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

480
Verses 11 and 12. If the Prāk-Dṛk-graha has a
longitude greater than that of the Sun or the Paschima-
Dṛk-graha has one less than that of the Sun, the sum of
the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas converted into
minutes of arc will have to be used to compute the elapsed
days or the days after which the respective phenomenon is
going to occur.
If the Iṣṭa-Kālāṁśas y be greater than the prescribed
namely x, the reverse to what has been stated in the second
half of verse (8) and in the first line of verse (9) happens
ie. a phenomenon which has elapsed when y < x will happen
in the future and vice versa.
Comm. If the Prāk-Dṛk-graha has a longitude greater
than that of the Sun, (since the prescribed Kālāṁśas are
between the planet and the Sun, the planet being behind
the Sun, and the Iṣṭa-Kālāṁśas are now on the other side
of the Sun) the planet has advanced over its position of
heliacal setting be it a superior planet or inferior by the
sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence
the days that have elapsed after the heliacal setting have
to be calculated with the sum of the two Kālāṁśas. If it
be also a case of a retrograde inferior Prāk-Dṛk-graha
rising, having a longitude greater than that of the Sun,
even then the prescribed Kālāṁśas are behind the Sun and
the Iṣṭa-Kālāṁśas ahead of the Sun. So from the position
of the planet’s heliacal rising, the Sun and the planet have
advanced in opposite direction to a distance which is the
sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence
computation has to proceed with the sum of the two
Kālāṁśas to get the time that has elapsed after the planet’s
heliacal rising. Similar is the argument for the Paśchima-
Dṛk-graha.
We have commented on verses from the latter half of
(8) upto (10) where the Iṣṭa-Kālāṁśas y happen to be less

481
than x. If y>x, or again the Prāk-Dṛk-graha has a longi-
tude greater that of the Sun, the phenomenon of rising or
setting which happened when y<x, will have to be taken
as going to happen and that which was going to happen
when y<x, must have happened already. It is enough to
consider just one case instead of all the four cases since
the argument is similar as before. Let us take y>x and
it is an inferior planet in the east behind the Sun.
According to the latter half verse (8) the rising had to
take place y being less than x, whereas, now, y being
greater than x, rising had already taken place, in contra-
distinction to what happens when y<x.
Here ends the Udayāstādhikāra.

SṚṄGONNATYADHIKĀRA
Verse 1. Either in the last quarter of a lunation, or
in the first quarter, on the day when the elevation of the
cusps of the crescent Moon is to be determined, then either
at the moment of Moon-rise or Moon-set or (for the matter
of that during any part of the night) the Hsine of the
altitude of the Moon is to be computed by noting the time
from the moment of Moon-rise.
Comm. Either in the first quarter or the last quarter
of the lunation, ie. when the phase of the Moon according
to the definition of phase in modern terms is less than
half, the Moon will be a crescent. Also, generally, on the
back-ground of the horizon, we notice that one of the
cusps is more elevated than the other. This elevation
goes by the name Sṛṅgonnati.
Even in the middle half of the lunation, Bhāskara
mentions that Brahmagupta and some others (meaning
Sripati whom he closely follows) attempted at finding the
elevation (strictly speaking elevation during the second
quarter and depression during the third quarter) of the
dark horns. Bhāskara does not appreciate this since,
nobody would think about this as it does not appeal to the
eye at all.
Verse 2. To find the Hsine of the altitude of the Sun.
The Hsine of the altitude of the Sun is to be
com-puted, assuming the rising Sun to be in the opposite
hemisphere, south or north (ie. if he be originally in the
northern, assume him to be in the south) and using the
formula given in verse 54 of Tripras'nādhyāya "अथोन्नता-
दूनयुताच्चरेण" given the time measured in asus that has
elapsed after Sunset.

483
Fig. 112
Comm. During the early part of the night or during
the latter part thereof, when the Sun is below the horizon,
the Sun will be occupying symmetrical positions with
respect to the horizon at times which are equally removed
from Sunset and the next Sun-rise. This is clear from the
figure 112. Let A and B be two such symmetrical positions
where AM and BN are the Hsines of the altitudes.
Evidently AM = BN, C P̂ A = C P̂ B. Since the rising eastern
hour-angle of the Sun equals he setting western hour-
angle, and since C P̂ A = C P̂ B, the time elapsed after
Sunset when the Sun is at B, will be equal to the time
before Sun-rise in the position A. Hence by congruence
AM = BN = Hsine of the altitudes in the two symmetric
positions. Using the modern formula from the triangle
PZS, cos z = sin ϕ sin δ + cos ϕ cos δ cos h, when
C P̂ A = C P̂ B = h cos z will be the same in the two positions
A and B. Putting z = 90 + θ, cos z = - sin θ will be
the same ie. H sin θ will be the same in the two positions
ie. the Śaṅkus will be the same. In the formula

484
cos z = sin ϕ sin δ + cos ϕ cos δ cos h
Put z = 90 + θ, h = 90 + H, δ = δ in the positions A, B
and z = 90−θ, h=90−H. δ=−δ in the positions A', B'
We have then − sin θ = sin ϕ sin δ − cos ϕ cos δ sin H
and sin θ = − sin ϕ sin δ + cos ϕ cos δ sin H
which are identical. This means that the altitude θ below
the horizon with +δ in the positions A, B is computable
with −δ in the positions A', B'. This accounts for the
statement made ‘गोलविपर्ययेण’.
The second statement of the latter half of verse (2)
says Śankutala = Śanku × ˢ/₁₂ which we have proved in
Tripraśnādhyāya.
Verse (3) and first half of (4). To obtain the Bhuja
of the Sun.
The Śankutala of the inverse altitude or the altitude
below the horizon, is north (in contradistinction to what
it is above the horizon). The sum or difference of the
Agrā and Śankutala according as they are of the same
direction or of opposite directions, is the Bhuja. The sum
or difference of the Bhujas of the Sun and Moon, according
as they are of opposite or the same directions is what is
called the Spaṣṭa Bhuja whose direction is to be construed
as that of the Moon. If the Bhuja of the Moon falls short
of that of the Sun, then the direction of the Spaṣṭa Bhuja
is that opposite to that of the Moon.
Comm. In fig. 113, we have shown five different
positions of the Sun S₁ to S₅ the feet of the Śankus being
B₁ to B. From these feet of the Śankus draw perpendiculars
on the Udayāstasutras as well as on the East-west line their
points of intersection being respectively A₁ to A₅ and
C₁ to C₅. The perpendiculars from the feet of the Śankus
on the East-west line go by the name Bhujas, the perpendi-

485
Fig. 113
cular distances between the Udayāstasutras and the East-
west line are called Agrās and the perpendiculars from the
feet of the S'ankus on the Udayāstasutras are called
S'ankutalas which were all defined in the course of the
Tripras'nādhyāya. We have from the spherical triangle
PZS, sin δ = sin ϕ cos z + cos ϕ sin z sin a so that
sin δ / cos ϕ = tan ϕ cos z + sin z sin a which was shown as
A=S+B in the Tripras'nādhyāya ie. Agrā = Sankutala +
Bhuja. Changing δ into —δ and a into —a we have different
formulae, the standard form being A=S+B. We don't
propose to change ϕ into —ϕ, because in India this case
does not arise. Thus our latitude being north, the S'anku-
talas defined above as the distances from the feet of the
Sankus from the Udayāstasutras are always deemed as
south. It need not be reiterated that the Udayāstasutras

486

are the straight lines joining the rising and setting points
of the celestial body. As such, these Udayāstasutras are all
parallel to the East-west line. In fig. 114, in the horizontal
plane containing the points
A's, B's and C's in the
respective cases, according
to the Hindu convention,
we talk of Agrā and Bhuja
as being Uttara Agrā,
Dakṣiṇāgrā, Uttara Bhuja
and Dakṣiṇa Bhuja.
Śaṅkutalam is always
taken to be south except in
the fifth case shown in figs.
113 and 114 when the altitude happens to be below the
horizon. In this case the Śaṅkutala is spoken as north.
Thus the general formula A=S+B assumes the following
various forms, in the respective cases.
(1) Uttarāgrā—Dakṣiṇa Śaṅkutala = Uttara Bhuja.
This corresponds to A, B, C of fig. 114 where AC, is
Uttarāgrā, BA, Dakshina Śaṅkutala and BC, Uttara
Bhuja.
(2) In the case of A₃ B₃ C₃, Śaṅkutala—Agrā =
Dakṣiṇa Bhuja since B₃ A₃—C₃ A₃=B₃ C₃. This occurs
after the Sun's diurnal path has crossed the prime-vertical
and the Sun has a position on the south of the prime-
vertical, having a northern declination. In the case of
A₄ B₄ C₄, B₄ C₄ = B₄ A₄ + A₄ C₄ ie. Dakṣiṇa Bhuja =
Dakṣiṇa Śaṅkutala + Dakṣiṇāgrā. In the fifth case
when the Sun's altitude is below the horizon, B₅ C₅ =
B₅ A₅ + C₅ A₅ ie. Uttara Bhuja = Uttara Śaṅkutala +
Uttarāgrā. In the sixth case when the Sun has a southern
declination and an altitude below the horizon, B₆ C₆ =
A₆ C₆ + A₆ B₆ ie. Dakṣiṇa Bhuja = Dakṣiṇāgrā minus
Uttara Śaṅkutala. In the case of A₂ B₂ C₂, Uttarāgrā =

487
Dakṣina Sankutala + Uttara Bhuja, since A₂ C₂ =
B₂ A₂ + B₂ C₂. In the absence of a unifying convention,
all these formulae are loose and apt to create confu-
sion. So we shall unify all these formulae into the
standard form A=S+B which holds good universally, if
we have the conventions.
(1) Agrā shall be deemed positive if it be north and
negative if it be south ;
(2) Similarly with respect to the Bhuja. But, with
respect to the Śankutala, we shall deem it positive if it be
south and negative if north. These conventions compre-
hend all the cases and unify them into the standard form
A=S+B. The corresponding conventions with respect
to δ, a are that +δ corresponds to Uttarāgrā and +a
corresponds to Uttara Bhuja.
Having the above conventions, and finding the Bhujas
of the Moon above the horizon and the Sun below the
horizon, the Spaṣṭa Bhuja required in finding the Śṛṅgon-
nati in the present chapter, is defined as the difference of
the Bhujas of the Sun and the Moon and the sum thereof
according as they are of the same direction north or south
or of different directions. Also this Spaṣṭa Bhuja is by
convention said to have the same direction as that of the
Moon. If, however, the Bhuja of the Moon falls short of
that of the Sun, then the Spaṣṭa Bhuja is said to have the
direction opposite to that of the Moon.
Verse (4). Definition of Koti.
I deem that the Koti should be taken as the sum
of the Śankus of the Sun and the Moon, the one being
below the horizon and the other above respectively.
Comm. We have defined above the Spaṣṭa Bhuja as
the north-south distance between the feet of the Śankus

488
of the Sun and the Moon. Herein Bhāskara defines the
Koti as the sum of the Saṅkus of the Sun and the Moon
which is the vertical distance between the points on the
armillary sphere which represent the Sun and the Moon.
Bhāskara says 'I deem' to signify that he differs
from Brahmagupta in this, who defines the chord joining
the Sun and Moon on the armillary sphere which is equal
to 2 Hsine of the SM on the sphere as the Karṇa. Brahma-
gupta defines the Bhuja and Karṇa from which he deduces
the Koti as √(Karṇa² — Bhuja²). Bhāskara argues that since
the Karṇa defined by Brahmagupta is not in the vertical
plane, since the Sun and the Moon are not in the same
vertical plane, so, the Koti defined by him throught he Karṇa
will not be in the vertical plane. The Karṇa defined by
Bhāskara is not on the other hand the chord joining the
Sun and the Moon but lies in the vertical plane in which
the perpendicular from the Moon's position on the armil-
lary sphere, on the horizontal plane containing the Sun's
position on that sphere. Also the Bhuja defined both by
Brahmagupta and Bhāskara is the projection on the north-
south line of the join of the Sun's position on the sphere and
the foot of the perpendicular from the Moon's position on
the horizontal plane through the Sun's position and it is
not actually the above join.
Bhāskara is evidently
guided by the right angled
triangle SMN' which is not
strictly a spherical triangle
as per its modern definition
as the arc SN is not that
of a great circle. This
figure guided him to take
the Bhuja horizontal and
the Koti vertical so that
his Karṇa is also in a
Fig. 115
vertical plane. Bhāskara's

489
Karṇa therefore has nothing to do with the arc SM but
only has the virtue of being in a vertical plane. Sripati
adopted Brahmagupta's method. Kamalākara neither
follows Brahmagupta nor Bhāskara but follows a method
of his own. In fact the methods adopted by these ancient
Hindu astronomers are not mathematically correct because
the question of determining the cusps as well as phase of
the Moon is concerned with the actual positions of the
Sun, Moon and the earth in space and not as seen on the
sphere. Hence M. M. Sudhakara Dwivedi has written a
small book by the name Vāstava-Sṛṅgonnati following
modern methods. The modern method which gives the
truth of the matter is depicted in standar modern texts.
Verse 5. The hypotenuse or Karṇa is the square-root
of the sum of the squares of the Bhuja and Koti. The
Bhuja multiplied by 6 and divided by the Karṇa gives
what is known as the Dik-valana of the Moon. The dire-
ction of the Valana has the same direction as the Spasṭa
Bhuja defined before.
Comm. The idea is that taking the radius of the
Moon's disc to be six angulas or units, representing the
Karṇa, the magnitude of the Bhuja on the same scale
gives a measure of what is defined as Valana. Thus Valana
= 6B/K where B and K stand for the Bhuja and Karṇa
defined before. The idea of this Valana will be clarified in
the ensuing verses.
Verse 6. The Hsine of the elongation of the Moon
is to be multiplied by the radius vector of the Moon
measured in Yojanas, and divided by the radius vector of
the Sun, also measured in Yojanas ; the arc of the Hsine
so obtained is to be added to the longitude of the Moon
in the bright half of the lunation and is to be subtracted
from the same in the dark half.
62

490
Comm. This is a correction to be made in the longi-
tude of the Moon to depict graphically the phase of the
Moon. Bhāskara says that many of the prior astronomers
took that the phase of the Moon was in direct proportion
to the elongation of the Moon. Taking the radius of the
Moon's disc to be six angulas or units, and assuming that
when the elongation is 180°, the entire disc being illumi-
nated, it was thought that 12 units of the diameter
correspond to 180° of elongation so that the Śukla of the
disc measured by the central width of the illuminated disc
increases at the rate of 1 unit for 15° of elongation. (The
word Śukla may be taken to correspond to the modern
word phase. Śukla is expressed in angulas, taking the
diameter of the disc to be 12 angulas. The measure of the
portion of the diameter of the Moon's disc perpendicular
to the diameter which is the join of the extremities of
the cusps, covered by the illuminated part of the disc,
(which may be defined as the maximum width of the
illuminated part of the disc) measured in angulas is said
to be the Śukla. The word 'phase' is used to signify the
ratio of the above width to the diameter. Śukla is
expressed in angulas, whereas phase is expressed as a ratio.
The Śukla is equal to twelve times phase). Bhāskara
rightly argues that this method of measuring the Śukla
is approximate because he says that six angulas of Śukla
is had when the elongation is not 90°, but only 85°-45'.
This may be substantiated as follows from fig. 116. Let
E, M, S stand for the earth, Moon and the Sun. Let
^
EMS = 90° so that it is a moment of dichotomy ie. the
moment when the phase is half and the Śukla 6 angulas.
Let θ be then the elongation of the Moon, so that
cos θ = EM / ES . Taking the average values given for EM
and ES by Bhāskara,
cos θ = 51566 / 689377 = 19 / 254 (obtaining a convergent) = .0748.

491
From tables, we find θ = 85°-43' which Bhāskara takes
to be 85°-45'.
Fig. 116
In the wake of this, Bhāskara tries to make amends
in the approximate formula prescribed to obtain the Sukla.
He prescribes addition of 4°-15' to the longitude of the
Moon in the bright half, and subtraction in the dark. In
between the moment of conjunction and the moment of
dichotomy, he derives the following formula (vide fig. 117).
Fig. 117
The deficiency of 4°-15' is had in the form of the angle
S₂ E S₁. When the Sun is at S₁ the Moon being at M₁, it

492
is the moment of conjunction. When the Sun has moved
from the point S₁ to S₃, S₁ Ê S₃ being 90°, there is a
deficiency of magnitude S₁ E S₂. In other words, when
S₂ N has assumed the position S₃ E there is a deficiency
of 4°-15′ ie. for an increase of 90° of elongation, there
is a deficiency of 4°-15′ in the longitude of the Moon.
Hence, for the Hsine to become the radius, there corres-
ponds a portion E M₂ or S₂ N, which is the Hsine of 4°-15′,
so that the following rule of three is adopted. 'If the
Sun's distance E S₃ corresponds to the radius, what does
E M₂ the distance of the Moon correspond to ?' The
result is (m / s) × R, m and s being the respective distances.
Then the following proportion is used "If by H sin ξ
equal to R, ξ being the Moon's elongation, we have mR/s,
what shall we have for an arbitrary H sin ξ ?" Thus the
answer is
(H sin ξ × mRs) / R = H sin ξ · (m / s) . H sin⁻¹ ((m / s) × H sin ξ)
where m and s are the distances of the Moon and the Sun,
is to be added to the longitude of the Moon or to be
subtracted as the case may be, to have the rectified
longitude of ths Moon from which the phase is to be
calculated according to Bhāskara.
Here we are to offer the following remarks. No doubt,
Bhāskara was correct in estimating the moment of
dichotomy to be that when ξ the Moon's elongation is not
90° but 85°-45. But the amended formula is not the
correct mathematical form. The modern formula to find
the phase is (1+ cos EMS) / 2 aud since from fig. 116, SM is
nearly equal to SE, so EMS is very nearly equal to

493
180-MES, so that (1+ cos EMS) / 2 = (1 —cos MES) / 2 which may
be taken to be an approximate truth. This ‘formula was
indeed given by Lallācārya in the following verse, long
before Bhāskara. “रविशीतकरान्तरांशजीवा विपरीता शशिखण्ड-
ताडिता च, विहृता त्रिमजीवया सितं स्यात् शशलक्ष्माङ्कवदङ्गुलानि तस्मिन् ”
verse 12 (Candra Sṛṅgonnatyadhikāra). Here विपरीता
रविशीतकरान्तरांशजीवा mean Hvers ξ = |R—H cos ξ. शशिखण्डता-
डिता = multiplied by the radius of the disc of the Moon.
विहृता त्रिमजीवया = divided by R. Thus the formula given
by Lallācārya amounts to r (R—H cos ξ) / R = d/2 (1—cos ξ)
where r gives the angulas in the radius of the Moon's disc
and d the diameter, and ξ = elongation of the Moon.
Since the definition of phase in modern terms is a ratio,
namely the ratio of the maximum width of the crescent to
the diameter, phase × d = Sukla of the Hindu astronomers
∴ (1 — cos (elongation of the Moon)) / 2 × d = Sukla
= r (1 —cos MES) as given by Lallācārya. So Lallā-
cārya's formula is quite correct. We shall trace Lallā-
cārya's steps in obtaining such an intricate correct
formula which was overlooked by Bhāskara. (Refer
fig. 118) Let E be the earth, S the Sun and M₁, M₂, M₃
etc. the positions of the Moon as the elongation
gradually increases. No doubt the Sukla increases with
elongation as known to all Hindu astronomers. But,
the question is, does it increase with the sine or versine?
We know both the sine and versine increase with the
angle. Lallācārya noticed that Sm₁, Sm₂, Sm₃ as the
Moon occupied positions M₁, M₂, M₃ etc., are the Hindu
versines which are increasing. So he postulated that the
Sukla increases with the Hindu versine. His formulation
was thus correct. But why Bhāskara overlooked this
correct formula was traceable to his getting prejudiced

494
Fig. 118
against Lallācārya's some other formulae which used the
Hversine where he ought to have used the Hsine. For
example in the case of the Valana Lallācārya's mistake
is traceable to his confusion as to whether he was to choose
the Hsine or Hversine when both increase as the angle
increases.
Verse 7. Graphical depiction of the cusps.
Fig. 119

495
Let in fig. 119, ST be the Bhuja, MT the Koti and
SM the Karṇa formerly defined. As per verse (5)
6B / K = Valana where B = ST, K = SM, and 6 = MG so
that GE = 6B / K = Valana, which is in practice drawn as E'G'
from the east point e, in the form of a Hsine. CG goes by
the name Valanasūtra. Compute the Śukla in angulas
after making the prescribed correction in the longitude of
the Moon as stated by Bhāskara and dividing the elon-
gation of the Moon, thereafter obtained, by 15. Mark off
the Śukla in angulas along the Valanasūtra from G.
Suppose GD is the Śukla. Draw the diameter AB
perpendicular to the Valanasūtra passing through M.
Draw the circle circumscribing D, A, B. Its centre lies
evidently on the Valanasūtra, say C. The circle drawn
is called Parilekha Vṛtta, its radius CA is called Pari-
lekhasūtra and the point C Parilekha Vṛtta Madhya.
In the triangle CAM, which is right-angled CA is
the Karṇa, AM the Bhuja and MC the Koti. CD is
equal to the Karṇa CA so that MD = CD – CM = Karṇa
– Koti = K – k (say). We have AM² = CA² – CM² =
K² – k² = 36 = B²; Hence K + k = B² / (K – k).
Here K – k = MD is known because GD the Śukla is
known. So B² / (K – k) = K + k is known. Thus knowing
K – k and K + k, by using what is called Saṁkramagaṇita
ie. by adding K + k and K – k, we have K and by sub-
tracting we have k. Here the Koti CM is called Vibhā
and the Karṇa CD the Swabhā. The Parilekhasūtra or
the radius of the Parilekha Vṛtta being the Karṇa, the
Swabhā is thus the Parilekhasūtra.
In the wake of this exposition, the translation of
verse (7) runs as follows. "Let the compliment of the
elongation (corrected as directed in verse (6)) divided by
15 be the denominator; let the numerator be 36; take the

496
result after division and put it in two places; with this
and the complement of the elongation divided by 15, using
Samkramagaṇita, we have successively Vibhā and Swabhā''.
Comm. Here, the meaning of taking the complement
of the elongation and dividing by 15 is to obtain the
magnitude of MD directly; for, (90 - e) / 15 = 6 - e/15 =
MG - e/15 = MG - GD = MD where e is the elongation
and e/15 is the Sukla. In other words, finding the Sukla
by dividing the elongation by 15 and subtracting it from
the radius to get MD is the same as taking the complement
of the elongation and dividing it by 15 to obtain MD.
Bhāskara quotes in the course of the commentary,
the verse from his Leelāvatī, namely "भुजाद्वर्गितात् कोटिकर्णान्त
राप्तम् द्विधा कोटिकर्णान्तरेणोनयुक्तम्। तदर्धे क्रमात्कोटिकर्णौ भवेताम्,
इदं धीमताऽऽवेद्य सर्वत्र योज्यम्" which means B² / (K - k) = K + k
since K² - k² = B². This situation arises when the Bhuja
B and the difference of Koti and Karṇa ie. K - k are given
and it is required to find K and k separately.
Verses (8) and (9). Draw a circle with radius 6
angulas to represent the disc of the Moon. Mark the disc
with the Cardinal points signifying the east, west, north
and south. Compute the Valana as directed in verse (5)
(which represents E' G' in fig. 119) and mark it off as a
Hsine from the west point w in the last quarter and from
the east point e in the first quarter. From the centre M
of the Moon's disc, mark off Vibhā MC (computed as
directed) along the join of MG'. With centre C and
radius Swabhā (already computed as directed) draw a circle.
The spherical sector of the Moon's globe thus demarked by
the arc of the circle ADB namely AGBD is found to have
the elevated cusp in the opposite direction in which
the Valana has been marked.

497
Comm. The directions are clear in the wake of the
previous commentary. In marking off the Valana, Bhās-
kara says ‘from the west in the last quarter and from the
east in the first quarter’. In the first quarter, the Moon
will be visible immediately after Sunset in the western
sky and the illuminated part of the Moon will be towards
the western side of the disc in which direction the Sun is
situated. The eastern point and the western point of the
disc are easily discernible on the background of the
horizon. Drawing first the diameter ‘WE’ of the disc
along the vertical circle in case the Moon is due west or
else along a small circle parallel to the prime-vertical called
Upa-Vṛtta, we have the east point of the disc namely e
vertically above the disc in the first quarter. (The figure
119 is shown for the first quarter). Then we are directed
to mark off the Valana E′ G′ in the form of a Hsine'.
Since the Valana is already computed as directed in verse
(5), to mark off this Valana we have to lay the scale
perpendicular to WE. Such a point G′ on the circum-
ference of the disc is to be marked from which the Hsine
equals the Valana computed. It is to be noted here that
it is not marking G′ E′ from G′, for, G′ is not known,
nor E′ for the matter of that, but locating G′ knowing
the magnitude of the Valana. Though we happened to
mention before that GE is the Valana, in the light of
what Bhāskara mentions in verse (8) we have to under-
stand that in the first quarter, the practice is to mark
off Valana from the point e marked on the disc before hand.
Though it does not matter when we take EG to be the
Valana E′ G′, we have to notice the practice followed.
The direction given in the course of the commentary
“केन्द्राद्वलनोपरिवृत्ताद् बहिरपि खटिकया सूत्रमुच्छाद्यम्” meaning
thereby that from the point M, we have to mark off
along MG′ where G′ is the point pertaining to Valana, a
line with the help of a chalk and a fine thread, namely
MC. This practice is still in vogue adopted by masons to
draw straight lines.
63

498
It is interesting to note Bhāskara waxing poetic in
the course of the commentary under these verses describing
how the illumination of the Moon's disc by the rays of
the Sun is effected. The description is quite apt, inter-
esting and scientific reminding us of Varāhamihirācārya's
description in the verse 'सलिलमये शशिनि etc.' in Bṛhat-
Saṁhitā. Of course, following Varāhamihira Bhāskara
also takes the Moon's globe to contain water within its
bosom which Modern Science has yet to confirm.
Verses 10, 11, 12. The Koti and Karṇa defined by
Brahmagupta, do not lead us to accordance between
computation and observation to locate the cusps. I
request good mathematicians to verify this carefully. In
a place where the latitude is (90—ω) = (90—24) = 66°,
when the ecliptic coincides with the horizon, and
when the Sun is in the beginning of Meṣa which is
then rising in the east, and the Moon in the beginning
of Makara, then the Moon is dichotomized by the
meridian and the illuminated part of the Moon's disc is
towards the east. This does not hold good according to
Brahmagupta's definition of Koti, because then the Bhuja
as well as Koti according to his definition is equal to R.
When the Bhuja is zero, the cusps will be horizontal, and
when the Koti is zero, they will be vertical. Brahma-
gupta's Bhuja and Koti both being equal to R, the cusps
therefore cannot be vertical which is against truth as
stated above. Why should I make this statement? May
my homage be to those great.
Comm. When the latitude of a place be 90—ω, and
when the ecliptic coincides with the horizon, it is clear
that in whatever positions be the Sun and the Moon
(assuming that the Moon is also approximately on the
horizon) the cusps of the Moon are always vertical, there
being Bhuja only and no Koti. But according to Brahma-

499

gupta's definition of Karṇa, it will be in this particular
case. √(R² + R²) = R√2 so that the Koṭi will be
√(2 R² — R²) = √R² = R, where the Bhuja also is R
(projection of the line joining the Sun and the Moon on
the north-south line. Though in the verse (11) above one
particular position of the Sun and one of the Moon, are
contemplated, the same argument holds good, says
Bhāskara in the course of the commentary, whatever
positions are occupied by the Sun and the Moon on the
ecliptic. In this case there is only Bhuja existing and
no Koṭi, so that the cusps will be vertical. But in all
these cases, there is Koṭi aecording to Brahmagupta's
definition, so that the ousps will not be vertical according
to him, which is clearly against truth.
Here we have to note that in the example cited by
Bhāskara the Bhuja equal to R is along the north-south
direction, and even though the Koṭi according to Bhās-
karajs definition is conceived to be vertical, the Karṇa
according to Brahmagupta's conception being ES where
E and S are the east and south points, his Koṭi will be
horizontal coinciding with OE, O being the centre of the
horizontal ecliptic. Further according to Brahmagupta,
the join of the ousps will be perpendioular to the Karṇa
whioh does not therefore go against truth. Similarly in
all the cases wherever be the Sun and the Moon on the
horizon, Brahmagupta's Karṇa being a line joining the
centres of the disos of the Sun and the Moon, it will be a
horizontal line and so the ousps could be vertical. It is
not clear whether or not Bhāskara recognized this namely
that the Karṇa and Koṭi of Brahmagupta in the cases
cited above are all horizontal lines so that the cusps could
be vertical even according to Brahmagupta. That is why
he pays homage to Brahmagupta in the last line of (12).
Probably Bhāskara expected that Brahmagupta's Koṭi also
should have been vertical as his. The fact that he
recognized that Brahmagupta's Koṭi will not be vertical