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

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Aries, Taurus etc. The names of the Rasis in Sanskrit
and the modern English words we use for them have the
same meaning, which raised a suspicion in the minds of
many orientalists that the Hindu Astronomy drew upon
the Greek. Many scholars of India assert that the Greeks
derived this knowledge from the ancient Hindus; but we
shall not enter into the controversy here. It may be noted
also that the Sanskrit names of week-days have the same
meaning as Sunday, Monday etc.
On the basis of taking trijyā equal to 3438', the other
H. sines or half-chords are also expressed in minutes.
Generally twenty-four H. Sines are given in a quadrant
and to obtain the H. sine of an angle intermediate, a for-
mula for interpolation also is given. Also the method of
calculating the H. sines for every degree is given, as we shall
see shortly. In the table of 24 H. sines, the first is H. sine
3°-45' or H. sine 225' and this is approximately taken
as 225' because in fig. 6 if AÔB=3°-45', the H. sine BC
will be almost equal to the arc AB. The H. Vers-sines
are also given to get the corresponding H. Cosines easily,
for, H. Vers sine 3°-45 = R — H. Cos 3°-45 = 7' means
H. Cosine 3°-45'=3431=H. sine (90°-3¾°) = H₂₃ where we
use the notation H r to mean the rth H. Sine.
Now we propose to give here some essential formulae
used by the Hindu astronomers, as given in the golādhyāya
by Bhāskara under the caption jyotpatti-krama. Inciden-
tally it may be noted that H. sine θ = R sine θ where sine θ
is the modern sine of the angle θ°. Similarly H. Cos θ=R
Cos θ and H vers θ = R vers θ. Thus when we have an equ-
ation of the type. Sin δ = Sin ϕ Cos z + Cos ϕ sin z sin a
in modern astronomy arising out of the famous spherical
triangle PZS where P is the Celestial Pole, Z the Zenith
and S the position of the Sun or a Star, the Correspond-
ing Hindu formula would be R³ H Sin δ = RH Sin ϕ
H cos Z + H cos ϕ H sin Z H sin a. Occasionally the

107
radius is taken to be 120, and the corresponding H sines
are called Laghu-jyās or simpler H sines used where great
accuracy is not required. Śrīpati took the radius to be
3270 units in addition to 120 as did Brahmagupta. Muñjāla
took 488′ and some others some other values also. Out of
these 3438′ alone has a right significance (Vaṭēśwara took
3272)
Bhāskara says under verses 1 to 5 under Jyotpatti-
Vāsanā in the Golādhyāya that the Hindu astronomers got
the values of the main H sines of 30°, 45°, 60°, 18° and 36°
by inscribing regular polygons in a circle. They are
called the pancha-jyakās or the fundamental H sines.
From these the others were calculated according to the
methods given by Bhaskara as follows.
To start with, we have the fundamental formula
H sin²θ + H cos²θ = R² (from fig-6) I
In addition to this formula, Bhāskara gives another
formula (verse 10, 11 Ibid) H Sin θ/2
= √(H sin² θ + H vers² θ) = √(½ R. H vers θ) II
In the commentary under the above verses, he has given
the method by which II was obtained (Ref. fig. 7) BM =
H sin θ where AÔB = θ ; also AM = H vers θ and AB² =
AM² + MB². Let N be the mid-point of AB.
½AB = AN = H sin θ/2
∴ H sin θ/2 = ½AB = ½ √(AM² + MB²)
== ½ √(H sin² θ + H vers² θ) which
proves the first part of II. Again from the right—angled
triangle ABC, AB² = AM · AC = H vers θ × 2R
∴ H sin θ/2 = ½AB = ½ √(2R H vers θ) = √((R H vers θ) / 2)
which proves the second part.

108
Fig. 7
In the Commentary under verses 1—25 ibid, Bhāskara
tells us how formulae I and II are used to construct the
table of 24 H sines. To start with, the four H sines of 30°,
45°, 60° and 90° which may be denoted by the symbol Hr
where r=8, 12, 16 and 24, are known. Now using the
formula II, H₄ is obtained from H₈, H₂ from H₄ and H₁
from H₂. Similarly from H₁₂, H₆ and H₃ are successively
obtained. Now using formula I, H₂₀, H₂₂, H₂₃, H₁₈, H₂₁ are
obtained respectively from H₄, H₂, H₁, H₆ and H₃. Now
again from H₂₀, H₂₂, H₁₈, we obtain using formula II H₁₀, and
H₅, H₁₁, H₉ respectively. Formula I gives again H₁₄, H₁₉, H₁₅,
H₁₃ from the above. H₁₄ gives H₇ and H₇ gives H₁₇ using
formula II and I respectively. Thus the table is Completed.
Then Bhāskara poses the problem as to how a table of
the H sines could be computed when a quadrant is divided
into 30 equal parts. He says that formula I and II do not
suffice in this behalf and shows how they do not, as follows

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in the same commentary cited above. To start with, the
H sines of 18°, 30°, 36°, 45°, 54°, 60° are known. They are
respectively H₆, H₁₀, H₁₂, H₁₅, H₁₈, H₂₀. Also H₃₀ i.e. H sin
90° = R is also known. Formulae I and II will help us to
derive,
from H₆, H₃ and from H₃, H₂₇ ; also from H₆, we derive
H₂₄.
From H₁₀, H₅ and from H₅, H₂₅ and again
from H₁₈, H₉ and from H₉, H₂₁ are derived.
The remaining H sines sixteen in number cannot be got
from either of the formulae.
To meet the situation Bhāskara gives other formulae
of his own discovery as he says “ प्रवक्ष्येऽथ विशिष्टमस्मात् ”
i.e. “ I shall tell something more than this. These formulae
he gives in the verses 12 to 15. They are
H Sin ((90 ± x) / 2) = √((R² ± R H Sin x) / 2) III
H Sin ((x − y) / 2) =
√((H sin x + H sin y)² + (H cos x − H cos y)²) IV
√(((H Cos x − H Sin x)²) / 2) = H Sin (45 − x) V
R − (2 H Sin² x) / R = H Sin (90 − 2x) VI
These formulae correspond to the modern formulae
Sin (45 ± x/2)° = √((1 ± Sin x) / 2)
Sin ((x − y) / 2) = √(((Sin x + Sin y)² + (Cos x − Cos y)²) / 2)
√(((Cos x − Sin x)²) / 2) = Sin (45 − x)
1 − 2 Sin² x = Cos 2x respectively

110
These formulae imply a knowledge of the expansion of
Sin (x ± y) which is given in verses 21, 22 in the form
H Sin (x ± y) = (H Sin x H Cos y ± H Cos x H Sin y) / R VII
The formula H Cos (x ± y) is got from VII by putting
90 - (x ± y) for x ± y.
To construct the remaining sixteen H Sines Bhaskara
directs us to use his formula IV wherein taking x = 27°,
and y = 15°, we have H₁₂ which gives H₇₈. From H₇₈ we
have H₁₄, H₇, and H₁ from H₄. Then H₁₆, H₂₃ and H₂₉ are
have H₈, and H₄ which in turn give give H₂₂ and H₂₆. H₂₆
gives H₁₃ which in turn gives H₁₇. H₂₃ similarly gives H₁₁
which in turn gives H₁₉. Thus the table is complete.
Verses 16 to

111
δ (Sin x) = Cos x δx. Since H Sin (x+1)° = H Sin x +
(H Cos x × H Sin 1°) / R approximately, H Sin (x + 1)° —
H Sin x° = H Cos x × (H Sin 1° / R) = H Cos x × a constant.
Hence Bhāskara could see that the variation in the
function H Sin x is proportional to H Cos x. Let it be now
required to find the increment in H Sin x for an increment
δx in x where δx < 60'. Let H Sin (x+1)° — H Sin x =
(60' × H Cos x) / R = y where y is called the Bhogya-Khanda.
Then Bhāskara argues “If for an increment of 60', there
is an increment of y, what shall we have for δx ?”.
The answer is yδx / 60 = (60 × H Cos x / R) × (δx / 60) = (H Cos x × δx) / R.
Hence H Sin (x + δx) — H Sin x = δ

112
षष्ठात्पञ्चदशादपि, सप्तमात् द्वादशात्सप्तदशाज्ञाधोत्तरं मतम् ” quoted
from the Brahma Siddhānta by Ranganātha in his com-
mentary of Sūryasiddhānta were alluded to in the articles
cited but no satisfactory mathematical explanations were
given by them. We shall give hereunder a satisfactory
explanation of the matter discussed in the articles.
In the first place it may be noted that in the table of
those 24 H sines, the sixteenth as given by Bhāskara
namely 2977 is more correct than that given in the Sūrya-
siddhānta namely 2978 (Lakshmi Venkateswara press edi-
tion 1955 Bombay).
In the course of the Commentary under the verses 15,
16 of the Sūryasiddhānta, Ranganātha gives the hint which
must have been at the back of the mind of the author of
the Sūryasiddhānta when he gave the rule to construct the
table cited. Just as δ (H Sin θ) = (H Cos θδθ) / R , Similarly
the formula δ (H Cos θ) = - (H Sin θ δ θ) / R must have
been known to the author. The negative sign means that
the successive differences of the H sines namely 225, 224,
222, 219 etc. are decreasing and also that the successive
differences of these differences are increasing according to
the H sine. Just as Bhāskara could see that the H sines
were increasing and the successive differences of the
H sines were in Kotijyānupāta i.e. in direct ratio to the
H Cosine at their respective place, similarly, the author of
the Sūryasiddhānta could see that the second differences
cited above were in Kramajyānupāta as hinted by Ranga-
nātha.
From the formula δ ((H cos θ δ θ) / R) = - (H sin θ δ θ²) / R²
putting θ = 90°, we have the second difference numerically

113
equal to δθ³ / R = (225 × 225) / 3438 = 14' — 43" — 30". Here
Ranganātha made a mistake in taking this to be
3438 / 222 = 15' — 16" — 48" — Even δθ³ / R is approximate
and a more correct value of the second difference would be
14' — 47" approximately. Ranganātha then argues that
taking this second difference to be 15 for the H sine 3438
‘what will it be for the H sine 225' ?’ The answer would
be (15 × 225) / 3438 = (15 × 25) / 382 = 375 / 382 = 1' approximately.
So, the second difference in the beginning of the table
happens to be 1' ie 225 / 225. This led the author of the Surya
siddhānta to use the words “तद्विभक्तलब्धोन”. This being
an approximate formulation, naturally necessitated a second
formulation where the approximation led to an error of 1'
through the verse “एकविंशाच्च विंशाच्च etc.” This second
formulation intended to make a correction, was done in
the wake of a correct calculation through the formulae I
& II which were known even prior to Bhaskara.
Verses 10, 11. To find the H sine of an intermediate
angle. Suppose it is required to find the H sine of an
angle θ° ie θ × 60'. Divide this by 225; the quotient
gives the previous H sine. Then (R × D) / 225 where R is the
remainder, and D the difference between the previous and
next H sines, added to the previous H sine gives the H sine
required.
Comm. The formula is evidently based on an applica-
tion of rule of three.
Verse 11. To find the angle when the H sine is given
Suppose the H sine of an angle is given to be x'. Subtract
the greatest H sine that could be subtracted from this.
15

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Suppose the H sine of θ° could be subtracted. Let the
remainder be r. Then (r × 225) / D where D is the difference
between the previous and next H sines, added to θ gives
the angle corresponding to x'.
Comm. Evidently this is the converse of the previous
process and this also is based on Rule of three.'
Verses 12—15. The H sine of the obliquity of the
ecliptic taken to be 24° is 1397. Now, the successive diffe-
rences of the H sines will be given (on the basis of taking
R = 120) which are known as Laghu-Jyās intended for
ease in Computations, namely 21, 20, 19, 17, 15, 12, 9, 5, 2.
These are given for intervals of 10°, so that if it be required
to find the H sine of x°, let q be the quotient and r the
remainder when x is divided by 10. q gvies the number
of the previous H sine. Then (r × D) / 10 where D is the next
difference or jyākhanda as it is called, added to the previous
H sine gives the required H sine. In this table the H sine
of 24° is 48' — 45". Also the H versines in this table are
got by the reverse differences. To get the angle θ° for a
given H sine say x' subtract the sum of as many differences
(Jyā—Khandas) as could be from x. Let the remainder
be r. Then (r × 10) / D where D is the next jyā—Khanda
added to the previous angle upto which the jyākhandas
have been subtracted, gives the required angle. The H sine
will be more accurate if the Bhōgya-Khanda or the next
H sine—difference is rectified (as per the rule of interpola-
tion next given).
Comm. H vers θ = R—H Cos θ = R—H sin (90—θ)
so that H verse 3¾° = 3438—H sin (86¼°) = 3438—3431=7
as given in the previons table. Similarly in the above table
of Laghu-Jyās, H vers 10° = R — H cos 10° = 120 — H
sin 80° = 120 — (21 + 20 + 19 + ... + 5) = 2 so that the

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above differences in the reverse order give the H versines.
The rest of the contents of the verses is simple, the proces-
ses being based on the ‘Rule of three’.
Verse 16. Rectification of the next H sine difference
known as Bhōgya—Khanda.
The difference of the previous and the following H
sine —differences being multiplied by the remaining degrees
and divided by 20, the result is subtracted from the arith-
metic mean of the previous and following H sine-differences
to give the rectified H sine —difference, in question.
Comm. This is a formula for interpolation which
agrees with the interpolation formula given by Ball in his
spherical astronomy on page 18 in the form y = yo + x/h
(y₁ — yo) + x (x — h) / 2h² (y₂ — 2y₁ + yo). This formula is
a re-statement of the formula enunciated by Brahmagupta
in his work Brahma Sphuta Siddhānta as well as Uttara-
Khandakhādya in the form “गतभोग्यखण्डकान्तरदलविकलवधात्
शतैर्नवभिराप्तैः, तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम्” wherein in
the place of ten-degree-interval, a fifteen-degree-interval
ie 900'-interval was taken. Rule of three is a linear for-
mula of interpolation, whereas the above is a quadratic
formula reflecting much credit on the mathematical genius
of Brahmagupta.
We shall now see how the formula is applied and what
mathematical significance it has. Suppose it is required to
find the H sine of 24° from the previous table of H sine —
differences given for intervals of 10° — from the table H
sin 10° = 21, H sine 20 = 41, H sine 30 = 60 where R is
taken to be 120. Now to find H sin 24°, we are asked to
rectify the next H sine —difference namely 19', where the
table is 21, 20, 19 etc. As a first approximation, applying
rule of three H sin 24° = 41 + ⁴/₁₀ × 19=48·6=48'—36"

116
This is a crude approximation, the actual value being
48' — 48" — 14"'. Application of rule of three is justified
if the H sine—differences are uniform, but they are not so,
being in a decreasing order. So, the following H sine—
difference namely 19' is to be rectified so as to be applicable
at 24°. In other words we have to take such a H sine—
difference which will hold good at 24°, not at 20° or 30° —
from 10° to 20°, the H. S. d. (H Sine—difference) is 20',
and from 20° to 30° it is 19'. If that be so what will be
exactly at 24°? It should be less than 20' and greater
than 19'. Now the argument advanced by Bhāskara is that
the H Sine—difference at the mid-point of the 2nd and 3rd
differences namely 20 and 19 should be (20 + 19)/2 = 19·5.
The H sine—difference at the end of the third interval is
19. Then by the rule of three ‘If there is a decrease of
19·5 — 19 = ·5, during the course of the 10° of the third
interval, what should the difference be for 4°? (where we
have to find the H sine at 24°) The result is 4/10 × ·5 = (4 × 1)/20
which is given by the words “यातैष्ययोः खण्डकयोर्विशेषः
शेषांश निघ्नो नखहृत् ”. This decrease makes the H sine—
difference at 24°, 19·5 — 4/20 = 19·3 at 24°—Now the argu-
ment to find the H sine at 24° is ‘If for an interval of 10°,
the H sine—difference is 19·3, what should it be for 4°?’
The answer is 4/10 × 19·3 = 7·72. Hence the H sine of 24° is
21 + 20 + 7·72 = 48·72 = 48' – 43" – 12"' which is nearer the
truth 48' – 48" – 14"' than what was obtained by the crude
rule of three namely 48' – 36".
Here we have to explain Bhaskara’s words more elabo-
rately, because, Kamalākara happened to criticise Bhas-
kara’s words in this context. Bhaskara Says “The H sine
difference at the end of an interval is the arithmetic mean
of the preceding and succeeding differences, whereas the
succeeding one is that which holds good at end of the suc-
ceeding. In between, we have to apply the rule of three to

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obtain the rectified difference." What Bhāskara means is
this. At the end of an interval, to obtain the H sine, it is
enough to add the H sine difference belonging to that inter-
val to the preceding differences. But when it is required
to find the H sine in the interior of an interval, we have to
construe that the difference at the end of the previous inter-
val is the arithmetic mean of the previous and succeeding
differences. There is apparently a self-contradiction in
Bhāskara's words; for, at the end of the interval, according
to his own words, the difference is that belonging to the
previous interval and not the arithmetic mean as postula-
ted. The contradiction will not be there when we read
Bhaskara's mind that he means "When we require to find
the H sine in the interior of an interval only, the difference
at the beginning of that interval is to be taken as the arith-
metic mean of the previous and the current differences, and
that at the end of the interval the current difference holds
good."
The truth of Bhāskara's statement could be seen
analytically as follows. The arithmetic mean of the prece-
ding and succeeding H sine differences is (The context is to
rectify the third H sine difference namely 19, for, we were
finding the H sine of 24°) (AB + BC) / 2 (Ref fig. 8) where OA,
AB, BC etc are the successive differences.
(AB + BC) / 2 = (H sin 20 - H sin 10 + H sin 30 - H S 20) / 2
= (H sin 30 - H S 10) / 2 = R ((Sin 30 - S 10) / 2)
= R × Cos 20 Sin 10 = (H Cos 20 H Sin 10) / R
The numerical difference of the preceding and succeed-
ing H sine differences is (i.e. यातैष्यखण्डविशेष). AB - BC
= (H sin 20 - H S 10) - (H sin 30 - H sin 20) =

118
(2 H Cos 15 H sin 5) / R — (2 H Cos 25 H sin 5) / R = (2 H sin 5) / R
(H Cos 15 — H Cos 25) = (2 H sin 5) / R × (2 H sin 20 H sin 5) / R
Let now X be the point where we are to find the H sine
(Here let us take it as x° after the previous interval for
generalisation). Then Bhaskara's formula would give
(H Cos 20 H Sin 10) / R — (2 H sin 5) / R² × (2 H sin 20 H sin 5) / 20 × x
= (H Cos 20 H sin 10) / R — x / (10 R²) × 2 H sin 20 H sin² 5
= (2 H Cos 20 H sin 5 H Cos 5) / R² — 1 / R² × x / 10 × 2 H sin 20
H sin² 5
= (2 H sin 5) / R { (H Cos 20 H Cos 5 — x / 10 × H sin 20 H sin 5) / R }
put now successively x = 0° and 10° to get the rectified
differences at B and C respectively; then those rectified
differences would be respectively H Cos 20 H sin 10 and
(2 H sin 5) / R × H Cos 25. But (AB + BC) / 2 = (H Cos 20 H sin 10) / R
(found above) and BC = H sin 30 — H sin 20 =
(2 H Cos 25 H sin 5) / R In other words the rectified differences
at B and C are respectively what exactly has been stated by
Bhāskara. Hence Kamalākara's condemnation of Bhāskara
is quite unjustified.
Verse 17. To rectify the arcual difference to obtain
the arc for a given H sine.
Subtract as many H sine-differences as could be sub-
tracted from the given H-sine. Half of the remainder
multiplied by the difference of the preceding and succeed-
ing H sine-differences and divided by the succeeding and

119
the result being subtracted from or added to as the case
may be (added in the case of Hversines) the arithmetic
mean of the preceding and succeeding H sine-differences
gives the rectified H sine-difference while finding the arc
for a given H sine.
Comm. Let the given H sine be that of 24° found
before i.e. 48·72. We could subtract 21 and 20 from this
and the remainder is 7·72; half of this is 3·86 which
multiplied by (20-19) is 3·86. This divided by the succeed-
ing difference namely 19 is ·2 approximately. The arith-
metic mean of the preceding and succeeding si (20 + 19) / 2
= 19·5. If the above result is subtracted from this, we
have 19·5 − ·2 = 19·3. If for 19·3 we have 10° increment
what shall we have for 7·72. The answer is (10 × 7·72) / 19·3
= 10 × ·4 = 4°. Hence the required arc is 20° + 4° = 24°.
The proof is analogous to the previous proof. Having
subtracted 21 and 20, the remainder is 7·72, (Ref. fig. 8)
Fig. 8
so that BX = 7·72. Now during the course of the succeed-
ing interval of 19, there has been a decrease of 19·5 −
19 = ·5 or to put it in general terms, during the course of
y the succeeding interval BC there has been a decrease

120
(x + y)/2 - y = (x - y)/2 where x is the previous interval AB
and (x + y)/2 is the H sine difference at B. Hence the argu-
ment is "If for y, there has been a decrease of (x - y)/2,
what will it be for δ (Here δ = 7·72) "? The answer is
δ ((x - y)/2) × 1/y = δ/2 (x - y) × 1/y = अवशेषकार्धम् × गतैष्यान्तरम्
÷ एष्यम्. This result is to be subtracted from (x + y)/2
i.e. ·2 is to be subtracted from 19·5 to give the rectified
H sine-difference.
Verse 18. Definition of Kēndra and assignment of
sign thereto.
The excess of the longitude of the mean place over
that of the apogee or aphelion as the case may be is called
the mean anomaly. The excess of the longitude of the
point called Sīghrōccha over that of the planet rectified
by the first equation known as Manda-phala or equation
of centre is known as the Sīghra-anomaly. The equation
of centre is positive or negative according as 180 < m < 360
or 0 < m < 180 where m is the mean anomaly. The case
will be reverse in the case of the sign of Sīghraphala, the
second equation.
Comm. “चन्द्रसूर्यौ स्फुटौ स्यातां मान्देनैकेन कर्मणा” i.e.
The Moon and the Sun could be rectified by the equation
of centre alone. This means that the Moon revolving
round the Earth directly and that the Sun revolving rela-
tively round the Earth are subject to only one correction
namely the equation of centre for rectification. In the
case of the Sun, though the fact is that he is revolving
round the Earth relatively, assuming as Hindu astronomy
does that he is revolving directly round the Earth and
subjecting him to the correction of the equation of centre

121
does not alter the mathematics that goes into his rectifi-
cation. According to modern astronomy the Sun and the
Moon, one relatively and the other directly go round the
Earth, in ellipses, the Earth being in one focus whereas in
Hindu astronomy both the Sun and the Moon are taken to
be going round the Earth in eccentric circles i.e. circles
whose centres do not coincide with the centre of the
Earth. In fact, Bhāskara says in so many words “ भुमेर्मध्यं
खलु भवलयस्याऽपि मध्यं यतः स्यात् , यस्मिन् वृत्ते भ्रमति खचरो नाऽस्य
मध्यं कुमध्ये, भूस्थो द्रष्टा न हि भवलये मध्यतुल्यं प्रपश्येत्, तस्मात् तद्ज्ञैः
क्रियत इहतत् दोःफलमध्यखेटे i.e. The centre of the celestial
sphere coincides with the centre of the Earth. The centre
of the circle in which a planet goes does not coincide with
the centre of the Earth. Hence an observer on the surface
of the Earth finds the True planet’s position differing from
that of the mean planet, so that what is called the correc-
tion of Bhujaphala is to be made in the mean position of
the planet to get the True position.” Here it has to be
noted that the Bhujaphala mentioned stands both for the
equation of centre and the second equation known as
Sīghraphala as well. One may wonder how it could be so,
but it may be noted that in the formulation of both the
equations, the centre of the eccentric does not coincide
with that of the Earth and also in both the cases the equa-
tion contains the term H sine of the anomaly where the
word anomaly whether it be of the first or second equation
is known as ‘Bhuja’.
. In the case of the other planets, the fact is that they
go round the Sun in elliptic orbits, the Sun being in one
focus. In Hindu Astronomy, we shall see that the centre
of the eccentric circle in which these planets are taken to
revolve coincides with the Sun. Hence, though the ancient
Hindu Astronomy postulates geocentric motion, the mathe-
matics that goes into the formulation of the second equa-
tion, makes the Sun’s centre the centre of planetary revolu-
tion. One may wonder again how the equation of centre
16

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formulated by Hindu Astronomy agrees with its formula-
tion in Modern Astronomy which enunciates elliptic
motion; but it will be seen that the eccentric-circle theory
also gives very approximately the same formula for the
Equation of centre. There is just one point of difference,
which does not matter. Whereas in Modern Astronomy the
mean anomaly is reckoned from the perigee or perihelion
as the case may be, it is reckoned in Hindu Astronomy
from the apogee or aphelion. The difference is made up by
prefixing the appropriate sign to the equation. The word
Mandōccha stands for the apogee in the case of the Sun
and the Moon and for the aphelion in the case of the other
planets and the word Manda kendra stands for the mean
anomaly in both the cases. In Hindu Astronomy the word
‘graha’ stands for not only the five planets Mercury to
Saturn but also for the Sun and the Moon. Why that
word is applied to the Sun and the Moon as well is, that
both the Sun and the Moon also while moving among the
stars along with the five other planets Mercury to Saturn,
wield an influence on the residents of the Earth. The
etymology of the word ‘graha’ is गृह्णातीति वा गृह्यते अनेनेति
वा ग्रहः ie. ‘that which seizes upon the fates of the resi-
dents of the Earth’, with this etymological significance
only, even the lunar orbital nodes known as Rāhu and
Kētu are also taken to be grahas in Hindu Astronomy.
Hence translating the word graha as a planet and criticis-
ing Hindu Astronomy for taking the Sun, Moon and the
lunar orbital nodes also as such is not right. In other words
the translation should be pronounced wrong. Uranus, Nep-
tune and pluto were not mentioned in Hindu Astronomy.
We shall now elucidate the eccentric and epicyclic
theories of Hindu Astronomy, which will be seen to give
identical position to the planets. How they came to be
postulated will be also elucidated. Incidentally we deal
with the ‘Bhagaṇōpapatti’ or the proof of the numbers of
sideral revolutions of the various grahas enunciated in the

123
beginning of the Madhyādhikāra, Bhagaṇādhyaya in verses
1 to 6. Even Bhāskara gave such a proof as appealed to
Āgama ie. 'Authority', which proof therefore will not be
acceptable to a student of Modern Astronomy, who is likely
to question how the Āgama came into existence.
How the Āgama came to formulate the number of
sidereal revolutions of the grahas, we shall now see.
The forefathers of Hindu Astronomy (have been re-
ported to be eighteen in number in the famous verses " सूर्यः
पितामहो व्यासो वसिष्ठोऽत्रिः पराशरः कश्यपो नारदो गर्गो मरीचिर्मनु-
रङ्गिराः लोमशः पौलिशश्चैव च्यवनो यवनो भृगुः, शौनकोऽष्टादशैतेस्युः
ज्योतिषशास्त्रप्रवर्तकाः " Of these Brahmagupta mentions
Brahma-Siddhānta, which he reports to have resuscitated.
Varāha Mihira gave a version of the old Sūryasiddhānta,
mentioning that it accorded with observations. Āryabhata
says that he revived his system from the then existing
ocean of knowledge both good and spurious. The fact that
none of these outstanding astronomers mentioned that
they had derived their systems from a foreign source, and
the reasonableness in presuming that all these three could
not be impostors, make the author of this work feel
strongly that there should have been some works in the
name of Āgamas extant long before these Acharyās. The
argument that the crude Vedānga-jyotiṣa alone should
have existed before Āryabhata, simply because, no other
work worth the name has been discovered, may not be
correct. It is quite possible that crude works could exist
side by side with advanced scientific works, just as even
nowadays we have thinkers and works of a primitive type
existing along with highly advanced thinkers as well as
scientific works).
Bhagaṇopapatti. In the first place, the forefathers of
Hindu Astronomy must have noticed very easily that the
Moon has a motion among stars, for, this could be detected

124
even by a lay man during the course of a single night. So,
the period of a single sidereal revolution, could be roughly
recognized by noticing the conjunction of the Moon with
a luminous star. Having thus observed a good number of
sidereal revolutions, which could be done even with the
naked eye, the average period could be arrived at with
sufficient accuracy within the course of a few years.
Having thus obtained almost accurately the average of a
sidereal revolution of the Moon, the sidereal revolution of
the Sun could have been arrived at as follows. The
moment of an eclipse solar or lunar could be observed with
the naked eye.-Observing a good number of eclipses within
the course of a few years, the average of a lunation could
be easily arrived at very accurately, for, in between the
eclipses of the same nature an integral number of lunations
elapse. That the Sun also has a motion among stars must
have been noticed clearly during the course of a few
months, for, observing at Sunset the star that was rising,
it should have been noticed gradually even during the
course of a month, that the Sun must have been approach-
ing the star or vice versa and as the stars were found to
keep the distances amongst them constant, it was the Sun
that was approaching the star and not the star it was that
was approaching the Sun. Having decided thus that the
Sun was moving among stars from west to East, the
approximate period that the Sun took to complete a side-
real revolution was arrived at. Then, as both the Sun and
the Moon were having east ward motion, and as the Moon
has a more rapid motion, the arc by which the Moon over-
takes the Sun during a day was roughly noticed. Thus
arriving at a rough estimate of a lunation within which
a conjunction of the Moon with the Sun recurs, it was
noticed that the excess of the sidereal revolutions of the
Moon over those of the Sun gave the number of lunations.
Since a correct estimate of both a sidereal revolution of the
Moon as well as that of a lunation were previously arrived
at, the number of sidereal revolutions of the Sun during

125
the course of a certain period were computed whereby a
correct estimate of a sidereal solar year was arrived at.
This period could also be checked simultaneously by obser-
ving the interval between the heliacal risings or settings of
a particular star of the Zodiac as well. Thus far, we have
seen how the sidereal periods of the Sun and the Moon
were determined very accurately. It may be noted that
these periods as determined by the Hindu astronomers
were correct to a good number of decimal places.
When once the Sun and the Moon were found to be
having eastward motion among stars, and when it was
discovered that there were other luminous bodies like the
Jupiter and Venus etc. moving among stars, it was
attempted to determine their sidereal periods. It must
have been done as follows. In the first place, it was
noticed that these other luminous bodies which were five in
number, namely Mars, Mercury, Jupiter, Venus and
Saturn, were found to be having retrograde motion also
unlike the Sun and the Moon. As these five bodies were
looking like stars they were named Tārā-grahas i.e. grahas
looking like stars. Also a distrinction could be drawn very
easily between Mercury and Venus on the one hand and
the other three on the other, for, the former were always
found oscillating about the Sun, never parting from him
through long distances. Thus during the course of a suffi-
ciently long interval, the geocentric sidereal periods of
Mercury and Venus coincided with that of the Sun. In
other words, it was taken that the geocentric sidereal
periods of Mercury and Venus also were taken to be an
year. This is clear from the statements made by all the
Siddhāntas that in a Kalpa of 4320000000 years the Sun, the
Mercury and Venus all the three make 4320000000 sidereal
revolutions. Then with respect to the other three planets
Mars, Jupiter and Saturn, it was noticed that they were
having a pre-dominantly longer period of direct motion,
though there was a retrograde motion for some time. This