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

ĀRYABHAṬA & BRAHMAGUPTA CONTROVERSY 93
already been said that perhaps to meet the popular demand,
Brahmagupta in this treatise took upon himself the task of sim-
plifying Āryabhaṭa's ardharātrika system or the system of
midnight day reckoning. Alberuni, the author of the Indika has
made several references or quotations from the Khaṇḍakhādyaka
proper and also its supplement, known as Uttara-Khaṇḍa-
khādyaka.
(a) There is a reference to the accepted circumference of
the Earth, as given in the Khaṇḍakhādyaka (Sachau's
Alberuni, Vol. I, p. 312)
Multiply the difference in longitude (from Ujjayinī) by
the (mean) daily motion of a planet (in minutes) and
divide by 4,800; apply the quotient taken as minutes
negatively in places east of the meridian line of Ujja-
yinī and positively in places lying west.¹
(b) The rules for finding the ahargaṇa as given in the
Khaṇḍakhādyaka in I. 3-5 (Sachau's Alberuni, Vol.
II, 46-47), to which Dr. Schram adds a valuable anno-
tation, the constants being taken from the later
Pauliśa Tantra as known to Bhaṭṭotpala. This Pauliśa
astronomy is derived from Āryabhaṭa I's ardharātrika
system.²
(c) A quotation from the Uttara Khaṇḍakhādyaka
(Sachau's Alberuni, Vol. II, pp. 84-86) which Sengupta
has given in his translation, Chapter X, pp. 148-152.
(d) A quotation also probably from the Uttara Khaṇḍakhā-
dyaka (Sachau's Alberuni, Vol. II, p. 87). These stanzas
are found in the Brāhmasphuṭasiddhānta, XIV, 47-52,
also quoted by Bhaṭṭotpala as occurring in the Brahma
Siddhānta in his commentary on the Bṛhat-Saṁhitā,
IV, 7. The manuscripts which Sengupta used did not
show them as occurring in the Uttara Khaṇḍakhādyaka.
These relate to the dimensions of the nakṣatras as seen,
as distinguished from the same as calculated.

1. उज्जयिनी-याम्योत्तर-रेखायाः प्रागृणं धनं पश्चात् ।
   देशान्तर भुक्तिवधात् ख खाष्टवेदैः कलाद्याप्तम् ॥

• KK. I, 15.

2. • KK, Pt. B. Misra's edition, p. 145.

94 BRĀHMASPHUṬASIDDHĀNTA & KHAṆḌAKHĀDYAKA
(e) Two quotations from the Uttara Khaṇḍakhādyaka rel-
ating to the celestial co-ordinates of Canopus and Sirius
(Sachau's Alberuni, Vol. II, p. 91). Present manus-
cripts do not show these stanzas, which are probably
the same as stanzas 35-36 and 40 of Chapter X of the
Brāhmasphuṭasiddhānta.
(f) Two quotations from the Khaṇḍakhādyaka proper as
alleged by Alberuni (Sachau's Alberuni, Vol.II, p. 116).
According to Āmarāja, the first is a couple of stanzas of
which the author is Bhaṭṭotpala and not Brahmagupa.
The second quotation cannot be traced. These relate to
finding the possibility of an eclipse whether of the Sun
or of the Moon.
(g) Two quotations from the Khaṇḍakhādyaka proper as
asserted by Alberuni (Sachau's Alberuni, Vol. II. p.
119). These relate to finding the Lords of the year and
of the month. According to Āmarāja the rules in
question were given by Bhaṭṭotpala and not by Brah-
magupta. Pṛthūdaka in his commentary on the first
chapter at its concluding portion says ;
“In this work the Khaṇḍakhādyaka, the teacher
(Brahmagupta) has not given the rules for finding
the Lords of the year and the month¹.”

1. अथाऽत्र खण्डखाद्यके वर्षाधिपमासाधिपा नयनमाचार्येण नाभिहितम् ।
   —: ० :—
   Reference
   P.C. Sengupta : The Khaṇḍakhādyaka, 1934.
   K.S. Shukla : Mahābhāskarīya, 1960.
   K.S. Shukla : Sūrya-Siddhānta, 1957.

Chapter IV
Brahmagupta’s Originality
in the Khaṇḍakhādyaka
Sengupta in his Indroduction to the Commentary of the
Khaṇḍakhādyaka has discussed this point. We shall reproduce
here some of the points mentioned by him.
Brahmagupta’s Khaṇḍakhādyaka
(i) Brahmagupta does not accept the system of Āryabhaṭa
but has simplified it in the Khaṇḍakhādyaka proper ; and here
he has given the system which he thinks to be correct.
Uttar Khaṇḍa Khādyaka
(ii) In the Uttara-Khaṇḍakhādyaka, he has further correc-
ted some of his results, given earlier in the Khaṇḍakhādyaka
proper. In the proper Khaṇḍakhādyaka, Brahmagupta assignes
to the longitude of the Sun's apogee the value 80º, whereas in
the Uttara text he corrects it to 77º (UKK. 4) :
As the process of finding the apparent places of planets
as given by Āryabhaṭa does not make them agree with
observation, I shall, therefore, speak of this process.
Of the Sun the apogee is at two signs and seventeen
degrees (2 signs 17º=60 plus 17 degrees=77º).¹
Compare this with the value given in the Khaṇḍakhādyaka
proper (I.13)² :
The longitude of the Sun's apogee is 80º [KK.I,13] (The
Sun's apogee is 80º or two signs plus 20 degrees) inocco means

1. न स्फुटमार्य्यभटोक्तं स्पष्टीकरणं यतस्ततो वक्ष्ये ।
   भानुमतो मन्दोच्चं राशिद्वयमंशकाश्च सप्तदश ॥ UKK. IX 4
2. भागाशीतिरितिनोच्चं शशिनः पादोनकृत शरकृतोनाद् ।
   भगणादि द्विस्त्रिर्दैर्वसुशुनव यम नव गुणैः सकलम् ॥ KK. I, 13

96 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA
mandocca of the Sun). The value given in the Pañcasiddhān-
tikā, IX.7-8) is also the same.
Let us compare it with the present value. According to
the astronomical constants as given in the Conn. des Temps., the
longitude of the Sun's apogee in 499 A.D. (i.e. 1,400 years before
1900 A.D.) was
—77° 19′19.44″ according to Conn.des Temps’
equation.
—76° 40′37.22″ according to Newcomb's equation.
The mean of these two values is very nearly 77⁰ as given
by Brahmagupta in the Uttara text. Thus the value given by
Brahmagupta is more correct than the value given by Āryabhaṭa.
The Āryabhaṭīya gives the value 78⁰ which is less correct.
Brahmagupta more correct than Āryabhaṭa
(iii) Brahmagupta detected that Āryabhaṭa had made the
Moon's apogee quicker and nodes slower, than they really are.
In both the cases, Brahmagupta made rather an over-correction.
We shall give the extract from Uttara-Khaṇḍakhādyaka in this
connection :
Multiply the ahargaṇa by 110, increase the product
by 511 and divide by 30, 31; subtract the result taken
as revolutions, etc., from the mean Moon; the final
result is the Moon's apogee.¹
Evidently Brahmagupta assumes that the anomalistic
month = 3031/110 days. This convergent to the anomalistic
month was known to the author of the Vasiṣṭha Siddhānta as
summarised in the Pañcasiddhāntikā² (II-2-6).
According to Brahmagupta, the length of the anomalistic
month
= 1582236450000—4320000000 days. (BrSpSi. I.15,16,18,
───────────────────────── and 20)
577533000000—488105858
= 27.55454641 days which is for 1900 A. D.
= 27.5545502 days according to Radau.
= 27.554602 according to the Āryabhaṭīya.

1. द्युगणात् ख रुद्र गुणिताद् भवशरयुक्ताच्छशित्रिखाग्नि हृतात् ।
   भगणादि फलं शोध्यं घसचन्द्राच्छशाङ्कोच्चम् ॥ (UKK. IX. 5)

UTTARAKHAṆḌA KHĀDYAKA 97
Here also Brahmagupta is more accurate.
Again, the length of the sidereal period of the Moon's
apogee
= 1577918450000 / 488105858 days
= 3232.732048 days.
Āryabhaṭa's value of the same is 3231.987844 days, and the
modern value is 3232.3754 days. Hence Brahmagupta's result
is by 0.3566 of a day out, while Āryabhaṭa's is by 0.3876 of a
day in.
Further in the Uttara-Khaṇḍakhādyaka(IX.10) we
have : Deduct 354½ from the ahargaṇa, divide the
remainder by 6792 ; subtract the quotient that is
obtained in revolutions etc. from the circle : the
result is the longitude of the ascending node.¹ (IX.10)
Here Brahmagupta gives the approximate period of the
sidereal revolution of the Moon's node to be=6792 days. This
according to his Brāhmasphuṭasiddhānta = 1577916450000 / 232311168 days
= 6792.25396 days, which according to Lockyer would be
6793.39108 days and according to the Khaṇḍakhā

98 UTTARAKHAṆḌA KHĀDYAKA
one-sixteenth.¹
This stanza says that in 499 A.D., Mars's aphelion point
had a longitude of 127⁰ ; of Jupiter the longitude of the aphelion
was 170⁰. ( KK.II. 6²)
According to Newcomb's rule, the longitude of the aphelion
point of Mars in 499 A. D. works out to have been = 128⁰28'12''.
According to the Conn. des. Temps' rule, the same was 128⁰27'51''.
Hence Brahmagupta's determination of Mars's aphelion is correct
within 1⁰30' and is therefore, quite satisfactory. According to
the Khaṇḍakhādyaka proper it was 110⁰, and according to the
Āryabhaṭīya 118⁰.
Of planets, beginning with Mars, the degrees of longitude
of the apogees are respectively 11, 22, 16, 8 and 24, each multiplied
by 10. (KK .II. 6)
Thus the longitudes of apogees of Mars = 110 (3 signs 20⁰);
of Mercury = 220⁰ (7 signs 10⁰) ; of Jupiter = 160⁰ (5 signs 10⁰);
of Venus = 80⁰ (2 signs 20⁰) and of Saturn = 240⁰ (8 signs).
Compare these values with those given in the Pañcasiddhāntikā
XVII. 2 (the Sūrya-siddhānta).
Again according to this stanza Jupiter's aphelion had a
longitude of 170⁰ in 499 A.D.
According to Conn. des. Temps' rule the same was 170⁰25'.
Thus here too, Brahmagupta is very accurate. According to
the Khaṇḍakhādyaka proper, Jupiter's aphelion had a longitude
of 160⁰ (KK. II.6) and according to the Āryabhaṭīya, the value
was 180⁰.
Brahmagupta First to Use Second Differences
All these illustrations reproduced here very well establish
the point that the great Indian astronomers from Āryabhaṭa I
to Brahmagupta were aware of the methods of separating the
two distinct planetary inequalities, viz., that of the apsis and
of conjunction in the cases of the five ‘star’ planets (PSi. Intro-
duction Lii). In the Khaṇḍakhādyaka, Brahmagupta having
given the “sines” and the equations of the Sun and the Moon

1. सप्तदशांशैरधिकं भौमस्योच्चं गुरोर्दशभिरंशैः ।
   सितशीघ्रात् कृतमुनयो लिप्ताः शोध्याः शनेः फलं मान्द्यम् ।।
   पञ्चांशोनं शैघ्यं षोडशभागाधिकं बुधस्य फलम् ।। —UKK. IX. 11
2. मन्दांशा दशगुणिता रुद्रा द्वियमाश्व षोडशाष्टजिनाः । —KK. II. 6

BRAHMAGUPTA FIRST TO USE SECOND DIFFERENCES 99
at the interval of 15° of arc of the mean anomaly, in the Uttara
Khaṇḍakhādyaka teaches, for the first time in the history of
mathematics, the improved rules for interpolation by using the
second difference. This very important feature I am reproducing
here from the translation by Senagupta of the verse ¹
(UKK. 8) :
Multiply the residual arc left after division by 900'
(i.e by 15°), by half the difference of the tabular
difference passed over and that to be passed over and
divide by 900' (i. e. 15°) : by the result increase or
decrease, as the case may be, half the sum of the same
two tabular differences ; the result which, whether less
or greater than the tabular difference to be passed, is
the true tabular difference to be passed over. (UKK. 8)
The rule given here applies to the case of all functions
hitherto considered in the Khaṇḍakhādyaka, which are tabulated
at the difference of 15° of arc of the argument. They are :
(i) the tabular differences of the Sun's equation,
(ii) the tabular differences of the Moon's equation.
(iii) the tabular differences of the 'sines'.
Sengupta has illustrated the rule by an example belonging
to the table of sines.
Illustration—To find the 'sine' of 57°.
Brahmagupta's table of sines in the Khaṇḍakhādyaka is as
follows :
Thirty increased severally by nine, six and one; twenty-
four, fifteen and five, are the tabular differences of
sines at intervals of half-a-sign. For any arc, the 'sine'
is the sum of the parts passed over, increased by the
proportional part of the tabular difference to be passed
over.² (*KK.*I.30 ; also III.6)

1. गतभोग्य खण्डकान्तरदलविकलवधात् शतैर्नवभिराप्त्या ।
   तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम् ॥ —UKK. IX. 8
2. त्रिंशत् सन्नवरसेन्दुजिन (तिथि) विषयागृहार्द्धं चापानां ।
   भर्द्ध्ज्या खण्डानि ज्या भुक्तैक्यं सभोग्यफलम् ॥ —KK. I. 30; also III. 6

100 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA
This can be shown in the tabular form thus :

Now 57° = 3420 minutes = 900' × 3 + 720'. Thus three of the
tabular differences are considered as passed over ; the last one
being 31 and the one to be passed over is 24.
The true tabular difference by the rule, for arc 57°,
= (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2
Hence the 'sine' of 57°
= 39 + 36 + 31 + (720 / 900) [ (31 + 24) / 2 - (720 / 900) × (31 - 24) / 2 ]
= 125.76
As worked out from the logarithm tables the same comes
out to be 125.80.
Again 'sine' of 57° from Brahmagupta's formula
= 106 + (720 / 900) × 24 + ((31 - 24) / 2) × (720 / 900) - [(720 / 900)] × (31 - 24) / 2
= 106 + (720 / 900) × 24 + (720 / 900) { (720 / 900) - 1 } × (24 - 31) / 2
This in fact is the modern form the interpolation equation
up to the term containing the second difference. Brahmagupta thus
takes a decidedly improved step here and is undoubtedly the first
man in the history of mathematics who has done this. One should
also remember that in the case where the function is not tabula-
ted at a constant interval, Brahmagupta's rule is remarkable.

BRAHMAGUPTA AND SINE RULE 101
Brahmagupta First to Introduce
Sine Rule in Indian Plane Trigonometry
In this connection, we shall reproduce the following verse
from the Khaṇḍakhādyaka :
Multiply the ‘sine’ of the (Śīghra) anomaly by the ‘sine’
of the maximum Śīghra equation and divide by the
‘sine’ of the corresponding Śīghra equation, the result
is the ‘Śīghra hypotenuse’ when the (Śīghra) anomaly
is half a circle, this śīghra hypotenuse is equal to the
radius diminished by the ‘sine’ of the maximum
equation ; when the anomaly is equal to the whole
circle, the same is equal to the radius increased by the
same ‘sine’ of the maximum equation.¹
Let S, E and P be the positions of the Sun, the Earth and the
planet, say Mars, respecti-
vely. Complete the paralle-
logram SEMP ; with M as
centre and MP as radius,
describe a circle. This circle
is the epicycle of conjunction
of Mars. Produce EM to cut
this circle at K. The ∠PMK
= ∠S′SP (the point S′ is on
[Fig. 4] ES produced ), the angle
gained by the Earth over Mars since the preceding conjunction.
The ∠PMK is called the śīghra anomaly or anomaly of conjunc-
tion. We take EM=360, and MP=234. The ∠PEM, which is
equal to ∠PES, the annual parallax of Mars, is called the śīghra
equation. The ∠MPE is equal to the ∠SEP, the elongation.
The ∠PMK is given, and PM and ME are also given. Hence in
the triangle MPE, we have
tan ½ (P−E) = (EM−MP)/(EM+MP) tan ½ PMK
= 126/594 tan ½ PMK
∴ L tan ½(P−E) = log [126/594] + L tan ½ PMK
We have also ½ (P+E) = ½ ∠PMK

1. केन्द्रज्याऽन्त्यफलज्या गुणिता फलजीवया हृताकर्णः ।
   त्रिज्यास्य फलज्योना चक्रार्द्धे संयुता चक्रे ॥ KK. VI. 1.

102 BRAHMAGUPTA'S KHAṆḌAKHĀDYAKA
Now log [ 126 / 594 ] = 1.3265841.
The values of the ∠PMK and the ∠PEM and Brahma-
gupta's values as given in the verse¹ are presented below in a
tabular form :
| ∠PMK = | 28° | 60° | 90° | 121° | 135° | 148° | 164° | 173° |
| ∠PEM = | 10°58′ | 23°1′ | 33°1′ | 39°56′ | 40°23′ | 37°31′ | 25°32′ | 12°35 |
| Brahmagupta's ∠PEM = | 11° | 23° | 33° | 40° | 40°30′ | 37°30′ | 25°30′ | 12°30′ |
It will be seen that Brahmagupta gives the values of the
equation within 1/8th of a degree. It seems inexplicable why
such discrepancies should remain in Brahmagupta's calculations.
It is probable that he wanted to state his equations to the nearest
half a degree.
Now we shall take up the Śīghra equations of Mars, and
then revert to the Sine Rule. We have in the Khaṇḍakhā-
dyaka :
Mars, by the degrees of Śīghra anomaly (i.e. anomaly of
conjunction) of 28 getting at the corresponding equation of 11°
rises (heliacally) in the east ; by the next 32° gets 12° more of the
equation ; by the next 30°, 10° more ; by the next 31°, 7°, more ; by
next 14°, half a degree ; these are positive ; by the next 13°, nega-
tive 3° ; by the next 16° ; negative 12° after this he is retrograde ;
by the next 9°, negative 13° ; by the next 7°, negative 12½°.
After this the parts of the equations occur in the reverse order¹.
On the basis of this we have the following table of the
Śīghra equations for Mars :
| Degrees of anomaly | Equation of | Phenomena |
| of conjunction | conjunction | |
| 0° | 0° | Motion direct. |
| 28° | +11° | Rises in the east. |

1. भौमोऽष्ट्यग्नौ रुद्रान् भुक्त्वा पूर्व्वौदितोऽदरैर्कान् ।
   ख_णैर्दशरूपगुणैः सप्तांशा मनुभिरर्द्धांशान् ॥
   धनम् 'मग्नि शशाङ्कैस्त्रिनष्ट्या भास्करात्नतो वक्री ।
   नवभिश्च योदशनगैर्द्वादशसार्द्धान् विलोमोऽतः ॥ KK.II.8-9

BRAHMAGUPTA AND SINE RULE 103
Degrees of anomaly Equation of Phenomena
of conjunction conjunction
───────────────── ─────────── ─────────
60° 11+12=+23°
90° 23+10=+33°
121° 33+ 7=+40°
135° 40+ ½=+40°30'
148° 40°30'—3°=+37° 30'
164° 37°30'—12°=+25°30' Retrograde motion
begins.
173° 25°30'—13°=+12°30'
180° 12°30—12°30'=0°0'
187° —12°30'
196° —25°30' Direct motion begins.
212° —37°30'
225° —40°30'
239° —40°
270° —33°
300° —23°
332° —11° Sets in the west.
360° 0°
Now we come back to our discussion on the verse VI.1.
The Śīghra hypotenuse spoken of here is EP, when SP or
EM is taken to be R ; when ∠PEM is a maximum, PM is its
'sine'.
It would be seen from the figure that
R sin PMK × PM
EP = ────────────────────
Rsin PEM
This may again be written as
EP PM
───────────── = ──────────────
sin PMK sin PEM
This is equivalent to the sine rule for a triangle in plane
trigonometry. Brahmagupta is here seen to be the first person
to give it in Indian mathematics. This expression reminds us
of the famous relationship in respect to triangle ABC :
a b c
─────── = ─────────── = ───────
sin A sin B sin C

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 :-