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

136 GREEK AND INDIAN METHODS
where Z is the zenith and Q is the point of intersection of the
prime vertical and its secondary passing through the Sun and
the north-south points.
Zeuthen¹ points out that later in the same treatise Ptolemy
finds the arc 2β described above the horizon by a star of given
declination δ' by a procedure equivalent to the formula.
(4) cos β=tan δ' tan ϕ.
With regard to the 'Analemma' of Ptolemy. it may be
noted, as Heath² says, that "the procedure amounts to a
method of graphically constructing the arcs required as parts of
an auxiliary circle in one plane." Many things may be, in
practice, done graphically far more easily than by the theoreti-
cal method. Besides, no theoretical calculations occur in the
'Analemma'. Zeuthen², following the method of this work, has
deduced in the general case, the two equations.
(5) cos Z=(cos δ, cos H+sin δ. tan ϕ) cos ϕ.
cos δ.sin H
(6) tan α=————————————————————————————————————————
sin δ
————— +(cos δ.cos H+sin δ.tan ϕ) sin ϕ
cos δ
These equations are suggested to a modern reader from a
study of the figures in the 'Analemma.' But neither in this work
nor in the 'Syntaxis' are they to be found. With regard to the
first four formulae, it is possible that they were recognised by
Ptolemy. With regard to the last two, Zeuthen³ remarks "mais
le texte nen contient rien,' and they were certainly not recog-
nised by Ptolemy.
Besides the tangent function is wholly absent in Greek
trigonometry. They are also different in form from those arrived
at by the Indian method as explained before. Thus, it is clear
that the Indian methods are in no way connected with the
method of the 'Analemma.'
Even taking for granted that the Indians followed a method
of projection much allied to the method of the Analemma' there
is no adequate reason for assuming that their method is derived
from any Greek source. Analogy and precedence do not neces-
sarily constitute originality—there is still the chance of a remoter
origin from which both the systems drew their inspiration. The
method of the 'Analemma,' as has been already stated, presents a
——————————————————————————————————————
1, 2, Bjornbo, loc. cit. p. 86.
3. Zeuthen, loc. cit. p. 27.

ANGLE BETWEEN ECLIPTIC AND MERIDIAN 137
graphical method for constructing the Sun's altitude and azimuth
from the hour angle when the Sun's declination is zero but such
a graphical method is generally complex as compared with the
elegant Indian method. An astronomer who constructs and uses
an armillary sphere to arrive at his equations in spherical astro-
nomy and who has not a well-developed spherical astronomy at
his command must have to draw perpendiculars from the positions
of the heavenly body, not only on the meridian plane, the hori-
zon or on the prime vertical, as the occasion arises, but also on
the line of intersection of the diurnal circle with the horizon.
Hence Braunmühl's statement that the Indian methods of spheri-
cal astronomy have their origin in the 'Analemma's, in spite of
his admitting that Indians were first to utilise its methods, is rather
far-fetched and tends to take away the honour from the great
Indian astronomers, who devised the beautiful methods. The
'Analemma' as it now exists is a Latin translation from an Arabic
version of the original Greek¹. We may reasonably doubt that the
Arabic version was greatly influenced by the ancient Indian system.
We now pass on to the consideration of other allied or
similar problems in the two systems of astronomy.
Problem V—
To find the Angle between the Ecliptic and the Meridian
Indian Method²
Let ♈SA be thee cliptic, ♈CE
the equator, E the east-point of
the horizon (Fig 10). Cut off
SH=90° and draw the great
circle HEAP' cutting the meri-
dian P'SCH at the points P' and
H. The aim is to find AP' but
it is enough to find EA since AP'
is the complement of EA'.
Both Āryabhaṭa and
Brahmagupta were unable to
find EA correctly. Let P be the
celestial pole and let PAE' be

1. On the influence of the ancient Indians on Arab mathematics and astronomy; see Alberuni's India, translated by Dr. E. Sachau, Vol. II, p. 304.
2. Āryabhaṭa, Gola, 45; BrSpSi, IV. 17; Sūrya-siddhānta, IV. 25; Bhāskara's Golādhyāya, VIII, 21-74, first example in his own commentary.

138 GREEK AND INDIAN METHODS
the secondary to the equator cutting it at E'. Both the above
astronomers were content with the idea that AE = AE', or that
AE = the declination of the point A of the ecliptic which is 90°
ahead of S in the above figure. This idea continued till the
time of Bhāskara II (1150 A. D.) who found out the correct
equation.
He recognised that CS, the declination of S = PP'; P'EH is
then the horizon of the station whose north geographical latitude
is CS. Also, the ‘sine’ of EA is the ‘Agrā’ or the sine of the
amplitude of the point A for the latitude CS.
∴ R sin EA = (R sin AE' × R) / (R cos CS) = [R sin (90° + γS) × R sin ω] / R
× R / (R cos CS)
or R sin EA = [R sin (90° + l) × R sin ω] / (R cos δ)
where l stands for γS and δ for CS.
Greek Method :
We give below the Ptolemy's method in a slightly modified
form¹. Let SHA be the triangle and γCE be the transversal ;
then we have,
(sin SC / sin CH) × (sin HE / sin EA) × (sin Aγ / sin γS) = 1
or (sin δ / cos δ) × (sin 90° / sin EA) × [sin (90° + l) / sin l] = 1
∴ sin EA = [sin δ × sin (90° + l)] / (cos δ × sin l) ,
which is readily transformed into Bhāskara's equation. The
originality of Bhāskara would be readily admitted.
Problem VI--
To find the Angle between the
Ecliptic and the Horrizon
Indian Method :
(A) Āryabhaṭa's method. It consists of the following²
steps :--
(1) Determination of the orient point of ecliptic.
(2) Finding the sine of its amplitude.

1. Manitius, ibid, Book I, pp. 104-06.
2. Āryabhaṭa, Gola, 33 : Sūryasiddhānta, V. 5-6.

ANGLE BETWEEN ECLIPTIC AND HORIZON 139
(3) Determination of the culminating point of the ecliptic
from the hour-angle of the Sun.
(4) Finding the declination of the culminating point of the
ecliptic.
Having obtained the above elements, his rule can be follow-
ed thus :
In this Fig. 11 NZH is the meridian, HMEAN the horizon,
CN'A the ecliptic. If N' be the nonagesimal or the highest point
of the ecliptic, the altitude of N' is the inclination of the ecliptic
to the horizon.
Let ZN'M be the vertical through N', meeting the horizon at M.
When the time is given, the longitudes of A and C can be
found out, from which CZ the zenith distance of C and EA the
amplitude of the orient ecliptic point can be determined.
Fig. 11
Here HM=EA.
According to Āryabhaṭa,
R sin CN' = (R sin CZ × R sin HM) / R
and R sin ZN' = √((R sin CZ)² - (R sin CN')²)
This is only an approximate rule. As expressed here,
R sin ZN' = (R sin CZ × R cos HM) / R approximately.
= (¹R sin CZ × R cos HM × R) / (R × R cos CN') accurately.
= (R sin CZ × R cos HM) / (R cos CN') .
(B) The method of Brahmagupta² :
Brahmagupta would also first determine the orient ecliptic

1. This correction was perhaps first noticed by Raṅganātha (1603 A. D.)
   in his commentary in the Sūryasiddhānta.
2. BrSpSi. V 3.

140 GREEK AND INDIAN METHODS
point A. Then he subtracts 90° from the longitude of A. Thus
having the longitude of N', he next finds the part of the day
elapsed of N' ; from which by the time-altitude equation discussed
above, he finds ZN'. This is of course more accurate than that of
Āryabhaṭa. Bhāskara¹ here follows Brahmagupta.
Greek Method :
Let the ecliptic CN'A cut the lower half of the meridian at
F. Ptolemy takes AK along the ecliptic=90° and AR along the
horizon=90°; then the great circle passing through R and K passes
through the nadir Z'. Now take Z'FK for the triangle and ANR
for the transversal, then by Menelaus's theorem.²
(sin FN / sin NZ') × (sin Z'R / sin RK) × (sin KA / sin AF) = 1
∴ sin RK = (sin FN / sin AF) = (cos FZ' / sin AC) = (cos CZ / sin AC) = (sin CH / sin AC)
or sin MN' = sin CH / sin AC.
Here Ptolemy's equation is simpler than that of Āryabhaṭa;
hence they must be independent of each other.
[Figure 12] Problem VII:—
To find the Angle made by the
Vertical through any Point of the
Ecliptic with the Latter
This problem is considered
by Ptolemy but it is not consider-
ed separately in Indian Astronomy,
but from the rule for parallax in
longitude, the rule for its calcula-
Fig. 12 tion can be deduced.
Indian Method :
In Fig. 12 S represents the true position of the
Sun and S' the Sun's position as depressed by parallax. N'SA is
the ecliptic. If from S', S'Q be drawn perpendicular to the ecli-
ptic, then, if P is the horizontal parallax,

1. Grahaganita; XII, 3-4.
2. Manitius, ibid, pp. 110-111.

ANGLE MADE BY THE VERTICAL 18
SQ = SS' × (R cos S'SQ / R) = (P × R sin ZS / R) (R cos S'SQ / R)
¹= (P / R) √[(R sin ZS)² – (R sin ZN')²]
²= (P / R²) × R sin N'S × R cos ZN', where N' is the nonagesimal
Thus R cos S'SQ is seen to be
= (R sin N'S × R cos ZN') / (R sin ZS)
The Indian method is fully described by Bhāskara in his
'Golādhyāya. VIII, 12-25. The truth of the Indian rule for R cos
S'SQ is easily seen from the spherical triangle ASN, where A is
the pole of the ecliptic.
Greek Method :
³Ptolemy takes SK and SL 90° each, along the vertical
circle ZSEK and the ecliptic N'SA. The great circle through K
and L cuts the horizon at R which is the pole of the vertical
circle. He takes SKL for the triangle and EAR for the trans-
versal, then
(sin SE / sin EK) × (sin KR / sin LR) × (sin LA / sin AS) = 1
or sin LR = (cos ZS × cos AS) / (sin ZS × sin AS)
or cos S'SQ = cot ZS × cot AS = tan SE × cot AS.
The Indian and the Greek rules are altogether different
both in form and method. There can, therefore, be no question
of any connection between them.
Problem VIII :—
To convert the Celestial Longitude of a Heavenly
Body into its Polar Longitude
If σ be the position of a (Fig.13), γK and σK are the celestial
longitude and the celestial latitude, respectively : γM and σM
are the polar longitude and polar latitude : γN and σV are the
right ascension and declination of the star.
Indian Method :
All Indian astronomers attempt at finding MK which, sub-

1. Āryabhaṭa, Gola, 34; Pañcasiddhāntikā, IX, 22 BrSpSi, XI, 23.
2. BrSpSi. V, 4-5 ; Sūryasiddhānta, V, 7-8 Bhāskara, Grahagaṇita, XII, 4.
3. Manitius, ibid, p. 119.

140 GREEK AND INDIAN METHODS
point A. Then he subtracts 90° from the longitude of A. Thus
having the longitude of N', he next finds the part of the day
elapsed of N' ; from which by the time-altitude equation discussed
above, he finds ZN'. This is of course more accurate than that of
Āryabhaṭa. Bhāskara¹ here follows Brahmagupta.
Greek Method :
Let the ecliptic CN'A cut the lower half of the meridian at
F. Ptolemy takes AK along the ecliptic = 90° and AR along the
horizon = 90°; then the great circle passing through R and K passes
through the nadir Z'. Now take Z'FK for the triangle and ANR
for the transversal, then by Menelaus's theorem.²
(sin FN / sin NZ') × (sin Z'R / sin RK) × (sin KA / sin AF) = 1
∴ sin RK = (sin FN / sin AF) = (cos FZ' / sin AC) = (cos CZ / sin AC) = (sin CH / sin AC)
or

ANGLE MADE BY THE VERTICAL 141
SQ = SS' × (R cos S'SQ / R) = (P × R sin ZS / R) × (R cos S'SQ / R)
¹= (P / R) √((R sin ZS)² - (R sin ZN')²)
²= (P / R²) × R sin N'S × R cos ZN', where N' is the nonagesimal.
Thus R cos S'SQ is seen to be
= (R sin N'S × R cos ZN') / (R sin ZS)
The Indian method is fully described by Bhāskara in his
'Golādhyāya. VIII, 12-25. The truth of the Indian rule for R cos
S'SQ is easily seen from the spherical triangle ΠZS, where Π is
the pole of the ecliptic.
Greek Method :
³Ptolemy takes SK and SL=90° each, along the vertical
circle ZSEK and the ecliptic N'SA. The great circle through K
and L cuts the horizon at R which is the pole of the vertical
circle. He takes SKL for the triangle and EAR for the trans-
versal, then
(sin SE / sin EK) × (sin KR / sin LR) × (sin LA / sin AS) = 1
or sin LR = (cos ZS × cos AS) / (sin ZS × sin AS)
or cos S'SQ = cot ZS × cot AS = tan SE × cot AS.
The Indian and the Greek rules are altogether different
both in form and method. There can, therefore, be no question
of any connection between them.
Problem VIII :—
To convert the Celestial Longitude of a Heavenly
Body into its Polar Longitude
If σ be the position of a (Fig.13), γK and σK are the celestial
longitude and the celestial latitude, respectively ; γM and σM
are the polar longitude and polar latitude ; γN and σV are the
right ascension and declination of the star.
Indian Method :
All Indian astronomers attempt at finding MK which, sub-

1. Āryabhaṭa, Gola, 34; Pañcasiddhāntikā, IX, 22 BrSpSi, XI, 23.
2. BrSpSi, V, 4-5; Sūryasiddhānta, V, 7-8 Bhāskara, Grahagaṇita, XII, 4.
3. Manitius, ibid, p. 119.

142 GREEK AND INDIAN METHODS
tracted from, or added to, γK the celestial longitude, gives γM
the polar longitude.
According to Āryabhaṭa¹,
MK = (σK × R vers γK × R sin ω) / R² .
Brahmagupta² makes a dis-
tinct improvement on Āryabhaṭa
and gives his rule for finding the
projection MK on the celestial
equator.
If P be the celestial pole, PKH
the secondary to the equator,
Brahmagupta says that,
Fig- 13
NH = (σK × R sin (γK+90°) × R sin ω) / R
If from σ, σR is drawn perpendicular to PKH, it is evident
that,
R sin σ R = (R sin σ K × R sin σ KR) / R
According to Āryabhaṭa and Brahmagupta, as explained
before,
R sin σKR = (R sin (γK+90°) × R sin ω) / R
Hence Brahmagupta intends that,
NH = σR = (σK × R sin σKR) / R
which is rather a big assumption. He then directs the finding of
the part of the ecliptic of which σR or NH is the projection on
the equator thus approximately to MK.
Āryabhaṭa, Brahmagupta³ and the modern Sūryasiddhānta
take the declination σN = σK + KH where σK is small. They
do not consider the case where σK is large.
Bhāskara alone gives us fairly correct rules for this trans-
formation of co-ordinates.

1. Āryabhaṭa, Gola, 36.
2. BrSpSi X, 17.
3. BrSpSi, X, 15, Sūryasiddhānta, II, 58.

CELESTIAL AND POLAR LONGITUDE 143
In order to find σN, he would multiply
σK by (R cos σKP) / R ; according to him,
σN = (σK × R cos σKP) / R + KH¹
This is a decided improvement on Brahmagupta's corres-
ponding rule. The declination σN obtained would be very
nearly accurate.
Having obtained σN, Bhāskara² then directs the finding of
NH, thus,
NH = (σK × R sin σKP) / (R cos σN)
He then directs the finding of MK on the ecliptic of which
NH is the projection by means of the times of rising of the signs
of the zodiac on the equator.
Thus, the Indian methods show a beginning and develop-
ment only. The Greek method as given by Ptolemy is mathe-
matically accurate.
Greek Method³
To transform the celestial longitude and celestial latitude to
right ascension and declination.
Let the great circle ΠσK meet the equator at Δ. Ptolemy
would then from the given value of γK, find γΔ and ΔK by
using his tables for the rising of signs of the zodiac on the equa-
tor. He then takes ΠPσ for the triangle and γNΔ Q for the
transversal. The Menelaus' Equation, then, is
(sin ΠQ / sin QP) × (sin PN / sin Nσ) × (sin σΔ / sin ΔΠ) = 1²
Here ΠQ = 90° + ω, QP = 90°, PN = 90°; σΔ = σK + KΔ.
ΠΔ = 90° + KΔ, whence Nσ is obtained.
He next takes PNQ for the triangle and ΠσΔ for the trans-
versal,

144 GREEK AND INDIAN METHODS
It is almost needless to say that neither in the method nor
in the rules is there any agreement between the Indian and
Greek spherical astronomy in the solution of this problem.
Kaye’s view¹
As to Kaye, it appears that he has not been able to find a
method in the translation of the Sūyrasiddhānta by Burgess. The
figures of his paper referred to before do not show the “Akṣakṣet-
ras” even in their projections on the meridian place. He refers
to Braunmühl’s History of Trigonometry but does not appear to
have been able to follow him in his “Methode der indischen
Trigonometrie.” Kaye, however, is not slow in belittling Indian
trigonometry when he says :—The Indian astronomers employed
the sine function principally and the versed sine occasionally ;
they never employed the tangent function; and generally, but not
always, preferred to employ the sine of the complementary angle
rather than the cosine functions.’’
It is evident that Kaye
never understood the meaning
of the Indian functions of
‘sine’ and ‘cosine.’ These
functions are fully explained
by Bhāskara² when he
says :—
“Of that point the dis-
tance from the east-west line
is the sine and the distance of
the point from the north-south
line is the “cosine”.
[Fig. 14: Circle with points N, S, E, W; P₁, P₂, P₃, P₄; N₁, N₂, N₃, N₄; M₁, M₂, M₃, M₄, A]
Fig. 14
In (Fig. 14), of the arc AP₁, P₁M₁ is the “sine”
and P₁N₁ is the “cosine” of AP₂, P₂M₂ is the “sine” and P₂N₂
is the “cosine”; of AP₃, P₃M₃ is the “sine” and P₃N₃ is the
“cosine”; etc. It is evident that a better definition of these fun-
ctions was never given.
We have thus seen that some of the solutions of Āryabhaṭa

1. J. A. S. B., N. S., XV, p. 154.
2. तस्य बिन्दोः प्राच्य-परायाश्च यदन्तरं सादोर्ज्या ।
   बिन्दोर्याम्योत्तरायाश्च यदन्तरं सा कोटिज्या ॥
   —(Bhāskara, Grahagaṇita, commentary. II. 88-21)

KAYE'S VIEW 145
are imperfect, of Brahmagupta the solutions are more accurate,
while those of Bhāskara are generally mathematically correct.
The date of the scientific ancient Indian Astronomy is indeed
499 A. D., while that of the Syntaxis is about 150 A. D. It is by
these shortcomings and differences in the methods, new ideas (e.g.,
the idea of the differential calculus)* and the like, that we
can safely say that Indian Astronomy in its scientific form,
although of a later date than the "Syntaxis" of Ptolemy, is origi-
nal and not borrowed from foreign source. There is evidence
that some crude form of Greek astronomy was transmitted to
India and went by the name of the "Romaka" or the "Pauliśa"
Siddhānta, prior to the time of Āryabhaṭa but our great Indian
astronomers, Āryabhaṭa with his pupils, Varāha-mihira and
Brahmagupta, had to construct a new science altogether.
(This Chapter is almost a reproduction of the paper by
P. C. Sengupta, as acknowledged earlier).
—: o :—
Reference
P.C. Sengupta : The Khaṇḍakhādyaka, 1934

The provided image is completely blank (empty white page), with no visible text or diagrams to transcribe. Please upload a clear image containing the text you would like transcribed.

CHAPTER VII
Epicyclic Theory
of Ancient Indians
We shall give here some details of the Indian concepts
regarding the motion of planets or wandering bodies among the
stars. The Vedāṅga Jyotiṣa (1400 B. C. or earlier) does not speak
of this, A comparison of the astronomical constants of the
Greek and the ancient Indian systems, points unmistakably to the
conclusion that the Indian constants as determined by Āryabhaṭa
I and his successors, are almost in all cases different from those
of Greeks. Indian astronomers were highly original in their con-
cepts and treatment. The originality of Āryabhaṭa I and other
astronomers would be seen from what we are discribing below.
Apparent Motions of the Sun and Moon
We have the following passages from Āryabhaṭa :
All planets move in eccentrics to their orbits at the mean
rates of angular motion, in the direction of the signs of the
zodiac from their apogees (or aphelia) and in the opposite
directions from their Śīghroccas.
The eccentric circles of planets are equal to their concentrics
and the centre of the eccentric is removed from the centre
of the Earth.
The distance between the centre of the Earth and the centre
of the eccentric is equal to the radius of the planet's epicycle;
on the circumference (whether of the epicycle or of the
eccentric) the planet undoubtedly moves with the mean
motion.
Here the central idea was that undou-
btedly planets moved unifomly in circles
round the Earth; if the motion appeared
to be variable, it was due to the fact that
the centres of such circle (i. e. the con-
centric circles) did not coincide with the
centre of the Earth.
Fig. 15

148 EPICYCLIC THEORY OF ANCIENT INDIANS
Let E represent the centre of the Earth(Fig.15).APM the Sun's
circular orbit or concentric; let A and P be the apogee and the
perigee respectively. From EA, cut off EC equal to the radius of
the Sun's epicycle. With centre C and radius equal to EA describe
the eccentric A'P'S cutting AP and AP produced at P' and A'.
Here A' and P' are the real apogee and perigee of the Sun's orbit.
Let PM and P'S be any two equal arcs measured from P and P'.
The idea is that the mean planet M and the apparent Sun S
move simultaneously from P and P' in the counterclockwise
direction along the concentric and the eccentric circles. They
move with the same angular motion and arrive simultaneously
at M and S.
Here EM and CS are parallel and equal, hence MS is
also equal and parallel to EC. Let SH be drawn perpendicular
to EM.
The angle PEM is the mean anomaly and the angle P'ES the
true anomaly; the angle SEM is the equation of the centre, is readily
seen to be plus (+) from P' to A' and minus (—) from A' to P'.
Thus as regards the character of the equation, the eccentric circle
is quite right. We now turn to exmine how far it is true as to
the amount.
Let the angle SEM denoted by E and the angle ∠PEM
=∠P'CS=θ; EP=CP'=a; EC=MS=p, then
tan E = SH / HE = p sin θ / (a - p cos θ)
∴ E = (p / a) sin θ - (p² / 2a²) sin 2θ + (p³ / 3a³) sin 3 θ.........
Now the true value of E in elliptic motion is given by
E = ( 2e - e³/4 ) sin θ + (5/4) e² sin 2θ + (13e³ / 12) sin 3θ*;...
It we now put p / a = 2e - e³/4, as a first approximation p / a = 2e.
Hence p² / 2a² = 2e², which is greater than 5/4 e² by 3/4 e². In the
case of the Sun if the value of p be correctly taken the error in the
coefficient of the second term becomes +3'; similarly in the case of
the Moon, the corresponding error becomes +8'.
*Godfray's Astronomy, p. 149.

APPARENT MOTION OF THE SUN AND MOON 149
Again if p/a=2e, what is the centre of the eccentric circle is
the empty focus of the ellipse or that the ancient astronomers
practically took the planets to be moving with uniform angular
motion round the empty focus. This was not a bad approxi-
mation.
Also ES=r=EH approximately,
∴ r=a ( 1 - p/a cos θ )
but in the elliptic motion
r=a (1-e cos θ).*
Hence the error is not very considerable here also.
This is the way in which the ancient astronomers, both
Greek and Hindu, sought to explain the inequalities in the motion
of the Sun and the Moon. In the case of the Moon, these astro-
nomers took the coefficient 2e - e³/4 = 300' nearly; the modern value
is 377' nearly. The reason for this has been pointed out to be
that the Moon was observed correctly only at times of eclipses.
At the eclipses of syzygies, the evection term of the Moon's equa-
tion diminishes (numerically) the principal ecliptic term by
about 76'.
We have thus far explained the idea of planetary motion of
the ancients under the eccentric circle costruction. The same,
however, is explained under the epicyclic construction.
Let AMP be the circular orbit of the Sun, having E the centre
of the earth for the centre. (Fig. 16)
Let the diameter AEP be the apse
line. A the apogee and P the perigee.
Let M be the mean position of the
Sun in the orbit. With M as the
centre, describe the epicycle UNS.
Let EM cut the epicycle at N and U.
Now the construction for finding S
the apparent Sun is thus given:—
Fig. 16
Make ∠UMS= ∠MEA, the arc US is measured clockwise
whereas the arc A to M is measured counterclockwise.
From this construction MS is parallel to EA. If EC be
measured equal to MS, the radius of the epicycle, along EA to-

• Godfray's Astronomy, p. 149.

150 EPICYCLIC THEORY OF ANCIENT INDIANS
wards the apogee, then CS is a constant length and C is a fixed
point. Hence the locus of S is an equal circle with the centre at
C. Thus both the eccentric, and the epicycle and the concentric
combined, led to the same position.
It was thus usual to explain the planetary motion under
both the assumed constructions; and both gave the position for a
planet. The eccentric circle construction appears to be the earlier
in the history of astronomy and the latter was later. If the
former construction can be traced to Apollonius of Perga who did
so much to develop the "conic sections" as science, the reason why
he preferred the eccentric circle to the ellipse, appears to be that
either that this planetary construction was always deep-rooted in
the minds of men or that he was carried by the idea that "the
circle was the most perfect curve." We are inclined to the view
that the eccentric circle idea was transmitted from Babylonia to
Greece. We now pass on to consider the Indian construction for
the possition of superior and inferior planets.
Superior Planets
With regard to the five planets, Mercury, Venus, Mars,
Jupiter and Saturn, the Indian astronomers give only one constru-
ction for finding the apparent geocentric position. Each of
these "star planets" is conceived as having twofold planetary
inequalities : (i) the inequality of apsis, (ii) the inequality of the
śīghra. With regard to the superior planets, the śīghra apogee or
the śīghrocca coincided with the mean position of the Sun. As
Varāhamihira observed, of the other planets beginning with Mars,
the Sun is the so-called śīghra. (PSi. XVII. 1)
Let AMSP be the concentric of which the centre E is the
same as that of the earth(Fig.17);
A'M₁P' the eccentric circle of
apsis of a superior planet, of
which the centre is C;A,M,S,P be
respectively the apogee, the Mars
.planet, the direction of the
śīghra and the perigee of the
concentric; A',M₁,P' be the
apogee, the planet as corrected
by the equation of apsis, P' the
perigee in the eccentric. The arc
Fig. 17

INFERIOR PLANETS 151
AM=arc A'M₁; MM₁ is parallel and equal to EC. As used be-
fore, both the concentric and the eccentric are of the same radius.
Here the mean planet M in the concentric is taken to be,
deflected to M₁ due to the true motion in the eccentric circle.
Join EM₁ cutting the concentric at M₂. Now let ES be joined
and let S' be taken along ES, such that
ES' / ES = (Śīghra periphery of the planet in degrees) / 360
= (Sun's mean distance from the Earth) / (Planets mean distance from the Sun or the Earth);
ES' thus determined is called the radius of the śīghra epicycle of
the superior planet.
With S' as the centre and the radius equal to ES or EA
describe another circle which is called the śīghra eccentric cutting
ES produced at S''. Now measure the arc S''M₃ in the concen-
tric=SM₂ in the concentric. The apparent superior planet is
seen in the direction EM₃ from the Earth. This is the construc-
tion used in Hindu astronomy calculating the geocentric longitude
of any star planet.
It is evident in the case of a superior planet that the eccen-
tric having S' for the centre and whose radius=EA=R the
standard radius for any circular orbit, is the mean orbit of
the planet and S' the mean position of the Sun. In other words,
in the case of a superior planet, the śīghra eccentric represents
the mean orbit round the Sun. If the parallelogram CES'C' be
constructed, then an equal circle described with C' as the centre
is the apparent eccentric orbit of the superior planet.
In the actual method of calculating the geocentric longitude
of a “star planet” there are four operations given, the first two of
which have the effect of changing the arc MA or rather the point
A. The last two operations relate to the two displacements MM₁
and M₂M₃. We have here followed solely the construction of
the eccentric circles; the same geocentric position of a superior
planet could be equally well obtained by the epicyclic cons-
truction. In describing the constructions for finding the position,
of an inferior planet we shall follow the epicyclic construction
only
Inferior Planets
Let E be the centre of the Earth (Fig. 18), AMS the orbit of

152 EPICYCLIC THEORY OF ANCIENT INDIANS
a mean inferior planet or the mean Sun. EA the direction of the
apogee of apsis and ES that of the
śīghra. The inequality of the apsis takes
the mean geocentric planet from M to
M₁, such that MM₁ is parallel to EA.
Let EM₁ be joined cutting the con-
centric at M₂ ; M₂ is taken as the
centre of the śīghra epicycle or the
real circular orbit in which the appa-
rent planet moves.
Fig. 18
With M₂ as the centre and the radius of the inferior pla-
nents' śīghra epicycle as radius, describe the circle NVU which
is here the śīghra epicycle or the real circular orbit. In it draw
the radius M₂V parallel to ES; then EV is the geocentric position
of the inferior planet.
Here the first displacement MM₁ is due to the inequality
of apsis and is for finding the position of M₂ the centre of the
real circular orbit. The idea was that the apparent planet moved
in a circular orbit of which the centre was very near the mean
position of the Sun, the first operation in this construction was
calculated to determine the centre of this so-called circular orbit
of an inferior planet.
The śīghra of an inferior planet moves round the Earth at
the same mean rate in which the inferior planet moves round the
Sun; hence the line ES in this figure is always parallel to the line
joining the Sun to the mean heliocentric inferior planet, and in
our construction, it is parallel to M₂V.
This in brief is an outline of the Indian idea of planetary
motion as taught by Āryabhaṭa I, Brahmagupta and Bhāskara II.
— : o : —
Reference
P.C. Sengupta : The Khaṇḍakhādyaka, 1934.

CHAPTER VIII
Brahmagupta
and Arithmetic
Scope of Gaṇita
The word Gaṇita means the science of calculation. The
term occurs in the Vedāṅga Jyautiṣa (c. 1200 B. C.) :
Just as the crest is to the peacocks, and just as the head-
gem is to the snakes, so the Gaṇita among the Vedāṅga
Śāstras stands at the head.¹ (VJ. 4)
In the ancient Buddhistic literature, we find mention of
three classes of gaṇita : (i) mudrā (finger arithmetic). (ii)
gaṇana (mental arithmetic), and (iii) saṅkhyāna (higher arith-
metic in general). In the Brāhmasphuṭasiddhānta, Brahmagupta
uses the word gaṇita in the sense of entire calculations. His
gaṇitādhyāya (Chapter XII) includes ;
(i) Miśraka (mixtures). (ii) Śreḍhī (series). (iii) Kṣetra
plane figures), (iv) Vṛtta-kṣetra (circles), (v) Khāta (ex-
cavations), (vi) Citi (piles of bricks), (vii) Krākacika
(sawn pieces of timber), (viii) Rāśi (heaps or mounds of
grain, and (ix) Chāya (shadow).
Brahmagupta also uses the term Dhūlīkarma (literally meaning
“ʇsnpwork”) for higher mathematics :
The one learned man who knows the dhūlīkarma or the
science of mathematics as propounded by Brahmagupta
would far excell them in learning who are taught the
calculations according to Āryabhaṭa, Viṣṇucandra and
others.²
In these ten chapters of the Brahmasiddhānta has been
given the dhūlīkarma or the science of entire calculations

1. यथा शिखा मयूराणां नागानां मणयो यथा ।
   तद् वद् वेदांग-शास्त्राणां गणितं मूर्धनि स्थितम् ॥ VJ. 4.
2. नावायों ज्ञातेरपि तन्त्रैरार्यभटविष्णुचन्द्राद्यैः ।
   यो ब्रह्म धूलिकर्मविदाचार्यत्वं भवति तस्य ॥ —BrSpSi. X. 62.