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

397
Fig. 88
Comm. Let M₁, M₂, M₃ be the positions of the centre
of the Moon's disc at the moment of first contact, at the
middle of the eclipse and the moment of last contact
respectively. Draw a circle with radius M₁ R₁ = r+p
which is called the Manaikyārdha Vritta. Let M₁ E₁
represent the Eastern direction known as the Sama-
mandalaprāchī. Draw E₁ V₁ equal to Hsine of the Valana,
so that M₁ V₁ is the Krānti-Vritta-prāchī ie. the point
of intersection of the Ecliptic with the Eastern horizon.
Draw N₁ R₁ perpendicular to M₁ V₁ in the form of Hsine,
which is the latitude (Vikṣepa) of the Moon at the moment
of first contact. With R₁ as centre draw the Grāhaka-
Vritta with p as centre ; this circle represents the Rāhu-
Bimba.
Similarly let M₃ be the centre of the Moon's disc at
the moment of last contact. Draw a circle with M₃ as
centre and r+p as the radius which is the Manaikyārdha-
Vritta. Let M₃ W₃ be the direction to the West, the
Samamandalapratichī. From W₃ draw W₃ V₃ the Hsine
of the Valana, so that M₃ V₃ is now the Krāntimandala-
pratīchī ie. the point of intersection of ecliptic with the
western horizon. Let, the Vikṣepa N₃ R₃ be drawn as a
Hsine of the Mānaikyārdha Vritta. With R₃ as centre
and radius p, draw the Rāhu-Bimba.
Let M₂ be the centre of the Moon's disc at the middle
of the eclipse. Let M₂ S₂ represent the South with respect
to the prime-vertical. From S₂, draw the Hsine of the
Valana S₂ V₂ so that M₂ V₂ is the Krānti-Vritta-Dakshiṇā
ie. the South with respect to the Ecliptic. Now the

398
Vikṣepa or the celestial latitude of the Moon M₂ R₂ is to
be drawn along this Valanasūtra M₂ V. With R₂ as centre
and radius p drawn the Rāhu-Bimba.
Fig. 89 Depiction of Fig. 88, keeping the Moon fixed
The slight flaw in this figure is that ML₁ L₃ implied
as the path of the Moon is taken to be parallel to the
ecliptic R₁ R₂ R₃ the path of the eclipsing body the
Grāhakamārga, in as much as latitudes are drawn perpendi-
cular to ML₁ L₃.
This figure depicts a total eclipse of the Moon. If M
coincides with R₂ at the middle moment of the eclipse,
then the eclipse is called central.
The duration of a central eclipse will be on the
average the time that the Moon's disc takes to cross the
diameter of the Rāhu-Bimba with its relative velocity.
Hence the mean duration of a central eclipse is
Average diameter of Rāhu + Average diameter of the Moon
──────────────────────────────────────────────────────────
Relative velocity of the Moon with respect to the shadow
= [(81+64) × 24] / [790′-35″ — 59′-8″] hrs = 145/732 × 24 = 290/61 =
4 hrs–45 minutes approximately.

399
Verse 30 and first half of 31. Geometrical depiction
of the eclipse at the beginning and end of totality and
also of the magnitude of the eclipse.
The Bhuja is to be laid from the centre of the Moon
along its Valanasūtra or the line indicating the direction
of the ecliptic; the latitude is to be drawn from the end
of the Bhuja and perpendicular to the Bhuja. The hypo-
tenuse is to be drawn from the centre of the Moon. Taking
the point of intersection of the latitude (Kōti) and the
hypotenuse, as centre and radius p equal to that of the
eclipsing body, if circles be drawn, from these circles could
be known the points where totality begins and ends as
well as the magnitude of the eclipse at any given moment.
Or these could be found in another way as follows.
Comm. The method given above for depicting the
phases of an eclipse geometrically, could be applied for any
moment during the course of the eclipse and depends
upon before-hand computed Bhuja and Kōti. Refer to
fig. 90. Let M be the centre of the Moon's disc. Mark
Eω the East-west line drawn through M. Compute the
Valana for the required moment, either for the moment
when totality begins or for that when totality ends or for
Fig. 90

400
any arbitrary moment whatsoever. With this Valana
primarily laid in the Manaikyārdha Vritta, decide the
Krānti Vrittaprachī or the East-West direction of the
eclipse. Thus in the figure V₁ V₂ is this direction. Then
lay off the computed Bhuja along this V₁ V₂ from M, say
MA for the Sammīlana moment or the moment when
totality begins or MC for the Unmīlana moment or the
moment when totality ends or MD for an arbitrary
moment. Draw AR₁ or CR₂ or DE equal to the latitude
at the particular moment, perpendicular to the Valana-
sūtra. In the figure drawn the Valanasūtra is shown to
be the same. This does not mean it will be the same
throughout. It will be changing because the position of
the Ecliptic changes from moment to moment. So Bhās-
kara uses the word ie. 'the respective Valanasūtra'.
Also the latitudes will be differing from moment to
moment as well as the Bhujas both of which are to be
computed for any moment along with the Valana. (The
method of computing the Bhuja was given in verse 15).
Computing the respective Karṇas or the hypotenuses from
the formula K = √(Bhujā² + Kōti²), (Kōti is here the lati-
tude) with centre M and radius equal to the Karṇas, if
arcs be drawn to cut the latitudes, the points of intersection
would be no other than R₁, R₂ or E. Join MR₁, MR₂ and
ME. With centres R₁ and R₂ and radii equal to p - r,
(where p is the radius of the Rāhu-Bimba, and r the
radius of the Moon's disc) if circles be drawn, they just
touch the Moon's disc at F and G which are the points
where totality begins and ends respectively. With centre
E and radius P, if a circle be drawn, that will show what
amount of the disc is shadowed as well as the measure of
the magnitude of the eclipse (defined in verse 11).
Note. In the above commentary and figure we
have depicted MD as the Iṣṭa-Bhuja or the Bhuja at a
given moment, taking a moment prior to the Unmī-
lanakāla, for showing the magnitude of the eclipse.

401
If a moment in between the Sammīlana and Unmīlana
were taken, the then Bhuja and Koṭi could be no doubt
computed, but the question of magnitude of the eclipse
does not arise as the entire disc has been plunged in the
shadow.
Second half of verse 31 verse 32 and first half of verse
33. Alternative method of depicting the eclipse geo-
metrically.
Joining the upper end of the latitude of the middle
moment of the eclipse to those of the first and last
contacts, we have what are called the Pragrahamārga and
Mokṣamārga ie. the path of the centre of the eclipsing
body from the first contact to the middle moment of the
eclipse and that from the middle moment to the last
contact. The lengths of these paths could be computed
and they could be drawn before hand. Then with the
centre of the Moon as centre and radius equal to p—r, if
a circle be drawn, it cuts the paths described above each
in one point. With those points as centre and radii equal
to p, if circles be drawn, they will touch the Moon’s disc
each in one point which are respectively the points of
Sammīlana and Unmīlana.
Comm. In as much as the latitude of the Moon
differs from moment to moment, the Pragrahamārga and
the Mokṣamārga are separated to achieve a little more
accuracy than could be got by joining the upper extre-
mities of the initial and final latitudes. The remaining
statement is evident, for, at the moments of Sammīlana
and Unmīlana, the distance between the centres of the
eclipsing body and the eclipsed will be p—r, so that the
points of intersection of the Pragrahamārga and Mokṣa-
mārga with the circle whose centre is the centre of the
eclipsed body and radius p—r will give the centre of the
eclipsing body.
51

402
Latter half of verse 33. To know the magnitude of
the eclipse at any given moment during the course of the
eclipse.
Let the product of the time elapsed from the moment
of first contact and the length of the path of the eclipsing
body traced from the moment of the first contact to the
middle of the eclipse divided by the time between the
moment of first contact and the middle of the eclipse, be
x. Similarly let the product of the time before the end of
last contact and the path of the eclipsing body traced
between the middle moment of the eclipse and the moment
of last contact divided by the time between the middle
moment and the moment of last contact be y. Lay off
x and y units of length from the first and last points of
the path of the eclipsing body along the path respectively.
Then we get the points of the centre of the eclipsing body
at the required moments. With these points as centre
and radius p, if circles be drawn, they represent the
eclipsing body. The length of the diameter of the eclipsed
body shaded, gives the magnitude of the eclipse called
grāsa.
Comm. Here rule of three is applied namely “If
during time T₁ or T₂ a path equal to l₁ or l₂ in length is
traced what length will be traced in times t₁ or t₂?”,
where T₁ and T₂ are the times called Sparsa-Sthiti-Khanda
and Mokṣa-Sthiti-Khanda respectively, l₁ and l₂ are the
times elapsed from the moment of first contact or before
the moment of last contact and t₁, t₂ are the times from
the beginning of the eclipse and before the end of the
eclipse respectively. Then x and y give the points where
the centre of the eclipsing body lies.
Verse 35. Given the magnitude of the eclipse at any
time to obtain the time elapsed after the first contact.
The time taken by the centre of the eclipsing body to
move through the segment of the path of the eclipsing

408
body which lies between the position of the eclipsing body
at the moment of first contact and the point of intersection
with the path of the eclipsing body of the circle drawn
with the centre of the eclipsed body as centre and radius
equal to the difference of p+r -g where g is the magni-
tude of the eclipse (grāsa) at the moment, or similarly the
time taken by the centre of the eclipsing body to move
through a similar and equal segment of the path of the
eclipsing body on the other side, gives the time elapsed
after the moment of first contact or the time before the
moment of last contact.
Comm. This is the converse of the above problem.
The method is clear being based on rule of three as above.
Both the problems could be algebraically expressed as
follows. Let T, t, l, g, and k, stand respectively for the
Sthiti-Khanda ie. the time between the moment of first
contact to the middle of the eclipse or the time between
the middle moment to the moment of last contact ; (2).
the time elapsed after the moment of first contact or the
time before the moment of last contact, as the case may
be ; (3) the length of the Pragrahamārga or Mokṣamārga ;
(4) the grāsa which is defined as p+r-k ; (5) the Karṇa
whose expression is √(B²+β²), B being the Bhuja defined
and β the latitude of the Moon.
Then the following working is stipulated (a) If in
time T, a path of length l is traced, what will be traced in
t? The result is lt/T (b) Then B = l - lt/T (c) B² + β² = K²
(d) p+r-k=g. Thus combining all the steps
{l (T-t) / T}² + β² = (p+r-g)² ie.
l² (T-t)² + β² T² = T² (p+r-g)² I
given t, this equation gives g and given g it gives t.

404
Again the following relation holds good between T
and l, l² = (p+r)² – β² II and
l / (m₁–s₁) = T with the nomenclature already employed which
means T = √(p+r)²–β² / (m₁–s₁) III
In the above working, the fundamental elements are
p, r, β, m₁ and s₁ with which the other elements could be
worked out. Replacing the other elements from equation
I, we have
{(p+r)²–β²} { √(p+r)²–β² / (m₁–s₁) – t }² +
β² (p+r²–β²) / (m₁–s₁)² = (p+r)²–β² / (m₁–s₁)² (p+r–g)²
ie. {(p+r)²–β²} [√(p+r)²–β² – t (m₁–s₁)]² +
β² (p+r²–β²) = (p+r²–β²) (p+r–g)²
ie. {√p+r²–β² – t (m₁–s₁)}² + β² = (p+r–g)² IV
Putting t=0 in this equation, we have
(p+r)² = (p+r–g)² ie. g=0 which means at the moment
of first contact, the grāsa is zero. Again putting
t = T ie. t = √(p+r)²–β² / (m₁–s₁) ie. t (m₁ – s₁) = √p+r²–β²
we have
β² = p+r–g² ie. g = p+r–β which gives the grāha
at the middle of the eclipse which was defined as the
Sthagita. In equation IV which we may take as a funda-
mental equation, the two unknowns are t and g one of
which being given the other could be got.
Verse 36. The colour of the eclipse.
When less than half the disc of the Moon is eclipsed,
the colour will be what is called Dhumra ie. of the colour

405
of smoke; when the disc is half eclipsed, the colour is
black; when more than half is eclipsed, the colour would
be a blend of black and red and when the entire disc is
eclipsed, the colour will be what is called pisanga or
reddish-brown.
Comm. Clear.
Verse 37. When declare the occurrence of an eclipse.
When even one-sixteenth of the diameter of the
Moon's disc is shadowed, the eclipse will not be visible
in as much as the shadowed portion is covered by the
illaminating rays of the disc. In the case of the Sun,
when even one-twelfth of the diameter is shadowed, the
eclipse will not be visible for the same reason. Hence we
shall not declare the occurence of an eclipse upto the
shadowing of the discs to the extents stated above.
Verses 38 and 39. Examples which disclose the
invalidity of construing Valana in terms of Hversine
instead of Hsine.
When the Sun is in the zenith, the Ecliptic being
vertical, the Valana is clearly seen to be the Agra of
(☉+90) where ☉ is the longitude of the Sun. If you
could show that the Valana will be the same on the basis
of Hversine-formula, then I would accept that what
Lallāchārya postulated in his work Śiṣya-Dhī-Vṛddhida
is correct.
Again, in a place of latitude 90—ω, ω being the obli-
quity of the Ecliptic (ω is taken to be 24°), when the Sun
being situated in Meṣa, Vṛṣabha, Mīna or Kumbha, the
Moon contacts him from the south at the moment of a
solar eclipse, in as much as the Ecliptic coincides with
the horizon. In this circumstance, how could the Valana
be equal to R, as made out by the Hversine-formula.

406
Comm. Lallāchārya gave the Valana in terms of the
following verses “स्पर्शादिकालजनितोत्क्रमशिञ्जिनीभिः, क्षुण्णाक्षभा
पलभवश्रवणेन भक्ता, चापानि पूर्वनतपश्चिमयोः फलानि, सौम्येतराणि
समवेहि पृथक् क्रमेण; ग्राह्यात् सराशित्रितयाद् भुजज्या व्यस्ता ततः प्राग्वदप-
क्रमज्या...” Verses 23, 25 Chandragrahaṇādhikāra, wherein
he formulated the Valana in terms of Hversine in the
place of Hsine. The reason for his slip, we have already
explained. Now Bhāskara gives two glaring examples to
substantiate his formula and to show up the flaw in
Lallāchārya's formulation.
In the first example, where the Ecliptic takes the form
of a Vertical, the Sun being in the zenith, the Spaṣṭa
Valana which is the angle between the Ecliptic and the
prime-vertical is the same as the arc between the East
point and the intersection of the Ecliptic with the horizon
known as Lagna. Since the Sun is then in the zenith,
the longitude of the Lagna is (90 + ☉) so that the said
arc is the Agra of the point whose longitude is 90 + ☉ as
stated. Hence Spaṣṭa Valanajyā = sin A = sin δ/cos ϕ
where A is the agrā (using Napier's rule from triangle
PNL where L is the Lagna N the north-point and P the
celestial pole). In the Hindu form, this is given by
H sin V = H sin A = (R H sin δ) / (H cos ϕ) where δ is the declination
of a point of the Ecliptic whose longitude is (90 + ☉). But
Lallāchārya's formula gives the Valanajyā as Hvers δ,
δ being the declination of a point of the Ecliptic whose
longitude is (90 + λ), λ being the longitude of the Eclipsed
body ignoring the latitude. In other words, in the case of the
lunar Eclipse when the Moon is in the zenith his Valanajyā
= Hvers δ (δ having the above value) the Ākṣa Valanajyā
here being zero. Since (R H sin δ) / (H cos ϕ) ≮ Hvers δ, the mistake
committed by Lallāchārya is evident even supposing
H cos ϕ = R when we ignore the latitude ie. take ϕ to
be zero.

407
In the second example cited by Bhāskara the Ecliptic
coincides with the horizon, the pole of the Ecliptic being
in the zenith. Then in a Solar Eclipse the Moon eclipses
the Sun from the south showing that H sin V = R. That
H sin V = R is also evident from the fact that the Ecliptic
makes 90° with the prime-vertical, having coincided with
the horizon. But here according to Lallāchārya’s formula,
H sin ξ = Hvers 90° = R and Āyana Valanajyā is Hvers δ,
where δ is the declination of a point whose longitude is
90° more than ☉. If ☉ = 30°, 60° H sin θ =
R/2 sin ω/R or (√3/2) (R sin ω)/R ie. (sin ω)/2 or √3/2 sin ω.
Evidently the sum of the two Valanas Āyana and Ākṣa
cannot be 90° as is also vouchsafed from geometry. So,
here also, the flaw is evident.
Note 1. Śrīpatyāchārya also followed Lallāchārya
vide verses 18, 19, 20 Chandragrahaṇādhyāya, Siddhānta
Śekhara. It will be noted that the commentator of
Siddhānta Śekhara, while reiterating Bhāskara’s stand as
the correct one, himself commits a mistake in saying
“सममण्डलीय नतांशज्यास्थाने नतकाकोत्क्रमज्या गृहीता” In fact
sin ξ = (sin ϕ sin h) / cos μ = (sin ϕ sin z) / cos δ as proved by us before.
The commentator cited above overlooked that sin ξ could
be also equal to (sin ϕ sin h) / cos μ , wherein natakāla also is
implied.
Note 2. It will be noted that even Pṛthūdakāchārya,
while commenting on Brahmasphuṭa Siddhānta, ignored
Brahmagupta and followed Lallāchārya blindly.
Note 3. The formula given by Lallāchārya and
followed by Pṛthūdaka as well as by Śrīpati is very rough
besides containing the flaw cited, in as much as both μ
and δ are taken to be zero, which are not so,

SURYAGRAHANĀDHIKĀRA
Verse 1. In as much as the observer situated on the
surface of the Earth and as such elevated by the radius
of the Earth from the centre there of, perceives not the
Sun and the Moon having the same longitude at the
moment of conjunction, to be in the same line of sight,
heyt being depressed unequally having different orbits, so
I proceed to elucidate what are called Lambana and Nati
ie. parallax in longitude and latitude, on which account
they are not in the same line of sight.
Fig. 91
Comm. (Refer fig. 91) Let E be the centre of the
Earth, M and S the centres of the discs of the Moon and

the Sun. Let A be the position of an observer on the
surface of the Earth, elevated by the radius EA from E.
Let M and S be in the same line of sight as seen from E.
But as seen from A, AS and AM are respectively the lines
of sight to the Sun and the Moon. Evidently these lines
of sight differ the Moon being depressed more than the
Sun. If a line AS' be drawn which is parallel to the central
line of sight namely EMS, we find that the Sun is
depressed by the angle S'AS whereas the Moon is depressed
by the angle S'AM'. These angles differ because the orbits
of the Sun and Moon differ.
Here the angle S'AS will be very very small, its
magnitude being in truth just about 8" only. But the
angle S'AM' will be sufficiently large since the Moon is
very near the Earth compared with the Sun. Taking ES
and AS to be almost parallel due to the largeness of the
Sun's distance, the angle SÂM will be almost equal to
AM̂S so that we could consider that the Moon is depressed
from AS the line of sight to the Sun by the angle SÂM' =
AM̂E. This angle AME is called the geocentric parallax
of the Moon and the angle AŜE that of the Sun M'ÂS =
angle of depression of the Moon over and above that of
the Sun = M'AS' - M'AS = EM̂A - EŜA = geocentric
parallax of the Moon minus geocentric parallax of the Sun.
Verse 2. The presence or absence as well as the
positiveness and negativeness of the parallax in longitude.
Compute the Lagna at the moment of conjunction of
the Sun and the Moon. There will be no parallax in
longitude when the Sun is situated at the point called
Vitribha or the point whose longitude is ≡ L - 90°, L
being the longitude of the Lagna ie. the ascendant which
is the point of intersection of the Ecliptic with the
52

410
horizon. If the Sun's longitude falls short of the longitude
of the Vitribha or exceeds it, there will be parallax in
longitude which will be positive in the former case and
negative in the latter.
Fig. 92
Comm. (Ref. fig. 92) Let SN be the horizon, Z the
zenith and VA the Ecliptic. A is the ascendant or Lagna.
Let V be the point called Vitribha which is 90° behind A.
Strictly speaking V is called Vitribhalagna or lagna from
which three Rāśis or 90° are subtracted (Bha = Rāśi.
त्रिभिर्विरहितम् वित्रिभम्; वित्रिभम् च तत् लग्नम् च वित्रिभलग्नम् ie.
a point whose longitude is got by subtracting three
Rāśis from that of the Lagna). Let ZV be the vertical of
V so that ZV̂A = 90°. It will be seen that AV = 90° as
follows. Let A' be the point where the Ecliptic intersects
the horizon on the west. One will construe that the
Ecliptic is bisected by the meridian; but it is not so.
Spherical triangles AVZ and A'VZ being right-angled at

411
V are congruent because AZ = A'Z and ZV is common
∴ AV = VA'. But AV + VA' = 180° because the Ecliptic
and the horizon being two great circles, they bisect each
other. Hence AV = 90°. Then a celestial body situated at
V will be depressed along ZV the vertical, say, to a point B.
Let ☉ be any arbitrary position of the Sun; then ☉ will
be depressed along the vertical Z☉, say, to a point C.
Draw CD perpendicular on the Ecliptic. Then ☉D is the
component of the parallax ☉C along the Ecliptic where
as DC is its component perpendicular to the Ecliptic.
Thus ☉D is the parallax in longitude and DC is the
parallax in latitude. The word ‘Lambana’ means
etymologically लम्बते अनेनेति लम्बनम् ie. that amount by
which the celestial body is depressed (along the Ecliptic).
In Hindu Astronomy the word Lambana is applied to
parallax in longitude alone whereas the word Nati is
applied to parallax in latitude. Hence to translate
Lambana as parallax alone is not correct. The word
Drik-lambana is applied to mean parallax along the
vertical, and the word Sphutalambana is occasionally used
to connote parallax in longitude.
As Bhāskara rapidly comments on the verses in this
Gaṇitādhyāya, he having dealt with the subject of parallax
elaborately under the caption, Grahaṇa Vāsanā, in the
Golādhyāya, to catch up his thought, we have to treat the
subject first from the modern view point and then elucidate
what he has said in the Golādhyāya, much matter of which
is reiterated by him under the commentary here in the
Gaṇitādhyāya.
(Ref. Fig. 91) From the △EAM,
(sin EMA) / a = (sin EÂM) / d = (sin ZÂM) / d where a is the radius of
the Earth and d the distance of the celestial body (here the
Moon)

412
∴ Sin EMA = a/d sin ẐAM = ÊMA expressed in radian
measure since ÊMA is very small
∴ ÊMA (expressed in radian measure) = a/d sin z I where
z is the apparent zenith-distance of the Moon ie. zenith-
distance as seen by the observer (in contradistinction to
the geocentric zenith-distance of the Moon namely ẐEM).
In particular, when z = 90°, ÊMA = a/d which is the
maximum parallax known as the horizontal parallax ie.
the parallax when the Moon is situated on the horizon of
the observer. Also the parallax is zero when M is situated
at Z as is seen from formula I and as is rightly remarked
by Bhāskara in the words 'खमध्ये नास्ति लम्बनम्'.
In fig. 91, ÊMA is the angle by which the line of
sight of the observer namely AM is depressed from the
geocentric line of sight EM. Since the plane of the paper
represents a vertical through the Sun and the Moon, the
depression of either the Sun or the Moon or the excess of
the depression of the Moon over the Sun are all in the
vertical plane. This depression is called Drik-lambana
because it is a lambana or depression in the Drik-maṇḍala
or vertical.
This Drik-lambana varies as sin z as is seen from
formula I where a, and d may be taken to be constants.
(Both a and d vary slightly a varying slightly from place
to place on the Earth, the Earth being an oblate spheroid,
and d varying from position to position of the Moon).
The maximum horizontal parallax is given by a/d in
radian measure which is equal to, according to Bhāskara's

418
estimate 1581/2 × 1/51566 × 3438 = 53′ approximately. Its
modern value is about 57′ so that the Hindu estimate is
not far from truth. Here 1581 and 51566 are the values
of a and d in Yojanas according to Bhāskara.
The Hindu astronomers do not, however, proceed
exactly as we have done in the para above to obtain the
maximum horizontal parallax. Their treatment is a little
different and is as follows. Whereas according to Modern
astronomy E M̂′ A (fig. 93) is viewed as the horizontal
[चित्र: Fig. 93]
Fig. 93
parallax, in Hindu Astronomy MÊM′ or the angular
measure of the Moon’s path equal to the radius of the
Earth is taken to be the horizontal parallax. Both,
of course, mean the same as is seen from the figure.
In as much as the Hindu astronomers knew very well
what they term कलीकरण or converting linear distances
into angular measure, converting a linear magnitude equal
to the radius of the Earth namely MM′ at the lunar orbit
into angular measure, they got
MM′/EM × 3438 = 1581/2 × 1/51566 radians = 52′-42″ as the
maximum horizontal parallax.

414
Having got this estimate, they reckoned this angular
measure in time as the time taken by the Moon to traverse
the distance MM' equal to the radius of the Earth as
follows. The Moon traverses 790′-35″, which is exactly
15 times 52′-42″. Hence they said that the maximum
horizontal parallax is 1/15th of the Moon's daily motion in
arc and expressing it in terms of time, that the maximum
horizontal parallax is 1/15 of a day or 1/15th of 60 nādis or
4 nādis.
That this horizontal parallax is 4 nādis as a maximum,
would have been also verified at the time of a solar eclipse
when the Sun was situated on the horizon at the time of
conjunction, by the fact that the eclipse occurred four nādis
in advance of the moment of geocentric conjunction
(which could be calculated very accurately by the Hindu
astronomers, as could be seen by the very correct estimate
of a lunation in Hindu Astronomy. In fact, the length
of a lunation must have been estimated correctly by noting
the time-interval between two solar eclipses or lunar and
by dividing that time by the integral number of lunations
elapsed in between the two eclipses).
The question then arises as to how the Hindu Astro-
nomers could know the distance of the Moon. From the
estimate of the horizontal parallax by actual observation,
and from the geometry of fig. 93, a correct estimate of the
distance of the Moon must have been arrived at.
Having thus known that the Moon traverses a
distance equal to the radius of the Earth in 4 nādis, his
daily linear motion was estimated to be 15 times the radius
of the Earth ie. (15 × 1581 1/6) / 2 = 11858 3/4 Yojanas.
The daily motion of the Moon having thus been
estimated almost correctly, an act of inexpedience on the
part of the Hindu astronomers was that they should have

415
presumed that all the other planets including the Sun would
be traversing the same linear distance during the course of
a day. This led to a wrong estimate of the Sun’s distance
as well as his spherical radius. Also, they supposed wrongly
that the parallax of the Sun also would be equal to 1/15th
of his daily arcual motion.
Their estimate of the spherical diameter of the Moon
was, however, very near the truth, for, they argued, that
if 790′-35″ angular motion per day corresponded to 11858¾
Yojanas in linear measure, to what linear measure did the
angular diameter of the Moon namely 32′-0″-9′′′ corres-
pond? The answer was
11858¾ × 32′ 1/400
------------------ = 480 Yojanas.
790′-35″
[चित्र: Fig. 94]
Fig. 94
It may be here pointed out that there is a relation
between the angular radii of two celestial bodies as seen
from each other and their mutual horizontal parallaxes
(Fig. 94). Let E and M bet he centres of the Earth and
the Moon respectively. E M̂ A = horizontal parallax of
the Moon = angular radius of the Earth as seen from the
Moon and B Ê M = Angular radius of the Moon = Hori-
zontal parallax of the Earth as seen from the Moon. Thus,
we see that the Earth will be seen from the Moon, as a
Moon with an angular radius equal to 57′. In other words
our Earth will be a Moon to our Moon, having nearly 16
times the area of our Moon’s disc.

416
The periphery of the Moon’s orbit was arrived at as
follows. “If 790′-35″ of the Moon’s angular motion
corresponds to 11858¾ Yojanas to what periphery must
360 × 60′ correspond ?” The answer is
(11858¾ × 360 × 60) / (790′-35″) = 324000 Yojanas. Reverting to the
subject of parallax on hand, the Drik-lambana or the
parallax along the vertical has the formula (4 H sin z) / R I
nādīs in Hindu Astronomy where 4 nādīs is the maximum
parallax obtained when H sin z = R.
From fig. 92, C☉² = CD² + D☉² ie.
Drik-lambana² = Nati² + Sphuṭalambana² II
CD = ☉C sin CÔD = (4 H sin z / R) ×
sin CÔD = 4 sin z sin VÔZ
= 4 sin ☉Z sin VÔZ = 4 sin ZV
= (4 H sin ZV) / R = VB III
Thus, the parallax in latitude at any point of the Ecliptic
is that at the Vitribha which is conveyed by Bhāskara
in the words “कक्षयोरन्तरं यत् स्यात् वित्रीभे सर्वतोऽपि तत्”.
Also ☉D = C☉ cos CÔD = 4 sin ☉Z cos Z☉V
= 4 cos ZV sin V☉ = (4 H cos ZV H sin V☉) / R² IV
= Maximum parallax × Vitribha-Sanku × H sine of the
arc V☉.
In the above working we proceeded in a modern way.
It is worth-hearing Bhāskara as to how these results were
arrived at elegantly and ingeneously from first principles.