, will be equal to the Iṣṭa-Karṇa K.

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Comm. Equation (23) under verse 41 is
(Iṣṭa Śaṅku / Iṣṭa Hṛti) = (D.S. / Hṛti) = (S.S. / Taddhṛti) = (U.S. / Kujyā) = cos φ I
and 12 / K = (H cos z) / R (under verse 40)
ie. K = 12R / Iṣṭa Śaṅku which means
Dinārdha Karṇa = 12 R / D.S. ; Sama Karṇa = 12R / S.S.
and unmaṇḍala Karṇa = 12 R / U.S.
Substituting for the numerators in I
12R / (K × Iṣṭa Hṛti) = 12R / (DK × Hṛti) = 12R / (S.K × Taddhṛti)
= 12R / (U.K. × Kujyā) II
∴ Iṣṭa Karṇa × Iṣṭa Hṛti = S.K. × Taddhṛti
= U.K. Kujyā
∴ Iṣṭa Karṇa = (Uumaṇḍala Karṇa × Kujyā) / Iṣṭa Hṛti
= (Sama Karṇa × Taddhṛti) / Iṣṭa Hṛti = (Dinārdha Karṇa × Hṛti) / Iṣṭa Hṛti
Verse 63. Just a caution.
If in any context where the word ūna-yuta has been
used, the quantity to be subtracted exceeds the quantity
from which it is to be subtracted, it goes without saying
that subtraction should be reversely effected and in the
place of addition subsequently prescribed subtraction should
be done and Vice-versa.
Comm. Bhāskara gives three examples to illustrate
his point. In verse 54, while defining Sūtra (H sin 90 – h)
we are asked to subtract chara from unnata when δ > 0

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Fig. 56
and add Chara to Unnata when δ < 0 and take the H sine
of the result. Let us first consider the case when δ > 0.
(Refer fig. 56) when the Sun is at ☉, Iṣṭa Śaṅku is H sin ☉L;
and Unmandala Śaṅku is H sin BM. When the
Sun is at ☉₁, Iṣṭa Śaṅku = H sin ☉₁N. In the first case
Iṣṭayaṣṭi = (H sin ☉L — H sin BM) which will be the
orthogonal projection of ☉B on the meridian plane.
Unmandala Śaṅku and the Iṣṭa Śaṅku in the position ☉₁
are similarly the orthogonal projections of BM and ☉₁N
on the same plane. In the position ☉ Iṣṭa Śaṅku =
Unmandala Śaṅku + Iṣṭayaṣṭi, whereas in the position
☉₁, Iṣṭa Śaṅku = Unmandala Śaṅku — Iṣṭayaṣṭi which is
now downwards. Thus in the place of addition we have
subtraction of Iṣṭayaṣṭi. This reversion has arisen out of
37

290
the fact that in the position ☉, Sūtra is the H sine of
(KA — KE) whereas in the position ☉₁ Sūtra is the H sine
of EC ie. H sine of (KE — KC) ie. in the former position
Sūtra = H sine (Unnata-Chara) and in the lattter Sūtra =
H sin (Chara-Unnata). Thus a reversion in subtraction
here, effects a reversion of addition of the Iṣṭayaṣṭi.
Similar is the case in the other cases cited by Bhāskara.
Analytically this happens so because cos h, when h > 90,
becomes negative and adding cos h tantamounts to
subtracting sin θ where h = 90 + θ.
Verse 64. Another point to be observed.
Hvers (90 + θ) = R — H cos 90 + θ̅ = R + H sin θ.
The Unmandala Sanku is not observable when δ is
south in as much as it is below the horizon ; none the less
it may be computed for the purposes of effecting proportion.
Fig. 57
Hvers (CG) = Hvers (CÔG) = GH
Hvers (EG) = Hvers (EÔG) = Hvers (90 + θ)
= R + OH' = R + EL = R + H sin ☉
as defined by Bhāskara,

291
Comm. Under verse 58, we had to form Hvers h; a
doubt might arise as to what this Hvers h would be if
h > 90. Hence Bhāskara defines it and the definition is
clear from fig. 57. The Unmandala S'anku has the
formula R sin δ sin φ so that either when δ is negative or
when φ is negative, it will be negative which means it will
be in the opposite direction ie. vertically downwards.
Since negative latitudes are not considered by the Hindu
astronomers, the other case alone is considered. Even
from a figure it is evident that when the Sun is in the
south of the Equator, the Unmandala S'anku is vertically
downwards. In proportions like I given under verse 62,
we can use the magnitude of this Unmandala S'anku also
and it does not vitiate the results, when we take its
numerical value.
Verse 65. The Sun crosses the prime-vertical when
his northern declination falls short of the latitude. Then
alone there is sense in giving the magnitude of his shadow
at that moment. When the Sun does not cross the prime-
vertical at all, the Sama S'anku which could be computed
out of its formulation, though it does not exist, under the
Sun, yet, it could be used in proportions (like I under
verse 62) and no blunder is committed.
Comm. This is a beautiful example cited by Bhāskara
where he intuitively uses the so-called principle of
geometrical continuity. We have formulated Sama
S'anku as (R sin δ) / (sin φ) ie. H cos z = (R sin δ) / (sin φ) . We have a real
value of z when δ > φ, for, then only H cos z < R. When
φ > δ, then H cos z > R which is impossible, for, no
Hindu sine or Hindu cosine could be greater than R just
as no modern sine or cosine could be greater than unity.

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TERMINOLOGY
N.B.—In as much this Tripraśnādhyāya has a good number of technical terms whose understanding is
necessary to understand the Hindu methods of solving diurnal problems, we shall collect here all
such technical terms under this heading for guidance.

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294

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