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

104 BRAHMASPHUṬASIDDHĀNTA'S KHAṆḌAKHĀDYAKA
Brahmagupta corrects
Dimensions of the Epicycle of Apsis
Brahmagupta corrects the dimensions of the epicycles of
apsis of the Sun and Moon by—1/42 nd part and 1/48th parts
respectively. The reference may be made to the following verse
in the Uttara Khaṇḍakhādyaka :
The Sun's equations are to be made less by dvikṛtāṁ-
śonam (1/42nd) and the Moon's equations, increased by
vasuvedabhāgayutam (1/48th). Multiply the Sun's equa-
tion a planet's daily motion in minutes and divide by
the number of minutes of a whole circle and this is
called Bhujāntara correction and applied in the same
way to the planet as the equation is applied to the Sun.¹
The Sun's epicycle of apsis has the dimension 14° in the
Khaṇḍakhādyaka proper. With the correction introduced here,
the value becomes 14° {1 - 1/42} = 13° 40'.
The correction to the Moon's equations would make the
epicycle's dimension changed from 31° to 31° {1 + 1/48} = 31°38'45''.
Pṛthūdaka's commentary further corrects it to 31°(1 + 1/52) = 31°35'
Brahmagupta 's correction to Saturn's epicycle of apsis
is—1/5th part and that to the Śīghra epicycle of Mercury 1/16th
part as seen in the verse :
Of Mars the apogee (the aphelion point) is to be
increased by 17°, that of Jupiter by 10°; from the
Śīghra of Venus 74' are to be subtracted ; Saturn's
equation of apsis should be decreased by its one-fifth
and the Śīghra equation of Mercury should be
increased by one-sixteenth.²
In the Khaṇḍakhādyaka proper (II.6), we have been given
the longitudes of the apogee of planets : Mars 11°, Mercury 22°,
Jupiter 160°, Venus 80° and Saturn 240°. Now with these correc-
tions introduced in the Uttara Khaṇḍakhādyaka in the above

1. द्विकृतांशोनं रविफलमिन्दोर्वसुवेदभागयुतम् ।
   अर्कफलभुक्तिघाताद् भगणकलाप्तं भुजान्तरं रविवत् ॥ UKK. IX. 9
2. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः ।
   सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् ।
   पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ॥ UKK.IX.11

DIMENSIONS OF THE EPICYCLE OF APSIS 105
verse, the aphelion point of Mars in 499 A.D. had a longitude
of 110 plus 17=127°; of Jupiter 160 plus 10=170°.
According to Newcomb's rule, the longitude of the aphelion
point of Mars in 499 A.D. works out to have been 128°28'12''.
The same according to Conn. des Temps rule would be 128°27'51''.
Hence Brahmgupta's determination of Mars's aphelion is correct
within 1°30', and may be, therefore, regarded as very satisfactory.
According to the Khaṇḍakhādyaka proper this value was, as
already said, 110°, while according to the Āryabhaṭīya, it was
118°.
The same may be said regarding Jupiter's aphelion. Accord-
ing to the Khaṇḍakhādyaka proper the value of its longitude is
160°, according to the Āryabhaṭīya it was 180°; according to
Conn. des. Temps' rule, it would be in 499A. D. 170°25', and the
value given by Brahmagupta in the Uttara Khaṇḍakhādyaka, it
is 170°.
We have thus shown by many illustrations the important
corrections introduced by Brahmagupta in his Khaṇḍakhādyaka
specially the Uttara part. Brahmagupta was highly original in his
methods of calculations, accuracies and interpolations. He intro-
duced new ideas in mathematics. He went much ahead Āryabhaṭa
in many details. He so many times did not follow Āryabhaṭa
in calculations. In the Khaṇḍakhādyaka proper, his treatment
of parallax in the calculation of solar eclipses is different from
that of Āryabhaṭa. The methods followed here are the same as
propounded by him in the Brāhmasphuṭasiddhānta.¹
Senagupta is right when he says : As has already been
remarked, these corrections and innovations in the Uttara
Khaṇḍakhādyaka paved the way for the acceptance of his great
work the Brāhmasphuṭasiddhānta as a standard work on astrono-
my by the western Indian school of astronomers. The directness
of the treatment of topics, and the simplicity of calculations
taught in the Khaṇḍakhādyaka made it very neat handbook
for the beginner. These two works of Brahmagupta were
perhaps the only astronomical works in circulations in western
India when the Arabs conquered Sind early in the eighth century

1. On Parallax—
   नाडी चतुष्कविधिना सर्वत्र समो यतस्ततः स्थूलः ।
   मानार्थं कर्म महत् कृतमार्यभटेन लघुनि सति ॥ BrSpSi. XI. 33

106 BRHĀMASPHUṬASIDDHĀNTA'S KHAṆḌAKHĀDYAKA
(712 A.D.) and the new conquerers learnt Indian astronomy and
mathematics from these works as has been observed by Sachau.
Alberuni who came to India early in the 11th centuty of the
Christian era, learnt Indian astronomy chiefly by studying the
Khaṇḍakhādyaka and the Bṛhat-Saṁhitā of Varāhamihira, and
both of them with the help of commentary of Bhaṭṭotpala.
—:c:—
Reference
P.C. Sengupta : The Khaṇḍakhādyaka, 1934.

CHAPTER VI
Indian Luni–
Solar Astronomy
In this chapter, it is proposed to give an account of astrono-
mical constants and the equations in Indian luni-solar astronomy
and to present a comparative view of these quantities with the
corresponding ones in Greek and modern Astronomy. This acco-
unt has been reproduced from P. C. Sengupta's. Appendix I of
the Khaṇḍakhādyaka It has been shown that in many cases
the Indian values of these constants are more accurate than the
Greek values. and in Indian lunar astronomy the equations
or inequalities discovered are the most startling.
Solar Astronomy
In solar astronomy the length of the year was determined
by Āryabhaṭa¹ from the heliacal risings of some bright star at
the intervals of 365 and 366 days.
(1) The year according to the Āryabhaṭīya
= 1577917500 / 4320000 days = 365.2586805 days,
= 365 da. 6 hrs. 12 mins. 29.64 secs.
(2) The same
= 1577917800 / 4320000 days = 365.25875 days.
= 365 da. 6 hrs. 12 mins. 36 secs., according to the
Khaṇḍakhādyaka, the Sūryasiddhānta of Varāha
and the modern Sūryasiddhānta.
(3) It is
= 1577916450 / 4320000 days = 365.2584375 days,
= 365 da. 6 hrs. 12 mins. 9 secs., according to the

1, P. C. Sengupta, "Aryabhata's Method of determining the Mean Motions
of Planets," Bulletin of the Calcutta Mathematical Society. Vol. XII, No. 3.

108 INDIAN LUNI-SOLAR ASTRONOMY
Brāhmasphuṭa Siddhānta of Brahmagupta.
Now the mean sidereal year
=365 da. 6 hrs. 9 mins. 9.3 secs. (Lockyer).
The mean anomalistic year
=365 da. 6 hrs. 13 mins. 49.3 secs. (Lockyer).
The mean tropical year
=365 da. 5 hrs. 48 mins. 46.054 secs. (Lockyer).
Though we take that Indian year was designed to be the
sidereal year, it approached most closely the anomalistic year ;
and its excess over the sidereal year was about 3 minutes. From
this consideration it appears that the Indian astronomers were
justified in taking the Sun's apogee to be fixed.
Against the error of +3 min. in the Indian sidereal year,
we may point out that—
(1) The Hipparchus-Ptolemy tropical year
=365 da. 14' 48" in sexagesimal units,¹
=365 da. 5 hrs. 55 min. 12 secs., which has an
error of about +6 min.
(2) Meton's sidereal year
= [ 365 + ¼ + 1/76 ] days²
=365 da. 6 hrs. 18 min. 57 secs. which has an
error of +9 min. 48 secs. nearly.
(3) The Babylonian sidereal-year was 4½ min. too long.³
Thus the Indian value of it is closer to the true value.
Again in 150 A.D. the longitude of the Sun's apogee
according to the Conn. des Temps was
=101° 13' 15". 17-6189". 03 [ (1900—150) / 100 ]
=1". 63 × [ (1900—150) / 100 ]²
=71° 16' 26". 37
while Ptolemy states it to be 65° 30' which was wrong
by—5° 36' 27" :

1. Syntaxis, Karl Manitius's edition, Vol. I. p. 146.
2. Ibid, p. 145.
3. Encyclopaedia Britannica, History of Astronomy.
4. Syntaxis, Vol. I. p. 148. The Romaka Siddhānta of the Pañca-siddhāntikā,
   VIII. 2, indicates the Sun's apogee to be at longitude of 75° ; this was per-
   haps a correction made by Lāṭadeva to the Greek constant.

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.