ब्राह्मस्फुटसिद्धान्त (ब्रह्मगुप्त - शून्य, कुट्टक, बीजगणित एवं सम्पूर्ण २१ अध्याय सान्वय सटीक)

LUNAR ASTRONOMY 109
In 500 A. D (Āryabhaṭa's time) the longitude of the Sun's
apogee by the same rule works out to be=77°19'19.44".
Āryabhaṭa states this to be 78° in the Āryabhaṭīya,
Brahmagupta in the Uttarādhyāya of the Khaṇḍakhādyaka states
the same to be 77°, while the Khaṇḍakhādyaka gives it as=80°.
Hence the Indian findings of the longitude of the Sun's apogee
were more accurate.
Again as to the Sun's equations of the centre we find that
the Āryabhaṭīya states the periphery of the Sun's epicycle to be
13°30', The Khaṇḍakhādyaka gives it as 14° ; while according to
the Indian form, Ptolemy's value of the same is 15°. Hence
according to these writers, the Sun's equations at 90° of the mean
anomaly are :—
According to the Āryabhaṭīya =2° 8' 54".
” ” Khaṇḍakhādyaka=2° 1' 40".
” ” Brahmagupta=2° 7' 20".
” ” Ptolemy=2° 23' 3".
The modern value =1° 55' 97".
Thus the Indian equations of the Sun are in general by more
correct than the Greek ones. The Indian constants in solar
astronomy are thus, generally, more accurate than the Greek
ones. We now turn to the Indian Lunar astronomy.
Lunar Astronomy
Before discussing the constants in Indian lunar astronomy it
is necessary to state something as to the time when the Moon
was observed by our ancient astronomers and the astronomers
from Āryabhaṭa I (499 A.D. to Pṛthūdaka Svāmī (864 A.D.).
The months were reckoned from the first visibility of the crescent
at the time of the Mahābhārata (1400 B. C.). We have a pass-
age in the Bhīṣmaparva where Vyāsa speaks of the evil omens
on the eve of the Kurukṣetra war thus—
चन्द्रसूर्य्यावुभौ ग्रस्तावेकामासीं त्रयोदशीम् ।
"That the Moon and the Sun have been both eclipsed on
the 13th days of the light and dark halves of the same month."
The eclipses could not take place on the 13th days of the
month unless the months were reckoned from the first visibility
of the crescent. This was the custom in Babylonia and it has
still survived in the Mahomedan world. Even in the Pañca
siddhāntikā of Varāhamihira ( 540 A. D. ), there is a special

110 INDIAN LUNI-SOLAR ASTRONOMY
chapter on शशि-दर्शनम् or the fiirst visibility of the crescent. It is
thus clear that the practice was to observe the Moon when very
near the Sun.
Again Āryabhaṭa says that 'रवीन्दुयोगात् प्रसाधितश्चेन्दुः',
"the Moon was determined from her conjunctions with the Sun."
The Moon was observed by him at the time of solar eclipses, or
at the time of the first visibility of the crescent.
Even up to the time of Pṛthūdaka, the accuracy in lunar
astronomy was chiefly aimed at the time of eclipses. Thus in
his commentary on the Khaṇḍakhādyaka. IV, he makes the
following introductory remarks :—
"All knowledge relating to (luni-solar) astronomy is desired
by the wise (or cultured) specially for knowing the right instants
of opposition or conjunction ; these instants are, however, not
visible to the eye. Of other things such as tithis, nakṣatras and
Karaṇas, as the planets, the Sun and the Moon, are not clearly
observed, their beginnings and ends are not visible. Men see
the agreement between calculation and observation at the times
of solar and lunar eclipses. Hence the word of the astronomers
is esteemed amongst men even in respect to such things as tithis,
etc."¹
Thus the chief aim of the ancient Indian astronomers was
to calculate the eclipses accurately and the Moon was observed
chiefly at lunar or solar eclipses, though the time for observation
related also to the fiinding of the first visibility of the crescent.
This latter phenomena did not perhaps lead them to directly
observing the Moon's position at such times by using instruments.
Moon's Mean Motion
The practice of observing the Moon at the time of the
eclipses alone led to the determination of the synodic month
with the following results :—
(i) Mean synodic month according to the Āryabhaṭīya
= 1577917500 / (57753336—4320000) days,
=29.530582 days.

1. बाहुल्येन पर्व्वज्ञानार्थं सकलं ज्ञानमिष्यते शिष्टैः । तेषां च पर्व्वणां दर्शनं नास्ति ।
   अन्येषामपि तिथिनक्षत्रकरणानां तस्मात् तेषां शशिभास्करयोरव्यक्तित्वात् । शशिभास्कर-
   ग्रहणयोर् 'ग् गणितैक्यं लोकः पश्यति । तस्मात् तिथ्यादिष्वप्यर्थेषु दैवज्ञं वाक्यं लोके आद्रियते ।

MOON'S MEAN MOTION 111
(ii) The same according to the Khaṇḍakhādyaka
=29.5305874 days.
(iii) The same according to the Brāhma-sphuṭa-siddhānta
=29.530582 days.
(iv) The same according to Ptolemy=29 da. 31′ 50″ 8″ 20″
in sexagesimal units=29.5305927 days
The modern value according to Newcomb and Radaú
=29.5305881 days.
Hence the Khaṇḍakhādyaka mean length appears to be
the closest approximation.
The mean sidereal month must have been deduced from the
mean synodic month and the year adopted. Hence no compari-
son need be made of this element here.
We will now consider the sidereal periods, the nodes and
the apogee of the Moon. These are shown below :-

According to | Sid. Per. of Moon's Apogee | Sid. per. of the Ascending Node

Āryabhaṭīya | 3231,987079 da. | 6794.749511 da.
Khaṇḍakhādyaka | 3231.987844 da. | 6794.750834 da.
Brāhma-sphuṭa - Siddhānta | 3232.73411 da. | 6792.25396 da.
Ptolemy | 3232.617656 da. | 6796.45571 da.
Modern values (Lockyer) | 3232.37543 da. | 6793.39108 da.

Here also the Indian values show a closer approximation
to the true values, Brahmagupta's figures representing the
nearest approach.
Other Constants
So far the Indian values of the constants have been more
accurate than the Greek ones ; but as to the inclination of the
Moon's orbit the Greek value is more accurate than the Indian
value.
Inclination of the lunar orbit
Indian value=4°30′.

112 INDIAN LUNI-SOLAR ASTRONOMY
Greek value=5°0'.
Modern mean value=5°8'43".427 (Brown)
This discrepancy confirms the conclusion, that the observa-
tion of the Moon was restricted to the time when she was near
a node, either at solar or lunar eclipses, where a small error of
observation magnified itself into about half a degree.
We now turn to the parallaxes of the Sun and the Moon :-

MOON'S EQUATIONS 113
vation was up to the time of Brahmagupta, restricted to the time
of eclipses perhaps also of syzygies.
The modern form of the Moon's equations is
=377' sin (nt—a)+13' sin 2(nt—a)+.........
+76' sin [2(nt—θ)—(nt—a)]+40' sin 2(nt—θ)..............¹
where nt=mean longitude of the Moon, a the longitude of the
perigee, θ=longitude of the Sun.
Here the first two terms, viz., 377' sin (nt—a) +
13' sin 2(nt—a), are due to elliptic motion about the Earth in one
focus; the term 76' sin [2(nt—θ)—(nt—a)] is known as the evec-
tion. We combine a part of the first term with the evection
term and the expression for the equation of centre becomes
=301' sin (nt—a)+13' sin 2(nt—a)+............+152 sin (nt—θ) cos
(θ—a)+40' sin 2(nt—θ).
Now at syzygies and eclipses sin (nt—θ) and sin 2(nt—θ)
will very nearly vanish. Hence according to modern astronomy
at the syzygies and eclipses, the chief term of the Moon's equa-
tion=301' sin(nt—a).
This according to the Āryabhaṭīya
=300' 15" sin (nt—a)
,, ,, Khaṇḍakhādyaka
=296' sin (nt—a)
,, ,, Uttara Khaṇḍakhādyaka
=301'.7 sin (nt—a)
,, ,, Brāhmasphuṭasiddhānta
=293' 31" sin (nt—a)
,, ,, Greek astronomy
=300' 15" sin (nt—a)
very nearly.
Hence both the Greek and the ancient Indian astronomers
were very near the true value of the Moon's equation at the syzy-
gies and eclipses. Godfray in his Lunar Theory, page 107,
observes, "the hypothesis of an excentric, whose apse has a pro-
gressive motion as conceived by Hipparchus served to calculate
with considerable accuracy the circumstances of eclipses; and
observations of eclipses, requiring no instruments, were then the
only ones which could be made with sufficient exactness to test

1. The accurate values of the coefficients appear to be 377' 19".06,
   12' 57".11, 76' 26" and 39' 30".

114 INDIAN LUNI-SOLAR ASTRONOMY
the truth or fallacy of the supposition." We next consider the
second inequality of the Moon.
Moon's Second Inequality or Equation
In ancient times it was Prolemy who first really found a
second inequality of the Moon. According to Godfray (Lunar
Theory, p. 107) "by dint of careful comparison of observations
he (Ptolemy) found that the value of this second inequality in
quadrature was always proportional to that of the first in the
same place, and was additive or subtractive according as the first
was so; and thus, when the first inequality was at its maximum or
5°1′, the second increased it to 7° 40′ which was the case when
the apse line happened to be in syzygy at the same time."
It is well known that though Ptolemy discovered the second
inequality in the Moon's motion he was not able to ascertain its
true nature. His corrections in this case are true when at the
quadrature the Moon's apse line passes through the Sun or it is
at right angles to the line joining the Earth and the Sun.¹ In
the general case his construction does not lead to the elegant
form of the evection term as we know it now, nor does it lead
to the nice form in which it was given by later Indian astrono-
mers from the time of Mañjula (or Muñjāla, 854 Śaka era = 932
A.D.).
As has already been pointed out, the early Indian astrono-
mers from Āryabhaṭa to Brahmagupta aimed at accuracy in
lunar calculation only for the eclipses and syzygies, and did not
interest themselves about the Moon's longitude at the quadra-
tures. Hence this second inequality is absent in the works of
these makers of Indian astronomy, as also in the Pre-Ptolemaic
Greek astronomy. This points to the conclusion that in both
the earlier Indian and Greek systems of astronomy, the modes of
observation of the Moon were copied from an earlier system of
astronomy whether Babylonian or Chaldean. Even in the
Romaka Siddhānta of the Pañcasiddhāntikā, there is no mention
of evection.² Thus inspite of the transmission of a vague system
of Greek astronomy, Indian astronomy as developed by Ārya-
bhaṭa and Brahmagupta must be regarded as independent and

1. Godfray's Lunar Theory. pp. 108-110.
2. Vide the Summary in P.C. Sengupta's paper, "Āryabhata the Father
   of Indian Epicyclic Astronomy." Journal of the Department of letters, vol.
   XVIII, Calcutta University Press.

MAÑJULA'S MOON'S EQUATION 115
original not only from this but also from other considerations.
It sought to correct the constants as were obtained from the
Babylonian and the Greek systems as has in some cases been
shown already.
Mañjula's Second Equation of the Moon (932 A.D.)
We now take up in detail Mañjula's second equation of
Moon. It is, however, necessary to say something about his first
inequality.
This is given in the form
—488 sin (nt—a')
————————————————————— degrees,
96+ (488/120) cos (nt—a')
where nt stands for the Moon's mean longitude, a'—that of the
apogee.
488°
Hence when nt—a'=90°, the equation = ———— =5°4'=304'
96
showing an excess of 4' over the modern value.
It is further necessary to modify the expression for
the Moon's modern form of the equation by chang-
ing a to 180°+a, as in ancient Indian astronomy anomaly
is measured not from the perigee but from the apogee.
The modified form is
=—301' sin (nt—a)+13 sin 2(nt—a)......
—152' sin (nt—θ) cos (θ—a)+40' sin 2(nt—θ)+...
Mañjula's lines giving the second equation are—
The (mean) daily motion of the Moon diminished by
11° and multiplied by the "cosine" of the longitude of
the Sun diminished by that of the Moon's apogee is the
multiplier of the "sine" and the "cosine" of the longi-
tude of the Moon diminished by that of the Sun,
divided severally by 1 and 5. The results taken as
minutes are to be applied negatively and positively to
the Moon and to her daily motion if the quantities
multiplied together are of opposite signs and in the
reverse order if they are of the same sign.¹

1. इन्दूच्चोनार्ककोटिघ्ना गत्यंशा विभवा विधोः ।
   गुणो व्यर्केन्दुदोः कोट्यो रूप पंचाप्तयोः क्रमात् ॥
   फले शशाङ्क--तद्गत्योर्लिप्तास्ते स्वर्णयोर्वधे ।
   ऋणं चन्द्रे धनं भुक्तौ स्वर्णं साम्य वधेऽन्यथा ॥ —11, 12

116 INDIAN LUNI-SOLAR ASTRONOMY
As to the positive or negative character of the "sine" and
the 'cosine' he gives the rule:—
The mean planet diminished by its ucca, the apogee,
aphelion or the Śīghra, is called Kendra or mean
anomaly; its "sine" from above six signs (180°) arises
from half circles and are respectively positive or
negative, and its "cosine" in different quadrants are
respectively positive, negative, negative, and posi-
tive.¹
The convention followed is that the "sine" is negative from
0° to 180° and positive from 180° to 360° of the arc and that the
cosine is positive between 0° and 90°, negative between 90° and
270° and positive between 270° and 360°.
We may now symbolically express Mañjula's second ineq-
uality thus :-
—(13° 11′ 35″—11°) × 8ᵖ 8′ cos (θ—α) × 8ᵖ 8′ sin (D—θ) where
D stands for the Moon as corrected by the 1st equation; we leave
out the correction to the Moon's daily motion as given in the
stanzas quoted above.
The moon's new equation comes out to be
=—143′58″ cos (θ—α) sin (D—θ).
This, it will be seen, is exactly the modern form of the
evection as combined with a part of the equation of apsis shown
before. The difference in the main is that Mañjula's constant
is 144′, a quantity less by 8′. In form the equation is most perfect,
it is far superior to Ptolemy's, it is above all praise. It is from
this inequality, we trust, that Mañjula should have an abiding
place in the history of astronomy. The next writer who gives
the second equation is Śrīpati (1028 A.D.).
Śrīpati's Second Inequality of the Moon
The following stanzas from Śrīpati's Siddhānta Śekhara, it is
said, were communicated to Sengupta by Pandit Babua Misra.
Though they are probably not very correct still the general
meaning is clear. They carry the following sense :
"From the Moon's apogee subtract 90°, diminish the
Sun by the remainder left; take the "sine" of the

1. ग्रहः स्वोच्चोनितः केन्द्रं षडूर्द्धादर्द्धजो भुजः ।
   धनर्णैः पदशः कोटिर्धनर्णै धनात्मिका ॥

ŚRĪPATI'S SECOND INEQUALITY OF THE MOON 117
result; multiply it by 160' and divide by the radius;
the result is called caraphala. Put it down in another
place, multiply it by śara (i.e., R vers (D—α) or versed
sine of the Moon's distance from the apogee) and
divide by the difference between the Moon's distance
(hypotenuse) and the radius; the result is called parama
(cara) phala, which is to be considered positive or
negative according as the hypotenuse put down in
another place is less or greater than the radius. Multi-
ply the "sine" of the Moon which has been diminished
by the apparent Sun, by the apparent paramaphala and
divide by the radius; the final result is to be called
caraphala to be applied to Moon negatively or posi-
tively as the Moon minus the Sun and the Sun minus
the Moon's apogee (diminished by 90°) be of opposite
signs; if these latter quantities be of the same signs, the
[L1

118 INDIAN LUNI-SOLAR ASTRONOMY
160' R sin [θ – (a – 90°)] R vers (D – a) × R sin (D – θ)
= ∓ -----------------------------------------------------------
R (H – R) × R
160' R cos(θ – a) × R sin(D – θ) R vers (D – a)¹
= ∓ ---------------------------------- × -----------------
R × R H – R
This equation is a slightly modified one but practically the
same in form as that of Mañjula, except that the constant here
is 160', greater than his by 16'. The constant is 160' also in
Candraśekhara's form as we shall see later on. We next consi-
der the Moon's inequalities as given by Bhāskara II in his Bījo-
panaya,* a special work on these inequalities composed in the
Saka year 1074 (=1152A.D.) two years after he had composed
the Siddhānta Śiromaṇi.
Bhāskara II on Moon's Inequalities
His preliminary statement runs thus .—
112' positive or negative representing the maximum
difference. have been found by me in the daily obser-
ved Moon (as calculated and as observed) at that
point of the ecliptic where the arc from the kadamba
(i.e., its pole) passing through the zenith cuts it.¹
Thus for observing the Moon he selected the nonagesimal
as the suitable point where the uncertainty about her parallax is
zero, and found ∓112' of arc to be the maximum difference bet-
ween her calculated and observed places.
Mallabhaṭṭa, perhaps a contemporary of Bhāskara II,
ascribed this difference to a supposed Śīghrocca of the Moon.
Bhāskara in stanzas 9-13, refutes the existence of the Śīghra in
the case of the Moon, the substance of his argument
begins (i) that it is against the teaching of the Sūrya-siddhānta
and other accepted authorities, (ii) that there is no variation
of the apparent angular diameter of the Moon corresponding to
this alleged Śīghra, and (iii) that planets having a Śīghra have
retrograde motion which is never the case with the Moon.

*There is some uncertainty, about this new fraction introduced by
Śrīpati.

1. लिप्ता विधोरर्के महीमिता मे दृग्गोचराः प्रत्यहमीक्षितस्य ।
   कदम्ब गोलगत-सूत्रपाते क्रान्तौ धनर्णत्वजुषो भमध्यात् ॥
   – Bijopanaya, stanza 8

BHĀSKARA II ON INEQUALITY OF THE MOON 119
The reasons for his new equations are stated as follows:—
When the Moon is situated at a quadrant ahead of
the apogee and with the Sun at half a quadrant ahead
of her, the maximum discrepancy (of 112′) is seen in the
negative character.
When the Moon is situated at three quadrants ahead
of the apogee and with the Sun at half a quadrant
behind her, the maximum discrepancy (of 112′) is seen
in the positive character.
When the eclipses of the Sun and the Moon take
place at the apogee or the perigee of the Moon, the
Moon as corrected by the equation of apsis is seen to be
without any new correction called bīja.
When the eclipses of the Sun and the Moon take
place at the ends of the odd quadrants of the Moon's
anomaly (measured from the apogee), the discrepancy
is seen to be less by 34′.
When the Moon is at the apogee, whether the Sun be
ahead or behind her by half a quadrant, the discre-
pancy amounts to be 34′.
The same discrepancy of 34′ is observed when the
Moon is at the perigee and the Sun is ahead or behind
her by the same distance.
Thus by analysis and synthesis, and by repeated obser-
vations, this variable correction has been devised by
me; let it be seriously considered by the learned.¹
Bhāskara here speaks of six cases and we consider them one
after another :—
The Moon's equations as modified to suit siddhāntas are
given by
−301′ sin (nt—α)+13′ sin 2(nt—α)......
−152′ sin (nt—θ) cos (θ—α)−40′ sin 2(nt—θ)+......
According to Bhāskara's Siddhānta - śiromaṇi. the Moon's
equation of apsis
= − (31° 36′ / 360°) × 3438′ sin (nt—α)
= −301′ 46″. 8 sin (nt—α),

120 INDIAN LUNI-SOLAR ASTRONOMY
this agrees well with the corresponding term of the modern
equation. As Bhaskara takes in all the six cases, nt-a=90°,
270°, 0° or 180°, the second term of the equation of apsis
vanishes.
Case I.
nt-a=90°, nt-θ=-45°, θ-a=135°;
Here the total equation of the Moon
=-301'-(76'+40')=-301'-116'.
This fairly agrees with Bhāskara's observation, the difference
being only of 4'.
Case II.
nt-a=270°, nt-θ=45°, θ-a=225°;
the total equation of the Moon
=301'+76'+40'=301'+116'.
This also agrees with Bhāskara's observation.
Case III.
nt-a=0° or 180°, nt-θ=0° or 180°, θ-a=0° or 180°,
the total equation=0', this also agrees with Bhāskara's
observation.
Case IV,
nt-a=90° or 270°, nt-θ=0° or 180°, θ-a=90° or 270°,

1. तुङ्गादाद्यपदान्तस्थाद् विधोरर्के पदार्द्धतः ।
   परमं चन्द्रवैषम्यं ऋणत्वेन समीक्ष्यते ॥ 20 ॥
   तत् तृतीय पदान्तस्थात् पृष्ठगोऽर्के पदार्द्धतः ।
   परं चन्द्रवैषम्यं धनत्वेन समीक्ष्यते ॥ 21 ॥
   चन्द्रतुङ्गे च नीचे च शशाङ्कार्कग्रहो यदि ।
   मन्दस्फुटगतश्चन्द्रो निर्धीजतुल्यमीक्ष्यते ॥ 22 ॥
   ओजान्तयोर्विधोस्तुङ्गाच्छशाङ्केऽर्क ग्रहो यदि ।
   चतुस्त्रिंशत्-कलाहीनं वैषम्यं तु समीक्ष्यते ॥ 23 ॥
   अग्रतः पृष्ठतो वाऽपि रवेश्चन्द्रे पदार्द्धगे ।
   तुङ्गतुल्ये चतुस्त्रिंशत् कलावैषम्यमीक्ष्यते ॥ 24 ॥
   एवं तन्नीचतुल्येऽपि वैषम्यं तावदेव हि ।
   एवं व्यासात् समासाच्च पौनःपुन्येन वेधनात् ॥
   चरबीजमिदं क्लृप्तं मया सद्भिः समीक्ष्यताम् ॥ 25 ॥

ŚRĪPATI'S SECOND INEQUALITY OF THE MOON 121
the total equation = ∓301′. This does not agree with Bhās-
kara's statement that the total equation
= ∓(301′ ± 78′).
Case V.
nt—a=0, nt—θ=±45°, θ—a=±45°,
the total equation
=0′—76′+40′=—36′ or 0′+76′—40′=+36′.
This fairly agrees with Bhāskara's observation.
Case VI.
nt—a=180°, nt—θ=±45°, θ—a=180°∓45°
the total equation
=0′+76′+40′=0′+116′, or 0—76+40′=0′—36′.
This does not agree with Bhāskara's statement.
Bhāskara then states his first system of 24 equations corres-
ponding to 24 sines in a quadrant to be 6′, 13′, 21′, 27′, 33′, 39′, 45′,
51′, 56′, 61′, 65′, 68′, 70′, 72′, 74′, 75′, 75′, 76′, 76′ 77′, 77′, 78′, 78′,
78′.¹
These equations, he says—"are negatively added to the
equation of apsis when that is negative and positively added to
the same when that is positive"². In other words his new equa-
tions are complements of the equation of apsis, the two together
being represented by
—301′ 46″. 8 sin (nt—a)—78′ sin (nt—a)
i.e., by—379′ 46″. 8 sin (nt—a).
Hence next states his second set of equations depending on
θ—D, to be 6′, 9′, 13′, 17′, 22′, 24′, 27′, 30′, 32′, 33′, 34′, 34′, 34′, 33′,
31′, 29′, 26′, 24′, 20′, 16′, 11′, 8′, 3′, 0′³ and says :
"These minutes are negative in the odd quadrants of the
argument and are positive in other quadrants.⁴"
When the value of the argument is 15°, the equation is 17′,
,, ,, " 45° " 34′,
,, ,, " 90° " 0′,

1. Bījopanaya, 26-28.
2. ˙˙˙फलं ऋणे ऋणं ।
   धने धनं मन्दफलेन संयुतम् ॥ 28 ॥
3. Bījopanaya, 29-32.
4. एताः कला ओजपदे ऋणं स्युर्धनं तदन्यत्र भवन्ति भूयः ।

122 INDIAN LUNI-SOLAR ASTRONOMY
Hence the new equation = - 34' sin 2 (θ - D),
= 34' sin 2 (D - θ).
Here the symbol D stands for the Moon as corrected by the
ancient Indian equation of apsis and its complement as given by
Bhāskara. It is readily seen that Bhāskara is the first of all the
Indian astronomers to detect the equation known as "Variation"
His constant, 34', is less than the modern value by about 6', and
cannot be considered as a serious error.
We now see that the sum-total of the Moon's equation as
given dy Bhāskara
= - 379' 46".8 sin (nt - a) + 34' sin 2(D - θ),
the evection term being totally absent. This is a serious
defect, and Bhāskara's new equations would make the Moon
generally more incorrect at the syzygies and eclipses than what
the ancient Indian equation of apsis would do.
Perhaps late in life when he was 69 years old in 1105 of
Śaka era (= 1183 A.D.) he discovered the inapplicability of his
new equations at the times of eclipses and in his Karaṇa-kutu-
hala he altogether omitted these new equations which he had
given in his Bījopanaya.
As to Bhāskara's second inequality which is really the com-
plement of the equation of apsis without the evection term, it is
far inferior to that of Mañjula and of Śrīpati; as we have seen
their form of the second inequality combines the complement of
the equation of apsis and evection in the mathematically cor-
rect form. For the discovery of such a form of the equation as
of these authors, very patient, careful and frequent observation
must have been coupled with very careful and nice comparison
of observed facts.
As to "variation" it was first discovered by Abul-Wefa in
976 A.D.¹ which was quite forgotten when Tycho-Brahe re-dis-
covered it in 1580 A.D. Hence Bhāskara, in 1152 A.D., re-dis-
covered it in India four centuries before Tycho.
Candraśekhara of Orissa on the Moon's Inequalities
In connection with lunar inequalities it is nesefsary here
to record what were the equations discovered or verified by
M.M. Candraśekhara Simha of Orissa in the later half of the
last century. He was educated in the orthodox Sanskrit fashion

1. Godfray's Lunar Theory, p. 114.

CANDRAŚEKHARA ON THE MOON'S INEQUALITIES 123
and had no acquaintance with English education. His work
Siddhānta-darpaṇa was edited by Prof. Jogeschandra Ray, late of
the Cuttack College, in 1899.¹ Candraśekhara in his work gives
four equations of the Moon which are :-
(1) The equation of apsis.
(2) The Tungāntra equation or the complement of the
equation of apsis in combination with evection.
(3) The fortnightly equation or variation.
(4) The Digamśa equation or the annual equation (i. e., 1/10
of the Sun's equation).
(5) The first equation is of the form
[31° 30' - 30' cos (nt - a)]3438 × sin (nt - a)
= ───────────────────────────────────────────────
360°
= -300' 49".5 sin (nt - a) + 4' 46".5 sin (nt - a) cos (nt - a)
= -300' 49".5(sin nt - a) + 2' 23".25 sin 2 (nt - a)
It is seen that Candraśekhara wanted to correct the equa-
tion of apsis to the second order of small quantities as in all the
Indian authors from Brahmagupta but Candraśekhara's form is
correct though his contant is wrong.
(2) His second equation is of the form
160' × 3438 sin[a - (θ + 90°)] 3438 sin(D - θ)
────────────────────────────── × ───────────────
3438 3438
Moon's appt. daily motion²
× ──────────────────────────
Moon's mean motion
= -160' cos (θ - a) sin (D - θ)
Moon's appt. daily motion
× ─────────────────────────
Moon's daily mean motion
Here the constant is the same as that of Śrīpati discussed
before. The symbol means the Moon as corrected by the
equation of apsis. It is readily seen that the constant of the
first term of the equation of apsis is increased by 80'. and that
the constant of evection is taken at 80'. In both the cases the
error is about +4'.
(3) Candraśekhara's third equation or Variation
3438' sin 2(D' - θ)
= ─────────────────── = 38' 12" sin 2(D' - θ),³
90

1. Siddhānta-darpaṇa, V, 100-114.
2. Ibid, VI, 7-9
3. Siddhānta-darpaṇa VI. 11-12.

124 INDIAN LUNI-SOLAR ASTRONOMY
where D' means the Moon as corrected by the 1st and the 2nd
equations. Here the constant is wrong by −1' 18".
(4) His fourth equation or the annual equation
= ± 1/10 of the Sun's equation of apsis,¹
= ± 1/10 × (12 × 3438) / 360 sin (Sun's distance from the
apogee).
= ±11'27". 6 sin (Sun's distance from the apogee).
The modern value of the constant is 11' 10". Tycho found
it to be 4' 30". Horrocks' (1639) co-efficient was 11' 51".
As Candraśekhara was aware of Bhāskara's Bījopanaya, as
also of the work of Śrīpati, his merit here lies in the discovery
of the annual equation, and correction to the constant of
variation.
Thus we have seen that so far as the luni-solar astronomy
is concerned Indian astronomy is independent of Greek astronomy
in respect of astronomical constants, that Indian astronomy is
generally more accurate than Greek astronomy and that Indian
astronomers were not mere "calculators"². There were observers
who verified and corrected the old astronomical constants as
they came down from Āryabhaṭa and Brahmagupta, who also
found independently all the principal equations of the Moon.

1. Siddhānta darpaṇa VI. 13.
2. G.R. Kaye Hindu Astronomy, p. 60
   — : o : —
   Reference
   P.C. Sengupta : The Khaṇḍakhādyaka, 1934
   Karl Manitius's edition : Syntaxis.
   Godfray : Lunar Theory.

CHAPTER VII
Greek and Hindu
Methods in Spherical Astronomy
Here we shall reproduce from Sengupta's paper a compa-
rative account of the Greek and ancient Indian methods in
Spherical Astronomy and to bring out the independence of the
Indian Astronomers on this subject. The views on this subject
would necessarily differ from those of many European scholars
such as Colebrooke and Bentley (early 19th century) to Kaye
(early 20th century). Kaye wrote as follows in the Journal of
Asiatic Society of Bengal 1919, No. 3.
The methods by which (the rules) were obtained are
buried in obscurity. Braunmühl¹ has stated "that the
Indians were the first to utilise the method of projec-
tion in the Analemma of Ptolemy." It is intended to
present the Hindu methods as clearly as possible and
to show that Braunmühl has not done sufficient justice
to the Indian astronomers.
As to Kaye, we shall show that his remark quoted above
is due to the fact that he had to rely mostly on the English tran-
slation of the Sūryasiddhānta of Burgess, and perhaps he had no
access to the works of Bhāskara II (1150 A.D.), who was the first
to explain the ancient Indian methods clearly.
Greek and Hindu Methods in Spherical Astronomy
Of the Greek methods in Spherical Astronomy, the
history begins with elementary principles only from Euclid (300
B.C.). Even in Theodosius' Sphaerica² (about 153 B. C.) "there
is nothing that can be called trigonometrical." Heath again says,

1. Heath, Greek Mathematics, Vol. II, p. 291. Braunmuhl, Geschichte der
   Trigonometrie, pp. 38-42.
2. Heath, Greek Mathematics, Vol. II, p. 250.

126 GREEK AND HINDU SPHERICAL ASTRONOMY
"the early spheric did not deal with the geometry of the sphere as
such, still less did it contain anything of the nature of the spheri-
cal trigonometry. (This deficiency was afterwards made good by
Menelaus's Sphaerica).¹ Hence the Greek spherical trigonometry
began with Menelaus (90 A.D.). His theorem in geometry is
well-known—"If the sides of a plane triangle be cut by a trans-
versal into six segments, the continued product of any three
alternate segments, is equal to the continued product of the
remaining three." From this proposition he deduced the so-
called "regula sex quantitatum" or the theorem, if the sides of a
spherical triangle be cut by an arc of a great circle into six
segments, the continued product of the chords of the doubles of
any three alternate segments is equal to the continued product
of the chords of doubles of the remaining three segments." In
plane geometry if the sides BC, CA, AB of a triangle be cut by
any transversal at L, M, N, respectively,
then we have
(BL / LC) · (CM / MA) · (AN / NB) = 1.
In spherics the theorem is :
(Chord 2 BL / Chord 2 LC) · (Chord 2 CM / Chord 2 MA) ·
(Chord 2 AN / Chord 2 NB) = 1
Both these theorems are proved in
Ptolemy's Syntaxis ( Karl Manitius's edi- Fig. 5
tion, Vol. I, pp. 45-51).
If R be the radius of the sphere on which the spherical
triangle ABC is constructed, then the chord of the arc 2 BL =
2 R sin BL. Hence Menelaus's theorem in spherics may be expres-
sed as follows :
(Sin BL / Sin LC) · (Sin CM / Sin MA) · (Sin AN / Sin NB) = 1
This theorem is true for any spherical triangle.
If ∠B = AN = AM = 90° and L the pole of AB, then LMN
is a secondary to the arc AB. There are four arcs of great
circles; taking any three as forming a spherical triangle and the
fourth as the transversal we readily get for the right-angled

1. A.A. Bjornbo, "Studien über Menelaos' Sphärik" in Abhandlun-
   gen Zur Geschichte der Mathematischen Wissenschaften for 1902, pp. 89 et
   seq.; also Heath, Greek Mathematics, vol. II p. 261-73.

GREEK METHODS IN SPHERICAL ASTRONOMY 127
triangle ABC, the relations :--
(i) sin a=sin b sin A
(ii) sin c=tan a cot A
(iii) cos b=cos a cos c
(iv) tan c=tan b cos A
The above are some of the Napier's rules for a right-angled
spherical triangle, deducible from Menelaus's theorem¹. They
are generally sufficient in the case of such triangles. In any
spherical triangle, however, this theorem of Menelaus does not
in any single step lead to any of the equivalents of the time-
altitude or altazimuth equations in spherical astronomy. The
ancient Indian methods, though none of them are so highly
finished as Menelaus's theorem, yet are not less powerful in
tackling the problems that arise in astronomy in connection
with the apparent diurnal motion of the heavens. The Greek
or Ptolemaic method presents no further points of interest except
in its application. We now proceed to illustrate the ancient
Indian methods and shall refer to the Ptolemaic method as
occasion arises.
Ancient Indian Methods
in Spherical Astronomy
In the Indian methods there is no general rule to follow.
It is by properties of similar right-angled triangles that a fairly
complete set of accurate formulae are obtained. These right-
angled plane triangles are classified under the names,—'Krānti-
kṣetras' (triangles of declination) and 'Akṣa-kṣetras' (triangles of
latitude). We consider the following problems :—
Problem :— To find the time of rising on the equator of a
length l, of arc of the ecliptic measured from the
first point of Aries.
Let ω be the obliquity of the ecliptic and R. A. the right
ascension corresponding to the longitude l, and δ the correspond-
ing declination. The Indian form of the equation is :

1. Three more can be deduced similarly, namely,
   (v) sin c=sin b sin C
   (vi) sin a=tan c cot C
   (vii) tan a=cos C tan b.

128 GREEK AND HINDU METHODS
*R sin R.A. = (R sin l × R sin ω) / (R cos δ) , where R is the radius of
the sphere.
Note:— If R be the radius of the circle of reference, the
Indian trigonometrical functions for the arc θ, are
(1) the 'sine,' (2) the 'cosine' and (3) the versed
sine. They are respectively equal to R sin θ,
R cos θ and R vers θ.
In the adjoining figure, O is
the centre of the armillary
sphere, YQ, YC are quadrants of
the equator and the ecliptic,
respectively. P is the celestial
pole, PCQ the summer solsti-
tial colure. Join OY, CQ, OP
and OC.
Let YS be=l, YM=R.A.,
CQ=∠SYM=ω, SM=δ.
Join OS, OM. PSM is the
secondary to the equator.
[Fig. 6]
From C draw CK perpendicular to OQ. From S draw Sm
and Sn perpendicular to OM and OY, respectively. Join MN
and from M draw MN perpendicular to OY.
Then the triangles Smn and CKO are similar. They are
called 'Krānti-kṣetras'¹ or declination triangles,—similar right-
angled triangles having one acute angle=ω.
∴ Sm : Sn = CK : OC
or R sin δ : R sin l = R sin ω : R
R sin δ = (R sin l × R sin ω) / R ......(I)

The Āryabhaṭīya, Gola, 25. Varāhamihira, in the Pañcasiddhāntikā
(IV. 92) states it in the form 2R √((R² Sin ²l)—R² Sin ²δ) / (2R cos δ)
= R Sin R.A., which is evident from figure. Brahmagupta's
equation is identical with that of Āryabhaṭa, (BrSpSi. III. 15, Sūrya-siddhānta
III. 40-41.
Also Bhāskara II, Grahagaṇita cap. VIII, stanzas 54-55, is in agreement
with Varāhamihira's forms.