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

529
when his longitude is equal to zero?" The result is
(Ap × 15) / (Δδ ± Δβ) = II
where rQ is looked upon as the longitude of the Moon at G
called the Goḷa Sandhi of the Moon.
But AP = the Sphuṭa Vikṣepa of the Moon when his
longitude is equal to zero where rA is the Asphuṭa Vikṣepa
at that point. Using his method of calculating AP from
Ar, Ap = (Ar × H cos δ) / R where
H sin δ = (H sin ω × H sin (90 + λ)) / R .
Here AP pertains to the longitude λ equal to zero, so
that H sin δ = (H sin 90° × H sin ω) / R = H sin ω.
∴ H cos δ = H cos ω = H cos 24° = 110 when R = 120'
Hence Ap = (Ar × 110) / 120 = (11 / 12) Ar.
But Ar is the celestial latitude of the Moon to be
calculated from RA taken to be λ
∴ rA = (H sin λ × 270) / 120 = (9 / 4) H sin λ (where 270' = 4½° = i)
taking R = 120 ∴ Ap = (9 / 4) H sin λ × (11 / 12). Hence
rQ from I = (9 / 4) H sin λ × (11 / 12) × 15 / (Δδ ± Δβ)
But (H sin λ / 4) × (135 × 11) / 12 = (123¾ / 4) H sin λ. This has to
be divided by Δδ ± Δβ. In the problem given. λ = 100°
and Δδ is taken as 362. To obtain Δβ Bhāskara adduces
the argument "If at λ = 0, the first Śarakhaṇḍa of 70
corresponds to H cos λ = 120' what amount of Śara-
khaṇḍa corresponds to an arbitray H cos λ?" The result
is (H cos λ × 70) / 120 = (7 / 12) H cos λ. This he calls Kotiphala
because it is based upon H cos λ which is the Kotijyā.
67

530
Thus in the given problem where l = 100°,
H cos 100° = 7/12 × 21 (since H cos 100 = - H sin 10 =
120 × .1735 = - 21 approximately where R = 120')
= - 12′-15″ ∴ Δδ ± Δβ = 362 - 12′-15 = 349-45.
Now 123/4 H sin λ to be divided above = (123 × 118) / 4
= 3628′-30″,
so that 123/4 H sin λ × 1 / (Δδ ± Δβ) = (3628′-30″) / (349-45)
= 10°-22′-28″ =
= rQ. Now r's longitude from the zero point of the
Hindu zodiac is 11 Rāśis 19° so that Q's longitude =
11 R-19° minus 10°-22′-28″ = 11 Rāśis 8°-37′-32″. This
gives us the longitude of G which is called the Moon's
Goḷa Sandhi. Adding 3, 6, 9 Rāśis, we get successively
the first Āyana Sandhi, the other Goḷa Sandhi and the
second Āyana Sandhi of the Moon.
(1) Bhāskara overlooked a crudeness in his proce-
dure namely that Δβ is perpendicular to the ecliptic
whereas Δδ is perpendicular to the celestial Equator, but,
since Δβ is small, he overlooked the nicety. Strictly
speaking Δβ should have been corrected for Āyanavalana.
(2) There is also crudeness in computing Δδ and
Δβ for arcs of 15°, whereas they should have been done
for increase of every degree in the longitude. He could
have done that, because at the end of the Goḷādhyāya he
gave the method or computing the H sines for every degree
under the caption प्रतिभागज्यकाविधि.
We shall now give a modern method of computing
the value of rQ. From the spherical triangle RrA, where
⁀rR = 100°, R̂ = i = 4½°, we have
sin 100 = (tan rA) / (tan 4½)

531
∴ log tan rA = log cos 10 + log tan 4½°
= 9·9934 + 8·8960 = 8·8894 ∴ rA = 4°-26ʹ.
From the spherical triangle rAG,
cos 90—ω = tan rA / tau Gr
log tan Gr = log tan rA —
log sin ω = 8·8894 — 9·6093
= 9·2801. Again from the spherical triangle rQG,
cos ω = tan rQ / tan Gr
∴ log tan rQ = log tan Gr + log cos ω
= 9·2801 + 9·9697 = 9·2498
∴ rQ = 10°-5ʹ whereas Bhāskara got 10°-22ʹ-28″.
This shows how Bhāskara was correct mathematically.
The small difference there is, due to his taking Hsines for
arcs of 15° instead of for smaller arcs.
Note. In the commentary called Sikhā of one Sri
Kedāra Datta Joṣī (page 357) we find a mistake committed
namely that he subtracted 3 R-10° the longitude of the
pāta which is in Rāśis and degrees from the Sphuṭakrānti
349ʹ-45″. What Bhāskara meant was, that since the
longitude 100° exceeds 90°, the cosine will be negative
which therefore entails △β to be subtracted from △δ.
Verse 7. How to know the occurrence of a pāta.
If the Sphuṭakrānti of the Moon, when it is maximum
be less than that of the Sun, then there could be no
occasion for their declinations to become equal in the
near future.
Comm. The situation in which the maximum decli-
nation of the Moon falls short of that of the Sun, arises

532
when the lunar orbit comes in between the celestial
Equator and the ecliptic. This situation, in its turn,
depends upon the position of Rāhu. Since Rāhu's sidereal
period amounts to nearly 18½ years, the lunar orbit
happens to take its position in between the celestial
Equator and the Ecliptic for sufficiently a long period as
shown below.
That Bhāskara could visualize this, reflects credit
to his genius. This is why he formulated Δδ ± Δβ
stressing upon the plus or minus sign.
Fig. 128
Bhāskara gives an example under the above verse
in the commentary to justify his finding given above.
Suppose Rāhu is at the autumnal equinox and the longi-
tudes of both the Sun and the Moon are equal to 79°.
Adding the Ayanāṁs'as 11°, their longitudes are each 90°
so that both of them are at the summer solstice ie. an
Āyana Sandhi. (Of course the Moon is not exactly at the
position of the Sun, but he being in his own orbit, his
longitude measured along the Ecliptic is 90°). Then
evidently the lunar orbit lies in between the celestial
equator and tho Ecliptic. Then the declination of the
Moon is 24°–4½°＝19½°＝1170′ and the declination of the
Sun is 24°＝1440′. Thus the Moon's declination when
it is maximum falls short of that of the Sun. After half
of the sidereal period of the Moon namely 13⅔ days, the

533
longitudes of the Sun, Moon and Rāhu are respectively
3-2-28-12, 8-19-4-26 and 6-11-43-28 (in Rāśis, degrees
etc), taking mean motions into account. When the Moon
has the maximum southern declination in the position M₁
of fig. 129 his longitude will be however 8-10-9-35 and then
his declination will be 1169. At that moment the decli-
nation of the Sun will be 1398. Even here the Moon's
declination falls short of that of the Sun. Again after
13⅔ days, we find the declination of the Moon falling short
of that of the Sun. Even after 2 months, the Moon
cannot have a declination equalling that of the Sun.
This phenomenon occurs when Rāhu of the lunar orbit
happens to be at Libra, and the Sun is at the summer
solstice approximately.
Verse 8. The definitions of Vyatīpāta and Vaidhṛti.
When the Sun and the Moon are in opposite Ayanas,
but in the same Goḷa, and if then their declinations be
equal, then that moment is said to constitute vyatīpātayoga.
If on the other hand, if both the Sun and the Moon be in
the same Ayana and opposite Goḷas, and if then their
declinations be equal, that moment is said to constitute
Vaidhṛti Yoga.
Comm. Explained before.
Verse 9 and first half of verse 10. To prognosticate
the occurrence of a pāta.
When the sum of the tropical longitudes of the Sun
and the Moon happens to be 180° or 360°, then the Vyati-
pāta and the Vaidhṛti respectively will occur or will have
occured. The number of minutes of arc by which the sum
of the tropical longitudes falls short of or exceeds 180° or
360° as the case may be, are to be divided by the sum of the
true daily motions of the Sun and the Moon, which will
give aqproximately the number of days after which or

before which the Yogas will occur or would have occured.
Compute the declinations of the Sun and the Moon for
that moment from the then true daily motions.
Comm. From fig. 124, it is clear that triangle ♈SL
and ♎ MN are congruent so that ♈S = ♎ M ∴ SS₁ = MS₁
∴ ♈S + ♈M = ♎ M + ♈M = 180°. Again triangle ♈SL
and ♈M'N' are congruent ∴ ♈S = M'♈
∴ ♈S + ♈M' = ♈M' + M'♈ = 360°. Thus in the former
case which constitutes Vyatīpāta, the sum of the tropical
longitudes of the Sun and the Moon is 180°; whereas in
the latter case, which constitutes the Vaidhṛtipāta, the
sum of those two longitudes is equal to 360°.
Hence we are asked to note when the sum of the two
longitudes is likely to amount to 180° or 360°. On a
particular day suppose the sum is 180 — θ or 360 — θ; so
that θ is to be made up by the then velocities of the Sun
and the Moon conjointly. Then θ / (u + v) where u, v are their
respective velocities gives the number of days or fractiou
of a day, by which the sum would be 180° or 360° as the
case may be. As the velocities change from moment to
moment, the above θ / (u + v) is only approximate. So, compute
again the respective positions and respective velocities.
Suppose the sum of the longitudes is 180 ± θ' or 360 ± θ'
and the velocities u' and v', Then θ' / (u' + v') gives the fraction
of a day after or before the moment in question when the
pātas occur Vyatīpāta or Vaidhṛti.
Verse 10. To know whether the Yoga is past or
future.
If the declination of the Moon situated in an odd
quadrant exceeds that of the Sun or falls short of the

535
Sun's in an even quadraut the moment of the occurence of
pāta has elapsed. Otherwise the pāta is to take place
shortly after.
Comm. This is clear because in an odd quadrant, the
declination of the Moon is on the increase. So if it be
already greater than the Sun's, in future it will be far
greater and as such cannot equal the Sun's declination.
In other words the pāta had already taken place.
Similarly in an even quadrant, the declination of the
Moon is on the decrease. As such if it be greater than
the Sun's it will become equal in future. Thus the pāta
is to take place. On the other hand if the Moon's decli-
nation is less than the Sun's, it will decrease further so
that prior to the moment concerned a pāta should have
occured.
Latter half of verse 11 and verses 12, 13, 14. To
compute the time of the occurrence of pāta through a
consideration of declinations.
Obtain the difference of the declinations if they be of
the same direction or their sum if they be of opposite
directions in the case of Vyatipāta ; obtain the sum or
difference of the declinations according as the declinations
are of the same or different directions in the case of
Vaidhṛti. Call this sum or difference 'the Ādya'.
After a lapse of an arbitrary time or before the moment
concerned, obtain the positions of the Sun, Moon and the
Rāhu ; also compute their declinations and form their
difference or sum as the case may be. Call this Anya. If
on both the occasions it is indicated that the pāta has
elapsed or is to elapse, obtain the difference of the Ādya
and Anya ; otherwise their sum. Divide the arbitrary
time taken, by the above sum or difference of the Ādya and
Anya and multiply by the Ādya. Take the result in
ghatis. Taking this as the arbitrary time, repeat the

536
process till an invariable quantity is obtained in ghatis.
This is the time by which the moment of the pāta has
elapsed or after which it is going to occur.
Comm. (1) It might be asked “why bother finding
the declinations at all, when it is defined that Vyatīpāta
happens when the sum of the longitudes of the Sun and
the Moon is equal to 180° and Vaidhṛti is defined when
the sum is equal to 360, since easily we could know when
this happens without taking recourse to declinations?”
The problem is not that simple. The above definition
in terms of the sum of the longitudes is an approximate
statement, because, the original and correct definition is
that their declinations must be equal in magnitude. If
they be equal in magnitude but opposite in direction, they
constitute Vaidhṛti Yoga. If they be both equal in
magnitude and direction, the situation constitutes Vyatī-
pātayoga. In other words, when the diurnal paths of
the Sun and the Moon coincide, the moment will be
Vyatīpāta. If on the other hand their diurnal paths are
of equal dimentions but one to the north of the celestial
equator and the other to the south, then the moment is
Vaidhṛti. Though the sum of the longitudes happens to be
180°, the diurnal paths may not coincide because the Moon
does not actually move on the ecliptic. So the situation
is more complicated.
Further the calculation of the declination of the Moon
is to be done by calculating the Asphuṭakrānti ie. the
declination of the point of the ecliptic which indicates
the longitudinal position of the Moon and his celestial
latitude ie. the Asphuṭavikṣepa from the known position
of the Node, and then by rectifying the Asphuṭavikṣepa to
obtain the Sphuṭavikṣepa and then adding the Asphuṭa-
krānti to the Sphuṭavikṣepa to obtain the Sphuṭakrānti.
(2) The latter half of verse (11) and the first half
of verse (12).

537
Let us consider the case of Vyatipāta. Suppose at
the moment concerned, the declinations are of opposite
direction. Find the sum of their numerical magnitudes.
Suppose they are of the same direction; find their differ-
ence. This sum or difference of the declinations gives the
distance between the planes of the diurnal paths of the
Sun and the Moon. This distance is to vanish in order
that the diurnal paths may coincide. So we are asked first
to find the above distance. Suppose we understand that
the Vyatipātha has elapsed by noting the deolinations.
This may be known easily by the criteria given previ-
ously. Supposing 𝑥 and 𝑦 to be the declinatious of the
Moon and the Sun and supposing that 𝑥 is on the decrease
and approaching 𝑦, then the Vyatipātha is to occur. But
suppose 𝑥 < 𝑦 and 𝑥 is on the decrease, then the Vyatipāta
has taken place. Thus knowing whether the Vyatipātha
has elapsed or is to occur, after an arbitrary time 𝑡, com-
pute the declinations of the Sun and Moon and form their
difference or sum as the case may be which gives the
distance between the diurnal paths. Let the first distance
found be called Ādya and the second distance the Anya.
Then the Anya will be less than the Ādya because we have
taken a time towards the occurrence of Vyatipāta when
the distance is to get nullified. Find the difference of the
Ādya and Anya. Then by the proportion “ If in time ‘𝑡’
taken in between the moments of the Ādya and Anya,
the distance between the diurnal paths is reduced by
Ādya — Anya, what time will be taken for the distance
Ādya to vanish ?” we have the result
T = (Ādya × 𝑡) / (Ādya — Anya) which gives approximately the time
that has to elapse for the occurrence of the pāta. This will
be approximate. After a lapse of time T from the Ādya
moment concerned, again compute the declinations and
repeat the procsss. We arrive at a particular point of
time, which gives the moment of occurrence of the pāta.
68

538
In the course of the commentary under these verses
Bhāskara solves two problems and points out that when
there is a pāta according to his exposition, the statements
made by Lalla, Brahmagupta Śrīpati and Mādhava all
indicate that there would not occur a pāta.
But we feel that Bhāskara read too much into those
statements on account of the following. Lalla states
सूर्योपमादोजपदोद्गवाच्चेद्युग्मादिजः चन्द्रमसो लघीयान्,
अपक्रमःस्यान्न तदास्तिपातः तदन्यथात्वेऽपमयोः समत्वम् ।
Brahmagupta states
त्रिनवगृहेन्दुकान्तिः मेषतुलादौ दिवाकरक्रान्तेः ।
ऊना यावदभावः तावत् भावोऽन्यथा चेति ॥
and Śrīpati says
त्रिनवभवनजाता क्रान्तिरिन्दोर्यदाऽल्पा
दिनकृदपमतः स्यान्मेषजूकादिजातात् ।
न हि भवति तदा च क्रान्तिसाम्यं रवीन्द्रोः
नियतमितरथात्वे जायते सम्भवोऽस्य ॥
In fact all these three statements mean one and the
same and are intended to be approximate statements, in
the first instance, having ignored the latitude of the Moon.
They are just statements like that of Bhāskara himself
when he says that there will be a pāta when the sum of
the longitudes will be 180° or 360°. The statements pur-
port to say that if the declination of the Moon when it is
on the decrease happens to be less than that of the Sun
which is on the increase, there could be no pāta. These
statements so far as they go, ignoring the latitude of the
Moon are perfectly in order. Bhāskara brings in a detailed
analysis of two critical examples, to prove that there is a
pāta, but which is negated by the rough statements of the

539
four Ācāryas. Mādhava's statement in his Siddhānta
Cūdāmaṇi is as follows which is also in similar terms as
those of the other three.
रवेरोजपदक्रान्तेः चन्द्रयुग्मपदोद्भवा
स्वल्पाचेन्नतयोःक्रान्त्योः साम्यं स्यादन्यथा भवेत्
Verses 15 and 16. To obtain the duration of the
pāta.
The semi-sum of the diameters of the Sun and the
Moon or what is the same the sum of their angular radii
being multiplied by the Spaṣṭaghaṭis (obtained in the
estimate as per the verses 11-14) and divided by the Ādya
in that context, gives the beginning and end of the pāta
from the moment of the computed time. The process
being repeated according to the method of successive
approximations we have the correct estimate of the
duration of the pāta.
Comm. This is a convention stipulated with respect
to the duration of the Pāta. Strictly speaking, when the
diurnal paths of the centres of the discs coincide in the
case of Vyatīpāta, that will be the middle moment of the
Vyatīpāta. The pāta is said to last for such a time as the
distance between the Centres of the discs (north-south
distance) is less than r + p. This will be clear from a
figure. The situation is akin to that of an eclipse.
The time obtained for the occurrence of the pāta
under verses (11) to (14) indicates the middle of the
duration of the pāta. The duration of the pāta is defined
as the time that lasts as long as the declination of the
highest point of one disc becomes equal to that of the
lowest point of the other beginning from the moment at
which the declination of the lowest point of the one
becomes equal to that of the highest point of the second.
In other words, just like in an eclipse, so long as the
distance between the diurnal paths traced by the centres

540
of the discs is less than the sum of their angular radii, the
pāta lasts. Extending the meaning of this to Vaidhṛti
also, so long as the numerical difference of the declinations
of the Sun and the Moon ignoring their direction happens
to fall short of the sum of the angular radii, the pāta
lasts.
To obtain this duration of the pāta the argument is
“If the sum or difference of the declinations according as
they are of opposite or the same direction (which was
taken to be Ādya under verses (11) to (14) ) was reduced to
zero in the time computed that time being known as
Spaṣṭaghaṭis, what time will be taken for a difference of
declinations equal to the sum of the angular radii?”
The result is ((r + p) × T) / Ādya where r and p are the
angular radii of the Sun and the Moon, T the time
calculated formerly known as Spaṣṭaghaṭis and Ādya is as
defined above. The above result gives the duration of the
pāta.
Note. An approximate estimate of this duration could
be obtained using differentiation. We have
sin δ = sin λ sin ω so that
cos δ Δδ = sin ω cos λ Δλ
Let Δλ be the motion in longitude of the Moon with
respect to the Sun per nāḍi which will be on the average
12′ approximately.
∴ Δδ = (sin ω cos λ × 12) / cos δ
If in one nāḍi, there be a variation in the declination
equal to the above, what time will be taken for 16′ + 15′
the sum of the angular radii approximately ? The result is
(31 cos δ) / (12 sin ω cos λ)

541
When λ approaches 90°, the duration will be very
much since cos λ → 0. This is true because the variation
in the declination when λ = 90°, is very slow even for the
Moon. The above formula holds good so long as λ does
not approach 90°.
Verse 17. It must be deemed that the declinations
will be equal so long as the difference in the declinations
is less than r + p numerically.
In the case of Vaidhṛti, we are asked to take the sum
instead of difference and difference instead of sum in
contradistinction. Why? We shall see. We know that
Vaidhṛti occurs when the declinotions of the Sun and the
Moon are of equal magnitude but of opposite directions
subject to the condition that the longitudes are such that
their sum is approximately 360°. This condition is
stipulated because when the Sun is in the first quadrant
and the Moon is in the 3rd quadrant and at the same time
their diclinations are equal, the moment does not constitute
Vaidhṛti, because the sum of the longitudes is then less
than 90° + 270° ie. less than 360°. Hence, for Vaidhṛti
one of the two celestial bodies must be in the first quadrant
and the other in the fourth quadrant, so that the sum of
the longitudes could equal 360° and at the same time the
declinations could be equal though of opposite sign. Now
compute the declinations of the Sun and the Moon at a
particular moment. Suppose they are a and b and both
are north. After a few ghatis or days compute again their
declinations, say c and d, both again being north. Suppose
then c + d < a + b, then Vaidhṛti is going to occur.
Construing northern declination to be positive and the
southern negative, for Vaidhṛti to occur a + b must be
reduced to zero ie. a and b are equal and of opposite
direction. Now the argument adduced is "If in time
t ghatis or days a̅+̅b̅ has become c̅+̅d̅ (calling a̅+̅b̅ as
Ādya and c̅+̅d̅ as Anya) there is a decrease of Ādya - Anya,
what time should lapse for a̅+̅b̅ or Adya to reduce to

542
zero ?" The answer is t (a + b) / Adya - Anya which gives the time
after which Vaidhṛti is likely to occur.
Suppose a and b the declinations observed at a particular
moment, are one north and and the other south. Compute
their difference a - b; after a few ghatis or days again from
the difference of the declinations c and d ie. c - d. If
c - d < a - b then Vaidhṛti is going to occur, otherwise it
has elapsed. This is so because the difference has to vanish
for Vaidhṛti to occur.
Viewing the situation algebraically, ie. construing
northern declination to be positive and southern negative,
we could have laid down only one criterion, instead of
separating the issues and stipulating separate criteria.
Proceeding on this basis, suppose the the Sun and the Moon
are one in the first quadrant and the other in the second.
Here both declinations are therefore positive. Compute
their difference a - b; compute again the difference of
c and d which are again northern declinations observed
after a few ghatis or days. If c - d < a - b then Vyatīpāta
is going to occur.
Now take a case when one of a and b is north and the
other south ; so also c and d. We are talking of a, c to be
in the first quadrant and b, d in the fourth quadrant. Even
now if c - d < a - b Vyatīpāta is going to occur. Thus
there is no need to stipulate "Sum or difference". Difference
alone counts now, because difference of one positive and the
other negative quantities means sum only. Thus in the
case of Vyatīpāta difference should tend to zero, whether
both the declinations are north, or one north and the other
south.
Similarly in the case of Vaidhṛti the sum should tend
to zero because the algebraic sum of one positive northern
declination and the other negative southern declination

543
shall be zero, if the declinations were to be equal and of
opposite directions.
Here ends the Pātādhyāya.
Note Bhāskara gives two examples in the course
of the commentary, one, to show the fallacy in asserting
that there would be a pāta when the Sun is in an even
quadrant, the Moon in an odd quadrant and the Moon's
declination is less than that of the Sun; and the other to
show the fallacy in asserting that the pāta has elapsed
when it is still to take place, and that it is still to occur
when it has already elapsed. He quotes the example given
in his Golādhyāya wherein the longitudes of the Sun,
Moon and Rāhu are 120°, 60° and 180° respectively, there
being no Ayanāṁśas. (Since at the time of Lalla, there
were no Ayanāṁśas, Bhāskara gives such an example),
Since the Sun is in an even quadrant and the Moon in
an odd quadrant, Bhāskara says, that as per the verse of
Lalla ‘सूर्यापमात् ओजपदोद्भवात् etc.’ there should be a
pāta. But since the longitude of Rāhu is 180°, Ketu
is at the equinoctial point so that the lunar orbit
lies between the ecliptic and the equator. Hence the
declination of the Moon remains smaller than that of
the Sun for a good length of time never equalling the
declination of the Sun. This, Bhāskara says, is a fallacy
regarding ‘भावाभावत्व’ ie. ‘Is there a pāta or not’ because
when there is actually no pāta, Lalla's criterion leads to
assert that there is one. It may be reiterated here, that
Lalla gave a general criterion ignoring the latitude of the
Moon ie. supposing the Moon to be moving on the ecliptic.
Bhāskara takes the latitude also into consideration and
proves that there is no pāta.
In the second example, which Bhāskara gives, he tries
to show the fallacy with respect to “गतेष्यत्व” ie. asserting
that a pāta has elapsed when it is still to occur, and that
it is to occur when it has already elapsed.

544
In this example the declinations of the Sun and Moon
are respectively 1416′, 1324′, when their tropical longitudes
are 80° and 100°. The Moon is in an even quadrant,
and his declination is less than that of the Sun. As per
the criterion of Lalla given in the verse “सूर्यापमात् etc.”
the pāta is not there at all. But Bhāskara computes and
shows that the pāta has occured 70 nādīs before (Rāhu
having a reverse tropical longitude of 100°). We shall
show here, how using differential calculus we could cut
short the process. We have
sin δ = sin λ sin ω ; differentiating
cos δ Δδ = cos λ δλ sin ω or
Δδ = (sin ω cos λ / cos δ) Δλ
= Āyanavalana × Variation in Bhuja.
This could be seen graphically also from fig. 129. Let
P and Q be two contiguous positions of the Sun (say) on
the ecliptic when arc pQ = Δλ. Let the increment in his
Fig. 129
declination in going from P to Q be NQ equal to Δδ.
Taking PNQ to be a plane Δ Δδ = Δλ sin v where
v̂ = QP̂N. We have named this angle to be v, because it
is no other than the Āyanavalana which was formulated
to be equal to (sin ω cos λ / cos δ) . Thus Δδ = Δλ × Āyana-
valanajyā.

APPENDIX
LIST OF TECHNICAL TERMS

1. Adhikamāsa : A month gained by the lunar reckoning
   over the solar. This is located in that
   lunar month which does not contain a
   Saṁkramaṇa,
2. Agrajyā : The Hindu sine of the arc of the horizon
   in between the rising point of the Sun
   and the east point.
3. Akṣa (or) Pala : Latitude (Terrestrial)
4. Ākṣa Dṛkkarma : The arc of the ecliptic between the point
   of intersection of the ecliptic with a
   secondary through the star to the prime
   vertical and the point of intersection of
   the ecliptic with the star's declination
   circle.
5. Akṣakarṇa : The hypotenuse of the gnomonic triangle
   when its shadow is equal to what is called
   Viṣuvat-chāyā
6. Akṣa-Kṣetra : A right-angled spherical triangle one of
   whose sides may be the arc of a small
   circle namely the diurnal circle but one of
   whose angles is a right angle and another
   equal to the terrestrial latitude.
7. Akṣavalanam : The angle at the point of the star in
   between the declination circle of the star
   and a secondary to the prime vertical
   through the star.
8. Antyā : The Hindu sine of an arc of the celestial
   equator corresponding to Hṛti.
   69

546
9 Asta : Setting or heliacal setting.
10. Ayanabindu : Solstice.
11. Ayana-Dṛkkarma: The arc of the ecliptic intercepted between
its point of intersection with the star's
declination circle and the secondary to the
ecliptic through the star.
12. Ayanāṁśam : The arc of the ecliptic in between the
vernal equinoctial point and the Hindu
zero of the ecliptic ie the first point of
the Zodiacal sign called Aśvinī.
13. Ayanavalanam : The angle at the point of a star, between
its declination circle and the secondary to
the ecliptic through the star.
14. Bārhaspatyamāna: The time taken by Jupiter to reside in a
Rāśi, on the average, is called a jovian
year. This falls short of a solar year.
15. Bhāga : A degree
16. Cāpa or : Arc.
Kārmuka
17. Carajyā . : The Hindu sine of the arc intercepted
between the east point and the declination
circle of a rising star or planet or the
Sun.
18. Cāndra-māsa : The time between two consecutive full-
moons or New moons.
19. Chāyā or Bhā : Shadow cast by the gnomon.
20. Chāyābhuja : The projection of the shadow on the east-
west line.
21. Chāyākarṇa or : The hypotenuse of the gnomonic triangle
Bhākarṇa whose two sides are the gnomon and its
shadow.

547
22. Chāyākoṭi : The perpendicular from the extremity of
a shadow on the east-west line.
23. Dhruva : The star near the celestial pole or the
celestial pole itself.
24. Dhruvaka : The celestial longitude.
25. Dhruva- : The declination circle.
protavṛttam
26. Digjyā : The Hindu sine of the azimuth measured
by the angle between the prime vertical
and the vertical of a star or a planet.
27. Dorjyā or : Hindu sine of celestial longitude.
Bhujajyā
28. Dṛgjyā : The Hindu sine of the Zenith distance.
29. Dṛg-lambana : Total parallax.
30. Dvāparayuga : Twice the period of a kaliyuga.
31. Dyujyā : The Hindu cosine of declination or the
radius of the diurnal circle taking the
radius of the celestial equator to be R
equal to 3438 units.
32. Dyujyā-vṛtta or : The diurnal circle of a star or a planet.
Ahorātra-vṛtta
33. Ghaṭi or Nāḍi : An interval of time equal to 24.' (minutes)
34. Grahaṇa : Eclipse.
35. Hṛti or Iṣṭahṛti : The Hindu sine of the arc of the diurnal
circle from a point of the same upto the
plane of the horizon.
36. Kadamba : Pole of the ecliptic.
37. Kadamba- : A secondary to the ecliptic through a star
protavṛtta or planet.

548
38. Kakṣāmaṇḍala : The deferent of a planet or the circle with
the earth as centre and radius equal to 3438
units.
39. Kalā : The Hindu sine in the diurnal circle
corresponding to the Sūtra (given below).
40. Kalā or Liptā : A minute of angle.
41. Kaliyuga : The period consisting of 4,32,000 mean
solar years.
42. Kalpa : Dvāparayuga is twice Kaliyuga; Tretāyuga
thrice and Kṛta four times. All these put
together constitute a Mahāyuga. 71 Mahā-
yugas make one Manvantara. Fourteen
Manvantaras with what are called Sandhi
periods on either side equal to a Kṛtayuga
or thousand Mahāyugas make a Kalpa. The
creation is supposed to last one Kalpa,
which is said to constitute the day time of
Brahma, the Creator. His night also is of the
same duration when there is no creation. It
is held that we are now in what is called
Śveta-Varāha Kalpa, wherein the seventh
Manvantara called Vaivasvata manvantara
is current. In this Manvantara, twenty seven
Mahāyugas are supposed to have elapsed
and in the twenty-eighth Mahāyuga, kṛta
Treta and Dvāparayugas have elapsed and
that we are now in the Kaliyuga. In this
Kaliyuga which began on the day when
Lord Kṛṣṇa gave up his mortal coil, 5080
mean solar years have elapsed by about
21 March, 1979.
43. Kramajyā or : Hindu sine of an angle.
simply jyā or
Jīvā or guṇa
44. Karaṇa : Half of the duration of a tithi.