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

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There is yet another subtlety in the commencement
of the week-day, arising out of the latitude of the place.
The former analysis pertains to the longitudinal difference.
The difference arising out of latitude between the local
Sun-rise and the Lanka-Sun-rise is given by what is called
Chara-Kāla. Since the week-day begins at Lanka Sun-rise
and the local Sun-rise differs from the Lanka Sun-rise not
merely by a longitudinal difference but also by a latitudi-
nal difference, to compute the actual beginning of the
week-day before or after the local Sun-rise, we have to take
into account both the differences cited above. In other
words, computing the local Sun-rise and also the Lanka
Sun-rise, we have to decide the beginning of the week day
before or after the local Sun-rise.
Verses 7, 8. The correction called Bijakarma for the
planetary positions.
The number of years from the beginning of the Kalpa
divided by 12000, the remainder, or the difference of the
divisor and the remainder whichever is less is to be divided
by 200. The quotient in minutes of arc, multiplied by
3, 5, 5, 15, 2 respectively is a negative correction in the
positions of the Sun, Moon, Jupiter, Venus, and the lunar
apogee and multiplied by 1, 52, 2 and 4 gives the positive
ocrrection in the positions of Mars, Mercury, the lunar
Node and the Saturn respectively.
Comm. By the phrase 'The remainder or the diffe-
rence of the remainder and the divisor', it is plain that the
corrections positive or negative increase for 6000 years and
decrease for the next 6000 years. Bhāskara gives no reason
for these corrections, but, we have to construe these correc-
tions on the following rational grounds. Bhāskara,
however, says that the corrections were accepted by him on
the basis of Āgama. This Āgama-stipulation was there
in Brahma-Sphuta-Siddhānta and was later incorporated
by Sripati also in his Siddhānta-Sekhara and as such was

98
accepted by Bhāskara also. However, in Brahma Sphuṭa
Siddhānta both as first published as an edition of M. M.
Sudhākara Dwivedi and later by the late Rāmaswarūpa
Śarmā in 1966, the verses 59, 60 of Madhyamādhikāra
suggest that the corrections are negative in the case of all
the planets; whereas both Śrīpati and Bhāskara make them
positive in the case of the latter four viz. Mars, Mercury,
the lunar Node and Saturn. By this we have to construe
that Śrīpati and Bhāskara must have had before them a
text which should have read ‘स्व’ in the place of ‘च’ in the
last pāda of verse 61. M. M. Sudhākara-Dwivedi did not
notice this anomaly of the positiveness of the correction
with respect to the latter four, but he remarked, however,
that there was a prosodial lapse in the last pāda of verse
61, for which he offered a suggestion that instead of वेदैर्,
we had better read वेदैः:—This suggestion, no doubt, rec-
tifies the prosody of the verse, but not the the anomaly
cited above which was not noticed by M. M. Sudhākara
Dwivedi. So, we have offered our own suggestion namely
that in the place of च as mentioned above if we read स्वं,
we not only rectify the prosodial error but also the anomaly
referred to. Rāmaswarūpa Śarma noted the anomaly but
did neither refer to the prosodial error nor offer a correc-
tion. It seems that Rāmaswarūpa Śarma did not verify
the corrections stipulated from the verses 91, 92, 93 of
Madhyamādhyāya of Siddhānta Śekhara. In this latter
work, there is another anomaly namely that in the case of
Mercury, the number 62 is the multiplier and not 52.
Makkibhaṭṭa, the ancient commentator had before him a
text which read 62 in the place of 52, in all probability, a
mistake of the scribe. M. M. Sudhākara Dwivedi is repor-
ted to have later pronounced that 52 must be the correct
figure when this was brought to his notice as this number
52 was found both in Brahmagupta and Bhāskara. As re-
ported by the editor of Siddhānta Śekhara Pandit Babuaji
Mishra, who mentions this latter pronouncement of Sudhā-
kara Dwivedi his teacher, also says that Sudhākara Dwi-

94
vedi suggested the reading द्विशर in the place of द्विरस of
verse 93 of Siddhānta Śekhara. The fact that Makkibhaṭṭa
commented द्विरससङ्गुणं as द्विषष्टिसङगुणं shows that he did
not consult Brahmasphuta Siddhānta in this place; also,
he must have had a manuscript before him which scribed
द्विरस in the place of द्विशर. Using श ष, स, indiscretely
is not uncommon in many books of North India, from a
long time and the scribe of the manuscript probably having
used स in the place of श and then by an oversight a
latter scribe having inverted सर as रस, Makkibhatta must
have commented like that.
Incidentally a remark may be made here about Makki-
bhatta. He was evidently a keralite because he used letters
to signify numbers as was a common practice among the
Kerala Astronomers, and as he also commented upon Brihad-
Bhāskarīya. Further, it is interesting to note that he
wrote in his commentary under verse 39 of the Sādhanā-
dhyāya of Siddhānta Śekhara viz. “भभ्रमोऽर्कमण्डलान्तरं
सावनानि कुदिनानि तानिवा”, “भूमेः प्राङ्मुखी भ्रमति” etc”. This
idea shows that he accepted Āryabhaṭa's verse “अनुलोम गतिः
etc” implying that the earth is rotating.
Bhāskara says that the Bīja correction mentioned was
purely based on Āgama and Upalabdhi (meaning authority
and observation'). M. M. Sudhākara Dwivedi seems to
have reiterated the same as reported by Babuaji Mishra, in
a foot-note. Kamalākara, condemned this Bījakarma as
it was unwarranted and had no proof.
A rational explanation as to why this Bīja-Karma was
prescribed either by Brahmagupta himself or some autho-
rity which he seems to have accepted may be given as
follows. The small differences in the numbers of sidereal
revolutions or what is the same the minute differences in
the accepted daily motions of the planets and the assump-
tion of a conjunction of all the planets and planetary
points at the beginning of Kalpa, which is beyond proof,

95
resulted in a difference between the computed planetary
positions and their observed positions. So, the originator
of this Bīja-Saṁskāra, noting the differences in his own
time devised a formula, which could account for those
differences. But this formulation was bound to go wrong in
later times as long as the daily motions are not corrected
to the minutest extent possible and as long as the funda-
mental basis of the conjunction of all the planets and pla-
netary points is not proved. This seems to be the reason
why so may texts were written incorporating small diffe-
rences in different times as reported by Gaṇēśa (1507 A.D.)
in his work Bṛihat-Tithi-Chintāmaṇi in the words "The
calculations of planetary positions according to the
methods indicated by Brahma, Vasishtha and Kasyapa
Siddhantas held good in their own times, but grew obsolete
later; Then Maya, the demon at the end of Krita obtained
the science from the Sun God, which again grew obsolete
in this Kaliyuga wherein parāśara began to hold the ground
for a good length of time. Then Āryabhata rectified the
methods; when even those methods grew obsolete, Durga-
Simha, Varāha Mihira and others set them right. Again
Brahmagupta came into the picture to rectify the methods
by his own observations. Then Came Kēśava (Gaṇēsa's
father) who rectified further. After a lapse of sixty years,
his son Gaṇesa has now to correct the Science. If this
also grows obsolete (as it is bound to) in course of time, let
others again rectify it by observing conjunctions of the
Moon and planets with the asterisms."
Obsoleteness arises out of two contexts, one a justifi-
able situation and the other based upon a wrong premise.
The first is as follows. Suppose as a first approximation
we take the length of an year as 365 days. We will have
committed an error nearly ¼ of a day, so that the error
accures to a day in 4 years. Thus the convention of the
leap year arose so that during four years we give a day
more to February. Here again we have overestimated the
error by nearly 1/100th of a day. Hence in 400 years the

96
above correction leads to an error of a day. So, it is
that we pronounced that out of the years 2000, 2100,
2200, 2300 A.D., the year 2000 A.D. alone is a leap year
and not the remaining, the convention being that the
number of the century, here 20, must be also a multiple
of four. On this back-ground, suppose we prepare a
manual called a Karaṇa grantha taking the length of the
year to be 365·25 days. It works alright for some time
but in the course of 400 years the error will have reached
to as much as one day. Thus a manual like the above
works only for a short time and the approximation made
gradually brings in a divergence on account of which such
a manual grows obsolete. That is why one Narasimha who
happened to prepare a manual in 1333 Saka year (1411AD)
opens his work with the words “तिथिचक्रं यत्प्रणीतं मल्लिका-
र्जुनसूरिणा, कालेन महता तस्मिन् खिलीभूते तदादरात्, नौपुरीसिङ्गयार्यस्य
नरसिंहेनसूनुना एतदेव स्फुटतरं क्रियते सौरसम्मतम्” i.e. “In as much
as a manual named Tithi-Cakra prepared by one Mallikār-
juna Sūri long ago, based upon the Sūryasiddhānta has now
diverged far from the Sūryasiddhānta (on account of the
approximations made transcending the limits of negligibi-
lity) I, the son of one Singaya belonging to a place named
Nau-puri (probably Vada-palle of the East Godavary Dt.)
am rectifying it and bringing it to accord with the Sūrya
Siddhānta again.”
This kind of obsoleteness arising out of inevitable ap-
proximations that have to be made in the preparation of
manuals is permissible. But Suppose the premise of the
manuals itself is incorrect, then the rectification of the
manuals is no good so long as the data given in the pre-
mise are not corrected. There are two fundamental
defects in the ancient works according to a modern analysis
namely (1) The Supposition that all the planets were in
conjunction at the Zero-point of the Zodiac in the beginn-
ing of a Mahāyuga (2) Small variations in the constants
like the daily motion of the planets and the like. Accord-
ing to the modern interpreters of Hindu Astronomy the

97
first premise was not correct. According to them, some
astronomers having observed the daily motions of the pla-
nets or what is the same the sidereal periods of the planets
to a sufficiently good approximation calculated back or
extrapolated a date on which these planets should have
been in conjunction at the Zero-point of the Zodiac. The
extra-polated date was naturally wrong to some extent be-
cause the sidereal periods found could not but be correct
only to a particular degree of approximation. Thus a little
alteration in the number of sidereal revolutions alone or
the number of days in a Mahayuga made to suit the obser-
ved positions at a particular epoch would be only a tinker-
ing of the problem and not a true solution. Thus Hindu
Astronomy could be saved and its methods could still be
followed provided instead of trying to presume a date at
which all the planets were in conjunction (No doubt in the
long bosom of time, such a presumption also could not be
ruled out) correctly observed positions of the planets by the
help of modern instruments were taken as the basis of an
epoch and thereafter using more correct values of the
constants such as the sidereal revolutions, maximum equa-
tions of centre and maximum Sīghraphala, obliquity of the
ecliptic etc. The second defect cited above thus being
removed, and the original premise being changed, the
methods of calculation still hold good and there would be
no necessity to be going on with tinkerings of the problem.
The Bīja-correction which we are commenting upon
was rightly criticised by Kamalākara as irrational though he
himself fanatically tried to uphold Surya Siddhānta. Even
today there are a good number of the traditional Hindu
Astronomers who do hold that the Sūrya Siddhānta was
revealed to Maya at the end of Kṛtayuga in spite of the
fact that scholars like M. M. Sudhākara Dvivedi pronoun-
ced that it was an extra-polated work shortly after the time
of Brahma-Guptāchārya. It is interesting to note that
Bhāskara, a very rational astronomer, had before him
the verse “त्रिंशत्कृत्वो युगे भानां चक्रं प्राक् परिलम्बते” of the
13

98
Sūrya Siddhānta (verse 9 ch. 3). He did not give it the
interpretation that was later put upon it through the
two subsequent lines "तद्दोः त्रिघ्ना" etc., which lines were
not there evidently in Bhāskara's time. Without these
two latter lines the rate of precession was too small to be
accepted by Bhāskara and so he chose to follow Muñjāla
rather than the Sūrya Siddhānta. Our Traditional astro-
nomers today have no reservation to accept the greatness
of Bhāskara and worship him though they do not question
what necessity Bhāskara had to write another treatise and
that too basing it upon the Āgama accepted by Brahma-
gupta and not Sūrya Siddhānta, when there existed Sūrya
Siddhānta before him and from which he had no objection
to quote verses like "अदृश्वरूपाः कालस्य मूर्तयः" etc. (verse I
ch. 2.)
The Bīja-correction first incorporated by Brahma-
gupta and later followed by a good number of astronomers
because Śrīpati and Bhāskara accepted it, will not be accep-
table to modern astronomers, though it might have worked
well at the time of Brahmagupta and for some years later.
The reason is that it is construed only as a tinkering of the
defect as explained before.
It is also to be noted that the originator of this Bīja-
correction did not make it secular i.e. valid for all time
increasing without a limit, for, then, the respective
corrections transcend all limits and render the corrections
meaningless. So, he said that the corrections would be
increasing for 6000 years and thereafter begin to decrease
to nothing. They were Zero at the beginning of the Kalī
because all the yugas are multiples of 12000 years. Also
the maximum correction is in the case of Mercury
6000/200 × 52 = 1560' = 26°. Let us see how far this is justifi-
able. The daily mean motion of Mercury as given by
Bhāskara is 4° - 5' - 32" - 18'" - 28"" whereas as per modern
astronomy it is 4° - 5' - 37⅞" approxly. So there is a posi-
tive error of 5" ²³⁄₄₀ which will accrue to 18' - 35" in the

99
course of 200 years. But as per the Bīja-correction it
should be 52'. Hence it is a fact that there is a positive
error but not so much as indicated. But it must be noted
that Mercury's orbit has the highest eccentricity of as
much as .2, and the observer who stipulated the correction
must have observed when Mercury was near its perihelion,
where the error could have been as much as indicated and
even more. Similarly on close analysis it could be proved
that the Bīja-correction should have been as indicated,
say, roughly about 3600 Kali era, which might be roughly
the date of its stipulation.
Verses 9, 10. Concluding verses of the Madhyādhikāra.
If the work is made more voluminous by describing
various methods which are easy and interesting to un-
intelligent people, learned men look down upon such a work
as indulging in unnecessary verbosity. Hence the volume
of a work does not add to its greatness; So I have made my
work neither voluminous nor brief-worded. The reason is
that both the intelligent as well as the unintelligent people
are to be enlightened.
For the sake of clarity of exposition, different ingeni-
ous methods being used in such a way that the work does
not exceed the normal limits of the previous works, and in
corporating as far as possible unit numerators, fractions
having numerators and denominators mutually prime, using
methods of interpolation and reduction, making use of
different kinds of denominators and numerators in many
ways, this kind of treatment must be given to a work of
this nature by an intelligent man.
Comm. Easy.
Before we proceed to the next chapter, we shall add
here tables of astronomical constants as given by different
authorities, which will help comparison and appreciation of
the work.

100
TABLE 1
Number of sidereal revolutions of planets and planetary points in a Kalpa

TABLE 2

101
[L

SPAṢṬĀDHIKĀRA — RECTIFICATION
OF PLANETS
Introduction. In the Bhagaṇādhyāya section of the
previous chapter Bhāskara gave under the Caption Bhaga-
ṇopapatti his proofs as to how the ancient is might have
obtained the number of sidereal revolutions of the planets
and the planetary points called apogees or aphelia and
Nodes. But in trying to give those proofs, he was aware
and he confessed also in so many words that some of his
proofs at least were obsessed by what is called Itarēta-
rāśraya-Doṣa i.e. “answer begging the question” It is
worth hearing his words in his commentary under verses
1–6 of the section cited above—“That the planets, and the
planetary points perform so many revolutions in a Kalpa,
is essentially conveyed by the Āgama i.e. the Śāstra (which
is to be taken on faith). That Āgama, got diversified i.e.
there are many versions of that Science, due to the defects
of scribes, the teachers and the students and due to a long
lapse of time from the originators of the Āgama. That
being so, the question arises as to which of the versions is
to be trusted as the right authority. If it be said so, in
mathematics only an āgama which could be proved also
should be taken as authority. Such a number of revolu-
tions as are obtained by proof, is to be accepted. Even
that could not be (a proof); for, a great scholar could just
understand the proof and by that proof alone, it is not
possible to know the exact number of revolutions (in a
kalpa), for, a man's longevity is not much. In the proof
that could possibly be given, the planet's position is to be
observed and noted every day, during the entire course of
its revolution. Thus Saturn Completes its sidereal re-
volution in about 30 years. The apogee of the Sun and the
aphelia of the planets have their revolutions running into
hundreds of years. Hence the observation of one complete
revolution (of such a planetary point) is beyond the capa-

103
city of a mortal. Hence great astronomers accept such an
āgama as would give results which accord with observations
during their times, and such a one as was formerly accepted
by a very intelligent astronomer. Then they produce their
own works exhibiting their own Skill in the Science and
refuting wrong notions of others. Their idea is 'Let the
Āgama we take as an authority be whatever it would be
Let us show our own skill in the course of our work', just
as in this work, the āgama accepted by Brahmagupta is
taken on faith as the authority. Then it might be argued
"Better not attempt at trying to prove how the numbers
of sidereal revolutions were arrived at. Even if a proof be
attempted, that proof would be obsessed by the ' Itarētara-
śrayadōṣa' (cited above). Nevertheless we shall give a
brief proof, That 'itarētarāśrayadōṣa' is apparently a
dōṣa i.e. an apparent defect; for, different proofs could
not be adduced simultaneously. The proof will now be
given".
These words indicate that even such a highly rational
and supremely intelligent astronomer like Bhāskara could
not set aside his faith in our āgama and attempt at a pure
and rigorous proof, which would not invoke the āgama-Let
us see where in his proof he does commit the so-called
itarētarāśrayadōṣa and where he invokes the āgama.
Also we shall try to construct a proof, of course to a good
extent on the lines on which Bhāskara tries to give his
proof, but at the same we shall not invoke the āgama.
where he does, but try to proceed purely on a rational basis.
We shall take up the proof under verse 18, in its appropriate
context. We shall now proceed with the text upto that
point, which gives a brief sketch of the Hindn trigono-
metry.

104
Verse 1. In as much as true positions of the planets
alone are required to decide auspicious moment; for
journeys, marriages, celebrations pertaining to temples,
astrology and the like, we shall now give the methods of
rectifying the mean positions of the planets so as to accord
with their observed positions.
Comm. Clear.
Verse 2-9. Obtaining the sines of the angles and
tabulation of the sines.
The planet deflected to the true position from the
mean lies at the end of a half-chord (which is the Hindu
sine of an angle) so that many processes pertaining to a
planet are carried through Sines of angles; hence the
word half-chord alone is connoted in this work wherever
the word Jyā meaning a chord is used.
The lengths of these half-chords (or the Hindu Sines)
for angles increasing from 0° to 90° at intervals of 3¾° are
as follows—225', 449', 671, 890, 1105, 1315, 1520, 1719,
1910, 2093, 2267, 2431, 2585, 2728, 2859, 2977, 3084, 3177,
3256, 3321, 3372, 3409, 3431, 3438. The ut-kramajyāis or
the Hindu versed-sines are respectively 7, 29, 66, 117, 182,
261, 354, 461, 579, 710, 853, 1007, 1171, 1345, 1528, 1719,
1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.
The word Tribhajyā or Trijyā is half-diameter. The
word jyā khandas used by pandits connote the differences
between successive sines.
Comm. There is a difference between modern trigono-
metrical sines and the Hindu sines as detailed below (Ref.

105
Fig. 6
fig. 6 overleaf). Let (O) be a circle i.e. a circle with centre
‘O’. Let AB be an arc called ‘Chāpa’; let BC be drawn
perpendicular on OA; then BC is half of the full chord
BD (known as jyā). The half-chord Ardha-jyā is itself
spoken of as jyā for convenience and is the Hindu-sine of
the arc or chāpa AB. In Hindu trigonometry ‘angle’ is
connoted by the arc corresponding to it and as such spoken
of as chāpa. OC is spoken of as the Hindu-cosine or Koti-
jyā and CA is called the ut-kramajya or the Versed-sine.
The radius O B is called trijyā and let us connote it by R.
To differentiate between the modern terms and the Hindu
terms, we use the words H. Sine, H. Cosine, H. vers-sine
for the Hindu sine, the Hindu cosine and the Hindu vers-
sine respectively. Also the radius R is generally taken to
be 3438′ which, we know to be the approximately the
minutes in a radian. To talk of a length in minutes ap-
pears rather odd but no confusion need be there, for, an arc
of length R subtends 3438′ at the centre. It is called Trijyā
for the reason that it is the H. sine of 3 Rasis or 90°.
A Rasi is equal to 30° because the ecliptic circle of 360°
is divided into 12 Rasis Mesha, Vrishabha etc. meaning
14

106
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

109
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