पञ्चसिद्धान्तिका (आचार्य वराहमिहिर - सूर्य, रोमक, पौलिश, वासिष्ठ एवं पितामह सिद्धान्त)

212 PAÑCASIDDHĀNTIKĀ IX.14
the previous day’s true Moon from the given day’s true moon. The daily mean
motion, multiplied by 120' and divided by the momentary motion per day is
the radius vector at the moment.
The following is given here:-
(A) (i) The daily change in sine Eq.C. for short interval = the interval in the tabular sine of
anomaly current × daily motion of mean anomaly × degrees of epicycle ÷ (225 × 360).
(ii) The daily change in Eq.C. in minutes of arc = (i) × 3438 ÷ 120. (since the sine is small and
there is no difference between sine and arc).
It should be noted here that it is possible to simplify the above work, because the daily anomaly
and the degrees of epicycle are fixed for each body, and the rest are constants. Only the interval
in the tabular sine of anomaly current varies. For instance, the Moon’s daily motion of anomaly is
783'.9, and epicycle 31°. Therefore, the daily change in Eq.C. in minutes of arc
= the interval in the tabular sine of anomaly current × 783.9 × 31 × 3438 ÷ (225 × 360 × 120)
= described interval × 8.6.
For the Sun it is, interval in the tabular sine of anomaly current × 59.1 × 14 × 3438 ÷ (225 ×
360 × 120) = interval etc × .29.
(ii) The daily rate of true motion for the moment = mean daily motion ± (ii) (additive if anomaly
is from rāśis 3 to 9 and subtractive if from 9 to 3.)
(B) The true daily motion = the given day’s true longitude – the previous day’s true longitude.
(C) The radius vector at the moment = 120 × mean daily motion ÷ the daily rate of true motion
for the moment.
The sine interval of anomaly used in computing the true Moon or Sun for noon can easily be
used to find the daily rate for the noon in question. It is this that should be used for finding the
motion during the interval between mean and true noons, as we have done already, and in correcting
for longitude. In eclipses also this true daily rate with radius vector, for the moment of syzygies
should be used, because this will give the circumstances accurately. This seems to be the author’s
idea in giving these here. As for the Sun, there is no distinction either in the rate or radius vector
between those for the day or for the moment. This is indicated by making the distinction in the case
of the Moon alone, using the expression śaśiviśeṣāt.
Example 8. For the Ujjain true noon of examples 5 and 6, find the true motion (a) for the Moon (b) for the
Sun.
(a) In example 6, the Bhuja, i.e. sin of Moon’s anomaly got is rā.0-24-7.5. Sines being given for
every 3 3/4 degrees, the current sine interval is the seventh, equal to 7' 9".
∴ By (A ii), the daily change in Eq.C. = 7' 9" × 8.6 = 61'.5
By (A iii), the true motion for the said noon = 790'.6 – 61'.5 = 729'.1, (subtraction since
anomaly is between rā. 9 to 3).
14a-b. A.B.C.D.ज्ञेयाहो
b. A.B2. भेंद्की; B1.2. र्भइकी; C.D. रात्रिकी d. B1.2.भक्ति A.B1.2हता:; B3.ऋता:

IX. 15 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 213
(b) The Bhuja or Sun's anomaly in example 5 is rā. 1-14-44.3. The current interval, 12th, is 5' 44".
∴ by (A ii), the daily change in Eq.C. = 5' 44" × .29 = 1' .7. By (A iii), the true motion = 59'.1 -
1'.7 = 57'.4, (subtraction since anomaly is between rā. 9 and 3).
Example 9. For the noon of example 8, find the Moon's radius vector.
By (C), the radius vector required = 120' × 790'.6 ÷ 729'.1 = 130'.1.
The following is the explanation of the rules: The true motion during any interval between two
moments is the difference between the true longitudes of the moments. The shorter this interval,
the more accurate is the motion. In the rule, all factors excepting the sine of anomaly are constants.
Therefore the accuracy of the motion depends on the sine of anomaly only. In the case of the
Moon, the motion of the anomaly being rapid, there is significant difference in the sine from time
to time even within the day, for four or even five sine intervals pass in a day, with the result that the
motion is got differently for different times. So the motion has got to be found for shorter periods
like the yāma in the day, and this is the motion for the time being. For every 225' of anomaly there
is one sine interval. So, during the time for which the anomaly interval is current, the change in sine
Eq.C. is caused by the corresponding sine interval current. Therefore, the change in sine Eq.C. is
got by multiplying the current sine interval by the degrees of epicycle, and dividing by 360, for the
period covered by the corresponding anomaly interval of 225'. From the change in sine Eq.C. the
change in Eq.C. is got in minutes (by multiplying by 3438 and dividing by 120, as we have done).
This change is for the time to which the current 225' interval of anomaly corresponds. It is
converted into the change per day by multiplying by the daily motion of anomaly in minutes and
dividing by 225. This is applied to the mean daily motion to get the daily rate.
Now, we know that the Eq.C. is zero when the anomaly is zero, that it is negative and increases
numerically in the first quadrant of anomaly, i.e. upto 3 rāśis, then decreases numerically, still being
negative, to the end of the second quadrant, i.e. upto 6 rāśis where the value becomes zero again,
then in the third quadrant it is positive and increases to a maximum at 9 rāśis and then decreases
in the fourth quadrant to zero at the end of 12 rāśis or zero. From this it can be seen that the Eq.C.
goes on decreasing as the anomaly passes from 9 rāśis to 3 rāśis i.e. the change is negative or subtrac-
tive, and goes on increasing as the anomaly passes from 3 rāśis to 9 rāśis, i.e. the change is positive
or additive, as instructed by the text. No harm will ensue from the instruction to multiply and divide
the sine interval first and then reduce it to the epicycle, since the sine is small. It is to be noted well
that the motion per day found is not actually the motion in the day, but only the rate during the
short interval or moment taken in the day. That is why the motion for the day is given by a separate
rule, for the Moon. In the case of the Sun there is no distinction between the two, since the daily
motion of the anomaly is small. The rule for the radius vector has been explained already when
dealing with the Romaka.
[रविचन्द्रकक्षे]
'मुनिकृतगुणेन्द्रिय'घ्नः स्फुटकर्णः 'खकृत' भाजितोऽर्कस्य |
'(स्वरवसु) मुनीन्द्रविषया' भानोः 'खकृतर्तु [व] सुगुणाः' शशिनः |

214 PAÑCASIDDHĀNTIKĀ IX. 15
Kakṣā of the Sun and the Moon
15. The Sun's radius vector multiplied by 5347 and divided by 40 is called its
kakṣā. The Moon's radius vector multiplied by 10 is its kakṣā.
It is to be noted that the kakṣa obtained here, depending as it does on the radius vector is also for
the moment taken and its neighbourhood. We can also derive it directly from the daily rate of
motion obtained from verses 3-14, above. Thus:
(a) The Sun's kakṣā = Sun's radius vector × 5347/40 = (120 × 59.13 ÷ Sun's daily rate of motion)
× 5347/40 = 9,48,558 ÷ Sun's daily rate of motion.
(b) The Moon's kakṣā = Moon's radius vector × 10 = (120 × 790.56 ÷ Moon's daily rate of
motion) × 10 = 9,48,680 ÷ Moon's daily rate of motion.
Example 10. Pudukkottai (Lat. 10° 24′), on a particular day, new moon falls at nā. 20-40 after sunrise. At
that moment, the longitude of the sun = the longitude of the Moon = rā. 2-0-0. The Rāhu-head, at that time is
rā. 7-29-24. The Sun's rate of motion for the time is 57' per day and the Moon's 810'. The daytime is nā. 31-
20. Compute the solar eclipse occurring.
For this, the kakṣā is found first:
(a) The Sun's Kakṣā for the time = 9,48,558 ÷ 57 = 16,641
(b) The Moon's for the time = 9,48,680 ÷ 810 = 1171.2
The author does not use the word kakṣā here in its usual sense of orbit, but for the actual distance
reduced by some factor, for the orbit is constant, while what we get here is a quantity varying with
the rate of motion. Since only the proportion of the distances of the Sun and the Moon from the
earth is significant, the reduction will not cause any error. That is why, the word yojana giving the
measure of distance, is not used here.
Now, the mean kakṣā, derived from the mean radius vector, 120', is for the Sun, 120 × 5347 ÷ 40 =
16,041. For the Moon it is 120 × 10 = 1200. These obviously are the respective reduced mean dis-
tances from the earth. In the Original Saura the Sun's orbit is given as 6,89,358 yojanas, and the
Moon's 51,566 yojanas, as we learn from the Ārdharātrika system etc. Since the mean distances are
proportionate to the orbits etc, if the Moon's orbit, 51,566, is reduced to 1200 as here, the Sun's
orbit, by the same factor, must be reduced to, 6,89,358 × 1200 ÷ 51566 = 16,042. This agrees very
closely with 16,041 got above. The difference of one may be due to giving the multiplier correct to
the nearest whole number, as 5347. Further, the orbits, which is the same for our present purpose
as saying distances, are inversely proportionate to the yuga cycles given in the Śāstras. Therefore,
from the Saura cycles of Sun and Moon in I. 14, by the proportion 1,80,000:24,06,389 :: 1200: x, we
get 16,042 for x, the Sun's reduced mean distance, when the Moon's is 1200.
This agreement is the justification for our correcting the reading drighna into digghna. But TS
correct it into gnighna. Also, they correct khaṛ into khārka though there is the alternate reading
khakṛta fitting correctly in the rule and adopted by us. By their corrections the Sun's mean kakṣā
will be 5347 and the Moon's 360. Thus the Sun's kakṣā becomes, 5347/360 ( = 14.85) times the
15a. A. स्पुट b. A. खऋभाजितो; C. खार्कभाजितो
B3. Has an unnecessary gap after c. B. कक्ष्येति. A. करणों; B. कर्णे
कर्णः; B1.2. do not have it. d. A.B. द्रिघ्नः; C. मिघ्नः

IX.16 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 215
Moon’s, against the fact that it is only 13.37 times from the Śāstras, and what they give is equal to
saying that there are 14.85 revolutions of the Moon in the year for the kakṣās vary directly as the
periods of revolution, i.e. inversely as the number of cycles in a given period. We shall also see what
havoc their wrong corrections play in the angular diameters following.
[बिम्बमानम्]
‘(स्वरवसु) मुनीन्द्रविषया’ भानोः ‘खकृतर्तु[व]सुगुणाः’ शशिनः ।
तात्कालिकमानार्थं स्फुटकक्षाभ्यां पृथग्विभजेत् ॥ १६ ॥
Measure of the orbs
16. Divide 5,14,787 by the Sun’s kakṣā, and 38,640 by the Moon’s to get the
respective angular diameters in minutes at the time.
It is to be noted that the angular diameters (of the orbs of the Sun and Moon) are always given
in minutes by our Śāstras. Thus,
(a) The Sun’s angular diameter in minutes = 5,14,787/Sun’s kakṣā.
(b) The Moon’s angular diameter in minutes = 38,640/Moon’s kakṣā.
Example 11. To continue the problem of example 10, find the angular diameters of the Sun and Moon for
the time given.
(a) The Sun’s angular diameter = 5,14,787' ÷ 16,641 = 31'.0.
(b) The Moon’s angular diameter = 38,640' ÷ 1171.2 = 33'.0.
The derivation of the rules for the angular diameters is as explained below. The angle formed
at the eye by the diameter of the orbs of the Sun and the Moon is their angular diameter and given
in minutes. We all know from experience that the nearer the orbs, i.e. the lesser the radius vector
(given in yojanas), the greater is the angle, and the farther away is the orb, i.e. the greater the radius
vector, the lesser is the angle. Thus the angle and the radius vector are in inverse ratio, as also the
kakṣā which is directly proportionate to the radius vector. So we have:
Sine angle at the eye = 120 × diameter in yojanas ÷ the radius vector in yojanas.
The angular diameter in minutes = 3438 × 120 × diameter in yojanas ÷ (radius vector in yojanas
× 120.)
Here, the author has reduced the diameter in yojanas by the same factor used in verse 15 to
reduce the radius vector in yojanas to the kakṣā, and multiplying by 3438, as explained already, to
convert the sine into minutes, he has given 5,14,787 for the Sun, and 38,640 for the Moon. The
mean kakṣā of the Sun derived by us in the explanations is 16,042. Dividing 5,14,787 by 16,042, we
have the Sun's mean angular diameter, 32'.1, and the Moon's is 38,640 ÷ 1200 = 32'.2.
16a. A.B1.2. खखवसुखमः; C.D. खवेसुखमुनीन्द्र;
(D. मुनीन्दु) c. B2. तत्कालिक; B1.2. तत्कलिका
b. B. खततर्तुः; A.B.C. om व; C. सुरगुणाः B2. Unnecessary gap after of शशिनः

216 PAÑCASIDDHĀNTIKĀ IX.18
The angular diameters derived from the Original Saura (and the modern Sūrya
Siddhānta) are 32'.3 and 32'.0, respectively. Though the difference is small, we must investigate
why there is a difference at all. The error is about 160th part of itself in each case. Either the author
has taken values slightly different from those of the Original to derive the numbers here, or there
are some errors in the readings here.
Now for the readings. The first foot of the verse is in excess by one mātrā. There are eight digits
in the number, khakhavasu-khamunīndraviṣayāḥ, though there should be only six digits. So we have
corrected khakhavasukha into svaravasu following the form of the letters also. TS correct it as
khavasukha etc., giving seven places in the number, as 5147080. By this, the dividend has become
ten times what it actually is. Since they give the divisor, viz. the Sun's kakṣā, as one third of the actual,
(as seen already), they have made the angular diameter in minutes, thirty times the actual. Unaware
of the mistakes they have made they wonder why the angular diameter comes thirty times the
actual, and make the following curious comment: The correct angular diameters can be got only
by dividing what we get here by 30. But the author does not say anything about dividing by thirty.
Perhaps in his days there was the well-known understanding that what is got is to be divided by 30
to get the angular diameter. That is why, we surmise, the author has not instructed the division by
30 !! (vide page 50 of the Sanskrit commentary.) All this is the result of the errors in their correction
of the readings. NP too have sensed the error here and emend the expressions as khavasukha
munīnduviṣayāḥ (5,17,080), with the result that "the radius of the Sun is about four times the radius
of the earth, the radius of the Moon about one third" (pt. II, p. 73).
In the same way, there is one mātrā less in the second foot. Therefore, supplying the lost letter
we have read khakṛtartusuganāḥ as khakṛtartuvasuguṇāḥ. NP too, do the same. By this we get the five
places required in the dividend. But TS read it as khakṛtartusuraguṇāḥ, thus making the dividend
nine times what it is. They have already made the divisor, the moon's kakṣa, three tenths of what the
author has said it is. By this the angular diameter of the Moon also has been made thirty times the
real value, by them, again rousing their wonder in the manner mentioned before.
[मध्यज्या]
मध्यार्कलम्बिततिथेरन[क्ष]राश्युद्गमैः प्रतीपांशाः ।
प्राक् समलिप्ताहानिः क्रमेण पश्चाद्धनं कार्यम् ॥ १७ ॥
तन्मध्यविलग्नाख्यं तस्माच्चापक्रमांशकाः क्रमश: ।
तैरक्षवियुतयुक्तैर्या ज्या (म)ध्याभिधाना सा ॥ १८ ॥
Sin Zenith Distance of Meridian pt.
17. Find the interval between midday and the moment of new moon. If the
Sun is east of the meridian (i.e. if new moon falls in the forenoon), find the
degrees of right ascension corresponding to this time using the ascensional
differences of zero latitude, (laṅkodayamāna.), backwards from the Sun.
Subtract these degrees from the Sun (= Moon) of the moment of new moon.
If the Sun is west of the meridian, (i.e. if new moon is in the afternoon), find
the degrees corresponding to the interval counting forward from the Sun,
and add to Sun (= Moon).

IX.18 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 217
18. The meridian point of the ecliptic (madhyalagnam) is got. Find its declina-
tion, north or south. If north, find the difference between the declination and
the latitude of the place. If south, add them. The sine of the result is called
madhyajyā, i.e. sin zenith distance of the point.
The madhyajyā is got thus:
(i) Interval between noon and new moon = time of noon ~ time of new moon.
(ii) The degrees of rise of the ecliptic, backward or forward from the Sun, using the corresponding
ascensional differences of zero latitude, for the interval in (i) in the forenoon or afternoon respectively
is to be found.
(iii) (Longitude of) meridian ecliptic point = The Sun (or Moon) at new moon ∓ (ii), ( - for fore-
noon, + for afternoon).
(iv) The declination north or south of (iii) is to be found.
(v) Sine zenith distance of m.e.p. in (iii) = sine (declination found in (iv) ± latitude), ( + if the
declination is south, and the sine found is directed south, ~ if the declination is north, and the sine
is directed north or south according as the declination or the latitude is greater.)
Example 12. To continue example 10 using the times given there.
(i) Interval between noon and new moon = nā. 20-40 - nā. 15-40 = 5 nā., afternoon.
(ii) Given, Sun = Moon = rā. 2-0-0, at new moon. Forward counting is to be done (because after-
noon) from the first point of Gemini where the Sun is. For successive 10° of rise, the times taken are,
in vināḍis, 105.4, 107.6, 108.8, 108.8. For the interval, of 300 vināḍis = 105.4 + 107.6 + 87, there
are 10° + 10° + 8° ( = 10° × 87 ÷ 108.8) = 28°.
(iii) Adding, the meridian ecliptic point = rā. 2-0-0 + 28° = rā. 2-28-0.
(iv) The declination of m.e.p. is 23° 58′N.
(v) Sine zenith distance of m.e.p. = sine (23° 58′ - 10° 24′) = sine 13° 34′ = 28′ 9″, north (∵
declination is greater).
It has been stated by us, above, in the context of the computation of the solar eclipse according
to the Paulīśa, that in order to get the parallaxes in longitude and latitude, the nonagesimal and the
sine and cosine of its zenith distance, (i.e. dṛgjyā and Śaṅku), are to be found. In the Saura, a diffe-
rent method is given to get the sine of the zenith distance of the nonagesimal (z.d.n.) for which the
m.e.p. and the sine of its zenith distance are necessary, and given here.
It has been said that the m.e.p. is the point of intersection of the meridian (NZS in the fig.) and
the ecliptic (OrO' in fig.) as M in fig. s is the position of the Sun at new moon, (occurring in the
17a. D. मध्याह्नलम्बित. B. लम्बितातीर्थे
b. A. ०रनराश्युः; B1.2. ०रतराश्युः; B3. ०रसत्तराश्युः; 18b. B1.3. तस्माच्च
C. ०निरक्षराश्यु; c. B1.3. विपुत
D. ०रन्त[र]राश्यु. B. प्रीतिपाशाः d. A. या ज्याकृतिं सद्याभि०; B1.2. या ज्या सधाभि०;
d. A.B. पक्षाधनकार्यः; A2. पक्षाधनकाकैर्यः B3. या ज्या सद्वाभि०;

218 PAÑCASIDDHĀNTIKĀ IX.19
Fig. IX. 2
afternoon in the example given to illustrate which the figure is drawn). HQ is the right ascension
corresponding to the segment of the ecliptic sM, which is found from the time to or from noon, by
using the ascensional differences at the equator. (In the figure the time is afternoon.) sr is the
longitude of the Sun at new moon. Mr the longitude of m.e.p. = sr - sM, as stated for afternoon.
Clearly, for forenoon, s being east of M, Mr = sr + Ms, as instructed.
MQ is the declination of M, (north in the fig.) ZQ is the latitude. MZ is the zenith distance. MZ
= MQ - ZQ (in the fig.), i.e. when north declination is greater than the latitude, the latitude is
subtracted from it, and the zenith distance is north. If M had been between Z and Q, then the north
declination would be less than the latitude, and MZ would be equal to ZQ - MQ, and south. In the
case when M is south of Q i.e. when the declination of M is south, it is clear that the zenith distance,
MZ = MQ + QZ, and it is south, as stated.
It is to bring in all the above three cases that we interpreted akṣaviyuta in two ways, 'subtracting
the latitude', and 'subtracting from the latitude'. But TS and NP take only the latter case, with the
result that the former case is shut off. In verse 17, NP has corrected madhyārka as madhyāhna, which
is not essential. The defect in the second quarter of verse 17 has been rectified here by the addition
of the letter kṣa as ranak(ṣa)rāśyudgamaiḥ, anakṣa meaning nirakṣa or zero latitude, to give the sense
required here. TS emends it as nirakṣa in the same sense and NP as rantararāśyudgamaiḥ.
[रविदृक्क्षेपः]
तिथ्यन्तविल(ग्र)ज्या काष्ठान्तज्याहता स्वलम्बहता ।
मध्यज्याघ्नी व्यासाऽर्धभाजिता वर्गिता सा च ॥ १९ ॥

IX.20 | IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE | 219
मध्यज्याकृतिविश्लेषितां पृथक् स्थाप्य मूलमेकस्याः ।
सवितुर्दृक्शेपाख्यं संस्मृत्यर्थं पृथक् स्थाप्यम् ॥ २० ॥
Dṛkṣepa of the Sun
19. Find the sine of the longitude of the Orient Ecliptic Point (o.e.p.) at new
moon, multiply by the sine of maximum declination (of the Sun, 48′ 48″) and
divide by the sine of the colatitude. (This is sine amplitude of o.e.p., called
Udayajyā.) Multiply this by the sine of the zenith distance (z.d.) of m.e.p.
already found, and divide by 120′. Square the result, and subtract from the
square of the sine z.d. of the m.e.p.
20. Set the remainder in two places. In one place, find its square root. This
is the sine of the zenith distance of the nonagesimal (z.d. of n.) called the Sun's
dṛk-kṣepa. Keep this safe aside for future work.
The following is to be done:
(i) Using the vināḍīs of ascensional differences of the place, the o.e.p. at new moon is to be
found.
(ii) Sine amplitude of o.e.p. = sine z.d. of m.e.p. × 48′ 48″ ÷sine colatitude.
(iii) Sine (m.e.p. ~ nonagesimal) = sin z.d. of m.e.p. (ii) ÷ 120.
(iv) Square of sine (z.d. of n) = (sine z.d. of m.e.p.)² – (iii)².
(v) Sine z.d. of n = √(iv).
Example 13. Continue example 10, given already lat = 10° 24′ and new moon is at nā. 20-40.
(i) Let us take it that using the ascensional differences of the place (given by chap. IV), the o.e.p.
found is rā. 5-27-56.
(ii) Sine amplitude of o.e.p. = sine rā. 5-27-56 × 48′ 48″ ÷ sin (90° – 10° 24′)
= sin 2° 4″ × 48′ 48″ ÷ sin 79° 36′
= 4′ 20″ × 48′ 48″ ÷ 118′,0 = 1′ 48″.
(iii) Sine (m.e.p. ~ ṇ) = 28′ 9″ × 1′ 48″ ÷ 120 = 25″.
(iv) Sin² (z.d. of n) = (28′ 9″)² – (25″)² = 792′ .25.
(v) Sin (z.d. of n) = √792.25 = 28′ 9″.
19a. A.B. विलग्ना ज्या | b. B. ०षितां; C.D. ०षिता. A.B.D. स्थाप्या; C. स्थाऽपि.
b. A. काष्टांत; B3. कापांत० | A.B. ०मेकस्या
c. B. मध्यमज्याघ्नी | c. A. ०दृक्षेपाख्यं; B. ०दृक्क्षेपाख्यं
20a. B1. ०तति०; B2.3. ०तति०. D. ०विशे० | d. B2. पृथक्थो य ||

220 PAÑCASIDDHĀNTIKĀ IX.21
The following is the explanation of the rule: See fig. 2. There, n is the nonagesimal, i.e. o.e.p.
minus three rāśis. Z is the zenith. n.z is the zenith distance of n, and sine nz, called the 'Sun's dṛk-kṣepa',
is wanted here, to get the Moon's parallax in latitude. This Siddhānta takes the spherical triangle Z
n M, right angled at n, to be approximately equal to a plane right-angled triangle, with the sines of
the arcs as straight lines and finds the dṛk-kṣepa by, (sine zn)² = (sine MZ)² – (sine nM)² = sin² z.d. of
m.e.p. – sin² (m.e.p. ~n). (The correct method has already been expounded by us in Chap. VI,
when dealing with the solar eclipse according to the Pauliśa.) Sin (z.d. of M.e.p.) has been got
already. The other quantity required, viz. sin (M.e.p. ~ n), is got by the well-known formulae relating
to spherical right angled triangles, (already given by us), sin (M.e.p. ~ n) = sin (z.d. of M.e.p.) ×sin
MZn ÷ 120'. But, MZn = O'ZW = OZE, which is the amplitude of the o.e.p. Its sine, Udayā, given
here = the declination of the o.e.p. × 120' ÷ sine colatitude = sin longitude of o.e.p. × 48' 48" ÷
sin colatitude, as given in chap. IV.
[शङ्कु:]
दृ[क्]क्षेपकृतिं जह्यात् त्रिज्यावर्गात् ततोऽस्य यन्मूलम् ।
लग्नाऽर्कविवरमौर्व्या गुणितं त्रिज्योद्धृतं शङ्कुः ॥ २१ ॥
Gnomon
21. Subtract from 14,400, the square of sin z.d. of n, (kept unused in the
other place in the previous work), and find its square-root. Multiply this by
the sine of the distance between the Sun and the o.e.p., and divide by 120'.
The result, which is the sine of the Sun's altitude, is called Śaṅku, i.e. the Sun's
Śaṅku.
∴ Śaṅku = √ 14,400 – sin² (z.d. of n) × sin (o.e.p. ~ sun) ÷ 120'. [sin² (z.d. of n) has already been
got, and kept apart]
Example 14. To complete example 10.
From Ex. 13, by (i), o.e.p. = rā. 5-27-56, and by (iii), sin² (z.d. of n) = 792.25, from Ex. 10, Sun =
rā. 2-0-0.
Śaṅku = √ 14,400 – 792.25 × sin (rā. 5-27-26 – rā. 2-0-0) ÷ 120'
= 116' 39" × sin (rā. 3-27-26) ÷ 120'
= 116' 39" × 106' 7" ÷ 120' = 103' 10".
The rule is derived as follows: From fig. 2 it can be seen that the Sun's altitude is 90° – Zs.
∴ Śaṅṅku = Sine Sun's altitude = Cos Zs = Cos ZW × Cos ns ÷ 120' (by the well-known formula,
already given) = √ radius² – sin² Zn × sin Os ÷ 120', (∵ ns is os – 90°). sin Zn is sine z.d. of n
already found, and its square has already been got and kept apart for use here.
∴ Śaṅku = √ 14,400 – sin² z.d. of n × sin (o.e. P ~ sun) ÷ 120' as given by the author.
21a. A. दृक्षेप; B. दक्षेप. B. कृति A. जह्या c. B.विवरे
b. A. वर्गात्रितोस्प. A. °वन्मूलं; B. °पन्मूलं d. B. गुणित. A. त्रिज्योदृवृतं; B. त्रिज्योधृत

IX.23 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 221
[लम्बितपर्वान्तः]
शङ्क्वङ्गुलाख्यविंशतिशतकृ(त्योर)न्तरेण विश्लेषात् |
स्थि(त)वर्गान्मूलं द्विनवकाहतं (त)द्वि(भ)ज्य कक्षाभ्याम् || २२ ||
भागविशेषा(त्ति)थिवत्तिथ्य(न्तनाम) पुनः पुनस्तत् स्यात् |
एवं मृग्यः कालस्तूत्पन्नो यावदविशेषः || २३ ||
Parallax-corrected New Moon
22-23. Subtract the square of the Sun's śaṅku got above from 14,400. From
the remainder subtract the square of the Sun's dṛk-kṣepa kept apart in the
previous work and find its square root, (technically called Dṛggati). Multiply this by
18 and divide by each of the kakṣās of the Sun and the Moon. Find the respective arcs
(in minutes) and get their difference. Treat this as the minutes of tithi and find the
tithi-nāḍīkās for this. Subtract the nāḍīkās from the time of new moon if forenoon, and
add, if afternoon. The parallax-corrected new moon (p.c.n.) is got. Repeat the oper-
ation of finding the p.c.n., till there is no difference (in time) in two successive opera-
tions. This is the p.c.n. (to be used in the subsequent work).
Though there is no doubt about the idea here, it is difficult to get the idea from the words used,
on account of several corrupt readings. In verse 22, a word is broken at the end of the third foot,
and there are 18 mātrās in the fourth, sinning against the Ārya metre. In the same verse, in the sec-
ond foot NP has emended viśleṣāt into viśeṣitāt against the manuscript readings, an emendation that
is not needed. In the 23rd verse, evam mṛgyaḥ kālaḥ is a repetition. TS have succumbed to this diffi-
culty and give the wrong interpretation that the difference between 14,400 and the square of the
śaṅku, should be subtracted from the square of the dṛk-kṣepa, unaware that this is impossible since
the latter would always be less than the former. We shall show this in the explanation. As for calling
the Sun's śaṅku as 'digits of śaṅku' we have seen it being technically called so in chap. IV.
The method enunciated here is as follows:
(i) Dṛggati = √(14,400 − śaṅku² − dṛk-kṣepa²).
(ii) (a) Sin Sun's parallax in long. = 18 × (i) ÷ Sun's kakṣā
(b) Sin Moon's parallax in long. = 18 × (i) ÷ Moon's kakṣā.
From the two sines, the arcs should be obtained in minutes. The Moon's minus the Sun's parallax
is the (effective) parallax in longitude.
22a. B. ॰ख्यं विंशति
b. A. शतकृशोनंतरेण; B. शततशोनन्तरेण (B2. त्तरेण;
B3. त्तरेण)
D. विशेषि[त]त्
c. A.B1.2. स्थिति
d. A1.B. हतं सद्भिभाज्य
A1. कक्ष्याभ्यां
23a. A. विशेषस्तिथि; B.C. विशेषास्तिथि
b. A. तिथ्यर्द्धान्तामतः; D. तिथ्यर्द्धातामनः;
C. तिथ्यन्तान्नामतः; D. तिथ्यन्तोऽतः पुनः
d. A. ऽक्षूत्पन्नो; B. तत्पन्नो
B. यावदवशेषः

222 PAÑCASIDDHĀNTIKĀ IX.23
(iii) Nāḍis of parallax = the parallax in longitude found in (ii) × 60 ÷ the motion of the tithi per
day. Subtracting the nāḍis from new moon in the forenoon, and adding in the afternoon, the p.c.n.
is got. Finding the m.e.p. etc. of the p.c.n., the work should be repeated upto getting the nāḍis in
(iii). These nāḍis are to be subtracted or added to the original new noon. A better p.c.n. is got. Using
this time the work may be further repeated for a still better approximation.
Example 15. To continue Ex. 10
In the last example the śaṅku got is 103′ 10″. In Ex. 10, the motion per day of the Sun and the
Moon found are 57′ and 810″, the Sun's kakṣā found is 16,641 and the Moon's 1171.2. The square
of the dṛk-kṣepa kept apart, is 792.25. From these:
(i) Dṛggati = √(14,400 − (103 1/6)² − 792.25) = 54′ 27″.
(ii) (a) Sine Sun's par. in long. = 18 × 54′ 27″ ÷ 16,641 = 3″.6.
(b) Sine Moon's par. in long. = 18 × 54′ 27″ ÷ 1171.2 = 50″.2.
The Sun's parallax is arc of 3″.6 = 1′.7.
The Moon's parallax is arc of 50″.2 = 24′.0.
The parallax in longitude = 24′.0 − 1′.7 = 22′.3.
(iii) Nāḍis of par. = 22′.3 × 60 ÷ (810′ − 57′) = 1-47.
Since new moon is afternoon, adding to the time of new moon, the p.c.n. = nā. 20-40 + nā. 1-47
= nā. 22-27.
We shall repeat the operation for a better approximation. (The motions of the Sun and Moon
per day, and their kakṣās need not be done again.)
The nāḍis of the p.c.n. after midday = 22-27 − 15-40 = 6-47 = 407 vināḍis. The right ascension,
corresponding to the interval of 407 vināḍis, using the ascensional differences of zero latitude = 10°

• 10° + 10° + 10° × 85.2 ÷ 108.8 (for vināḍis 105.4 + 107.6 + 108.8 + 85.2) = 37° 50′, after
  the Sun ( = rā. 2-0-2, 2′ more for the 2 nāḍis later). ∴ The m.e.p. = rā. 2-0-2 + rā. 1-7-50 = rā. 3-7-52.
  The declination of m.e.p. = 23° 46′ north. The zenith distance of the point = 23° 46′ − 10° 24′ =
  13° 22′, north. Sine z.d. of m.e.p. = 27′ 44″. The o.e.p. at p.c.n. = rā. 6-8-48 (using as before the
  ascensional differences of 10° 24′). Since amplitude of the point = 7′ 34″. Sine (m.e.p. ~ n) = 27′ 44″
  × 7′ 34″ ÷ 120 = 1′ 45″. Sine ² (z.d. of n) = (27′ 44″)² − (1′ 45″ )² = 765.89.
  Sine (z.d. of n) = 27′ 40″.
  Śaṅku = √(14,400 − 765.89) × sine (rā. 4-8-46) ÷ 120′ = 91′ 1″.
  Dṛggati = √(14,400 − (91′ 1″)² − 765.89) = 73′ 9″.
  Sine Sun's par. in long. = 18 × 73′ 9″ ÷ 16,641 = 4″.8. Parallax = 2′.2.
  Sine Moon's par. in long = 18 × 73′ 9″ ÷ 1171.2 = 1′ 7″.5. Parallax = 32′.2.
  Relative parallax = 30′.0.
  Nāḍis of parallax = 30′ × 60 ÷ 753′ = nā. 2-23.
  Adding to time of new moon, the closer p.c.n. = nā. 20-40 + nā. 2-23 = nā. 23-3.

IX.24 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 223
Repeating the work, the p.c.n. got will be about nā.23-20.
The rule is thus explained: It has been shown in the context of the Paulīśa solar eclipse that the
relative total parallax is obtained by multiplying the relative horizontal parallax (π) by sine zenith
distance of the Sun (dṛgjyā) and dividing by the radius. In this Siddhānta, the horizontal parallaxes
of the Sun and the Moon are got separately by dividing by their distances for the sake of exactness.
But the Sun's dṛgjyā is used for the Moon too, since the difference is very small in the neighbour-
hood of new moon, with the solar eclipse occurring. In fig. 2, dṛgjyā = sin Zs, and the relative
parallax = ss'. Its projection on the ecliptic, sl, is the relative parallax in longitude, by which (Moon
— Sun) has got to be increased or decreased to get their apparent difference in longitude. In the
figure, since s is west of n, it is subtractive, and p.c.n. is later, and therefore the nāḍīs of parallax are
additive. (When the Sun is east of n and parallax is additive, clearly the nāḍīs are subtractive.) Since
(Moon — Sun) is tithi element, the relative parallax is treated like tithi, and multiplied by 60 and
divided by the daily motion to get the nāḍīs of parallax.
Now, sl is found thus in this Siddhānta: sl² = ss'² - s'l². (∵ the triangle ss'l is right-angled at l, and
so small that it may be considered plane.)
= ss'² - ss'².sin² l ss'
= ss'² - ss'².sin²Zn ÷ sin² Zs (∵ triangle Zns is right angled at n)
= π² dṛgjyā² - π² sin² (z.d. of n) (∵ ss' = π × dṛgjyā = sin zs, and Zn is the z.d. of n)
= π² (radius² - Śaṅku² - sin² z.d. of n). (dṛgjyā² = radius² - Śaṅku²)
∴ sl² = π √(120² - śaṅku² - sin² z.d. of n)
= π × dṛggati, as given
But, π = the Moon's horizontal parallax - the Sun's horizontal parallax. ∴ the dṛggati is
multiplied by each and then subtracted.
It has been said already, in previous two solar eclipse contexts, that the sine of the horizontal
parallax is obtained by dividing the earth's radius by the respective distance. Since the author uses
as the divisor not the actual distance but the respective distance divided by 43, the earth's radius also
has to be taken divided by 43. The author takes 788 yojanas as the earth's radius, adopting the value
of the Āryabhaṭīya and multiplying it by 3/2 to express it in the yojana measure of the Ārdharātrika etc.
systems. (These systems give the earth's radius as 800 yojanas.) Dividing it by 43 we get 18.3, and
the author gives it as 18, corrected to the nearest unit's place.
Further, since Zn is perpendicular to the ecliptic, Zs, the zenith distance of the Sun at any position
on the ecliptic, is always greater than Zn, and, accordingly, their sines also, since the arcs are all less
than 90°, i.e. (radius² - śaṅku²) is always greater than sin² z.d. of n. Therefore, the interpretation
of TS that the former is to be subtracted from the latter is wrong, as mentioned already. The need
for successive approximation by repetition of work is plain.
[नति:]
अविशेषाद् (दृक्क्षे)पं 'वस्वेक'घ्नं विभज्य कक्षाभ्याम् ।
लब्धान्तरचापांशा मध्यज्यादिग्वशेन नतिः ॥ २४ ॥

224 PAÑCASIDDHĀNTIKĀ IX.25
Parallax in latitude
24. Take the sine z.d. of n last got in the successive approximation, multiply
by 18, and divide by the respective kakṣās. The respective sine parallax in
latitude is got. The arc of their difference is the relative parallax in latitude
and its direction is that of sine z.d. of m.e.p. (i.e. of M from Z.)
Since the sines are very small, it is immaterial whether the arcs are found first and their difference
is taken, or whether the arc of the difference of the sines is taken, both being the same practically.
But the latter will entail less work.
a) Sine parallax lat. of the Sun = 18 × sin z.d. of n ÷ Sun's kakṣā.
b) Sin parallax in lat. of the Moon = 18 × sin z.d. of n ÷ Moon's kakṣā.
(b) − (a) is the sine of the relative parallax in lat. whose arc is to be found, and its direction is that
of M from Z.
Example 16. To continue example 10.
The sine z.d. of n, last got in the successive approximation in the last example is 27′ 26″ say.
(a) Sun's sine par. in lat. = 18 × 27′ 26″ ÷ 16,641 = 1″.8.
(b) Moon's par. in lat. = 18 × 27′ 26″ ÷ 1171.2 = 25″.3.
Sin relative par. in lat. = 25″.3. − 1″.8 = 23″.5.
Rel. par. in lat. = arc of 23″.5 = 11′.2, north, since M is north.
The rule is thus derived: The zenith distance of the nonagesimal, Zn, is the Sun's dṛkkṣepa. In
the context of the solar eclipse, according to the Paulīśa, it was shown how the parallax in latitude
(p.c.lat) is to be got by multiplying this dṛk-kṣepa by the relative horizontal parallax and using it as
a correction to the Moon's latitude to obtain the corrected latitude. Here, the parallax is derived
separately for each of the Sun and the Moon, for the sake of greater accuracy. Now, the direction
of the nonagesimal and the sine of its zenith distance is the same as that of sine z.d. of M, and the
direction of the apparent shifting of the Sun and the Moon by parallax, as resolved on the line of
latitude, is the same as that of sine z.d. of n, (as ls' in fig. 2). Therefore, the direction of the parallax
correction is the direction of sine z.d. of M, as mentioned by the author. (In the fig. it is north.) Thus
everything is explained.
ज्याविधिना विक्षेपं तत्कालं प्राप्य तेन सहितोना ।
स्पष्ट[T] नतिः प्रमाणैः स्वैस्स्वैर्ग्रासं स्थितं च वदेत् ॥ २५ ॥
25. The Moon's latitude at the time taken is to be got by using the sine (of
Moon ~ Rāhu), and this is to be added to or subtracted from the parallax
correction in latitude (according to their direction). This is the parallax-
corrected latitude (p.c. lat.). This is to be got separately for each of the times
separately and from them the times of total obscuration and total duration are
to be got.
24a. A. दृक्षेपं; B. दृक्षेप 25a. B. विक्षेप d. A. स्वैस्वैग्रसं; B. स्वैस्वैग्रांमं
b. B. चस्वेकग्रं (B2.3.°ग्रं) d. B. ज्याद्विख b. A. प्राथ c. A.B. स्पष्टनति D. स्थिति

IX.26 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 225
The maximum latitude is given in verse 6 to be 270'. The use of sine (Moon − Rāhu) has been
already indicated in the context of the Romaka, and therefore only indicated here. In correcting,
like directions are additive, and unlike directions subtractive, the resulting direction being that of
the greater. The parallax-corrected latitude is to be got for each of the time of first contact, last con-
tact, and middle, the last serving for immersion and emergence too, as these times are near enough
to the middle. The separate computation of the corrected latitude suggests that the nāḍīs of parallax
also are to be computed separately for the different times. Thus, the following is to be done:
(i) The uncorrected latitude = 270' × sine (Moon − Rāhu) ÷ 120 = sine (Moon − Rāhu) × 9 ÷
4. (If Moon − Rāhu) is less than 6 rāśis, the latitude is north, otherwise south. Rāhu here means the
Head of Rāhu).
(ii) Parallax-corrected latitude = latitude ± relative parallax in latitude (+ if of the same direc-
tion, and ~ if of different directions, the resulting direction being that of the greater).
Example 17. (To continue Ex.10,) find the parallax-corrected latitude at the final parallax corrected new
moon, i.e. at nāḍī 23-20.
This time is nāḍīs 2-40 later than new moon. So, from the data given in Ex. 10, Rāhu-head = rā.
7-29-24, and Moon = rā. 2-0-0 + 810' × 2 2/3 ÷ 60 = rā. 2-0-36. Moon − Rāhu = rā. 6-1-12. From
this,
Moon's latitude = 9 × sine (rā. 6-1-12) ÷ 4 = 5'.7, south. (∵ Moon − Rāhu-head) > rā. 6-0-0.)
(ii) Parallax corrected lat. = 5'.7 ~ 11'.2 = 5'.5, north (∵ of different directions, north being
greater.)
These rules have been explained before.
[विमर्दकालः]
अवनतिवर्गं जह्याद् रवीन्दुपरिमाणयोगदलवर्गात् ।
तन्मूला(त्तु) द्विगुणात्(ति)थिभुक्तवदादिशेत् कालम् ॥ २६ ॥
Duration of the eclipse
26. Subtract the square of the parallax - corrected latitude from the square of
the sum of the semi-diameters of the Sun and the Moon and find the square
root. Double this, and find the time for it, treating it as the motion of tithi.
(The duration of the eclipse it got.)
This verse has already occurred as verse 16 of chap. VIII and fully explained there. The only
difference is two mis-readings here.
Example 18. To continue Ex. 10.
In Ex. 11, the angular diameter of the Sun has been found to be 31', and of the Moon, 33' and
the daily motion of the tithi 753'. The parallax-corrected lat. has been found to be 5'.5. From these,
Duration in nāḍikās = 2 × 60 × √(31 + 33)/2 − 5.5² ÷ 753 = 2 × 60 × 31.52 ÷ 753 = *nā.*5-1.
26b. B3. ॰न्दुः. B. परिपरिमाणग्दल (B3. ॰योगदल) c. A.B. मूलात् d. A.B1.2.तिथिभुक्ति. B. ॰वदादिकेत्कालं

226 | PAÑCASIDDHĀNTIKĀ | IX.27
It has already been mentioned that half the duration subtracted from the final parallax-corrected
new moon is the beginning and added to it is the end of the eclipse. Further, if the difference of the
semi-diameters is used in the work, instead of the sum, the duration of total eclipse is got. Here, if
the Moon's is greater, there is actual total obscuration. If the Sun's is greater, there is annular
eclipse. The author expects us to be conversant with these things.
तिथ्यवना(मो) ग्रहणादिना(म)विश्लेषि[तो यु]तः स्थित्याम् ।
गोलाऽन्यत्वे देयस्त्ववनामो[मौ]क्षि(क)स्यैवम् ॥ २७ ॥
27. Find the nāḍis of parallax for the time of the beginning. If the time of
beginning and the new moon are both in the forenoon or both in the after-
noon, find the difference of the nāḍis of parallax and add it to the half duration
to get the correct half duration to be subtracted from the time of the corrected
new moon. If one is before noon and the other afternoon, add the nāḍis of
parallax, and add it to the half-duration to get the correct half-duration (to be
subtracted from the time of parallax-corrected new moon). Do the same for
the time of the end of the eclipse, (to find the correct half duration to be added
to the parallax-corrected new moon, to get the correct last contact).
The following example will make the meaning clear.
Example 19. To continue Ex. 10.
We have already obtained, par.c. new moon = nā. 23-20, parallax-correction for new moon =
nā. 2-40, and the total duration nā. 5-1. Applying the half duration on both sides of p.c.n, the
approx. time of first contact = nā. 20-50, last contact = nā. 25-51. The parallax correction in time
for first contact using verses (22-23) is nā. 1-50. As both the new moon and time of first contact are
in the same part of the day, i.e. afternoon, the difference between their parallax correction = nā. 2-40
− nā. 1-50 = nā. 0-50. This is to be added to the half duration to get the first half duration. Adding,
nā. 2-30 + nā. 0-50 = nā. 3-20. Subtracting this from the parallax-corrected new moon, the correct
time of first contact = nā. 23-20 − nā. 3-20 = nā. 20-0, after sunrise.
Next, the parallax-correction for time of approx. last contact is nā. 3-30. As both new moon and
last contact are in the afternoon, subtracting the corrections, for both from each other, we have
nā. 3-30 − nā. 2-40 = nā. 0-50. Adding this to the half-duration we have, nā. 2-31 + nā. 0-50 = nā. 3-21,
for the correct second half duration. Adding this to the p.c.n, the correct time of last contact =
nā. 23-20 + nā. 3-21 = nā. 26-41 after sunrise.
The following is the explanation of the rules for the correction given here and the justification for
our interpretation. At first the duration is given neglecting the effect of parallax on the time. If the
parallax is taken into account, the duration will always be longer than otherwise, as we have said.
This can be seen from the following consideration. Let us take the case when the end of new moon
27a. A.B. नाम
b. A.B.C.D. ॰दिना च वि॰. A.B. विश्लेषित; c. B. ॰न्य चेदेय
C.D. विश्लेषितः d. A. स्त्वनामो A.B. Hapl. om of मौ

IX.27 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 227
is before noon. The first contact being earlier still, its interval from noon is greater, and therefore
its nāḍīs of parallax too is greater than those of the new moon. Since both are subtractive, the first
half duration is lengthened, the first contact happening earlier. Therefore the difference is to be
added to the duration. In the case taken, the last contact may happen before noon or after noon.
If before noon, the interval from noon upto the last contact is less than that upto new moon. There-
fore the nāḍīs of parallax of the last contact is less than those of the new moon, and both are sub-
tractive. Therefore the second half duration also is lengthened, the last contact happening later.
So the difference is, here too, additive to the duration.
If the last contact is afternoon, the naḍīs of parallax are clearly additive to the time of last
contact, and the last contact happens later. But the parallax-corrected new moon occurs earlier,
and so the second half-duration is lengthened both ways, and so the sum of the parallaxes is added
to the duration. Thus in all three possibilities of the first case, there is only additive correction.
Let us now take the second case, viz. that the new moon occurs after noon. Clearly what is said
for the first contact in the first case applies to the last contact in the second case, and vice versa, but
the additiveness and subtractiveness alone have to be interchanged. Therefore, here too, in all
three possibilities the differences or sums, have to be added to the duration, as we have said in
giving the meaning of the verse. Not understanding the above, TS have interpreted the verse in
such a way that the instruction will result in lessening the duration, which is contrary to facts.
Now, for the readings. In the second foot of the verse three mātrās are missing, and to restore
them we have read viśleṣita as viśleṣito yutaḥ in accordance with the meaning. The emendation, by TS
and NP, of the manuscript reading viśleṣitasthityām into viśleṣitaḥsthityā does not express the
intended idea fully. In the fourth foot two mātrās are missing, and to restore them we have read the
meaningless nāmokṣi as nāmo maukṣī.
To conclude: In the introduction to this chapter we said that the Sun, Moon, and Rāhu, together
with the methods of computing them are better in the Saura than in the Romaka. Now, we have seen
that in the computation of the solar eclipse also, the Saura excels. For instance, the mean angular
diameters of the Sun and the Moon are 30' and 34' according to the Romakas, while they are 32'
and 32' according to the Saura, very near the correct 32' and 31', respectively. Computing true
diameters and the parallax using the distance of the instant of eclipse, and using the true motion
of the time of eclipse for getting the duration etc. are commendable in the Saura. Getting the sine
z.d.n. by using the sine of the zenith distance of m.e.p. is a better method than that used by the
Romaka, as also the method of successive approximation for various things like parallax in time of
new moon etc. The abandoning of the Romaka's faulty correction of the Moon's position in its own
orbit, is itself praiseworthy. With such good features, the Saura is easily the best of the five Siddhāntas.
[इति पञ्चसिद्धान्तिकायां वराहमिहिरविरचितायां
सूर्यसिद्धान्तेऽर्कग्रहणं नाम नवमोऽध्यायः ]¹

1. Col. A.B.D. इति (B. om इति) सूर्यसिद्धान्तेऽर्कग्रहणं
   (B. णनाम) नवमोध्यायः |
   C. इति सूर्यसिद्धान्ते सूर्यग्रहणं नाम नवमोऽध्यायः |
   Thus ends Chapter Nine entitled ‘Saura-Siddhānta: Solar Eclipse’
   in the Pañcasiddhāntikā composed by Varāhamihira

Chapter Ten
SAURA-SIDDHĀNTA — LUNAR ECLIPSE
१०. दशमोऽध्यायः
सौरसिद्धान्तः — चन्द्रग्रहणम्
Introduction
In this chapter the method of computing the lunar eclipse according to the Saura Siddhānta is
given. Since the true Sun and the Moon and Rāhu, the true distances of the Sun and the Moon, and
the Moon’s angular diameter and latitude, have already been given in chap. IX, the angular diameter
of the Shadow alone is given here, as also the computation of the times of contacts etc. The last
three stanzas give the amount of eclipse at a desired time, as also the beginning and end of total
phase of the eclipses, both of the Sun and the Moon.
[तमोबिम्बमानम्]
रविकक्षा नवतिगुणा ‘षडष्टदस्रो’द्धृतेन्दुकक्षायाः ।
छेदः ‘षट्त्रि’घ्नाया ल(ब्धे) नोनश्च षड्वर्गः ॥ १ ॥
‘वियदर्क’गुणे शशिक(क्ष्य)या हृते कार्मुकं त(मो)व्यासः ॥ २ a ॥
Diameter of the Shadow
1-2a. Multiply the Moon’s true distances in its orbit by 36, and divide by the
Sun’s true distance multiplied by 90 and divided by 286. Subtract this result
from 36, multiply by 120, divide by the Moon’s true distance and get the arc
of the resulting sine. This is the angular diameter of the Shadow.
The following is asked to be done:
(i) ‘Result’ = 36 × Moon’s true dist. ÷ (90 × Sun’s true dist. ÷ 286)
= 36 × Moon’s true dist. × 286 ÷ (90 × Sun’s true distance).
(ii) Sine angular diameter of Shadow = (36 − ‘result’) 120 ÷ Moon’s true distance.
Or, simplifying, this is equal to:
{36/Moon’s true distance − (36/Sun’s true distance) × 286/90} × 120
= 4320/Moon’s true distance − 13,728/Sun’s true distance.
1a. A. कक्ष्या. B. नवतीगुणा
b. A. द्रुतेन्दु. A. B 1. 3. कक्ष्यायाः C. षडश्च
c. B 1. 3. षद्रिघ्नाया
d. A. लघोनो; B 1. 3. लधोनातश्च
2a. B 1. 3. वियदर्वगुणे
b. A. B. कक्ष्याया; C. D. कक्ष्या. A. तमोर्व्यांसः;
B 1. 3. तयोव्याघ्रः

X.2 X. SAURA-SIDDHĀNTA — LUNAR ECLIPSE 229
Or, (since the arc is small, multiplying this by 3438 and dividing by 120), the angular diameter
of the Shadow in minutes = 1,23,768 ÷ Moon's true distance − 3,93,307 ÷ Sun's true distance.
Example 1. On a certain day at the time of full moon (T) the true Sun is rā. 10-0-0, the true Moon is rā. 4-0-0,
Rāhu Head is rā. 3-25-0, the Sun's motion per day for the time is 60′, and the Moon's 780′. Compute the
lunar eclipse.
The Sun's true dist. = 9,48,558 ÷ 60 = 15,809 (by IX.15).
The Moon's true dist. = 9,48,680 ÷ 780 = 1216.3 (by IX.15).
The Moon's angular dia. = 38,640 ÷ 1216.3 = 31′.77 (by IX.16).
The Moon's lat. at T = 9 × sine (rā. 4-0-0 − rā. 3-25-0) ÷ 4 = 23′.5, north.
From the true distances got above, the angular diameter of the Shadow = 1,23,768 ÷ 1216.3 −
3,93,307 ÷ 15,809
= 101′.76 − 24′.88 = 76′.9.
The following is the explanation of the method for finding the Moon's angular diameter: The
actual diameter of the Shadow is the diameter of the circular section of the Shadow-cone (formed
by the earth intercepting the Sun's light,) at the Moon's orbit, at the time of full moon. This is
represented in fig. 1, below by U′U″.
Fig. X. 1
The angle subtended by this at the centre of the earth, E, is the angular diameter desired to be
computed here. In the figure, S is the centre of the Sun, E, that of the Earth, and U that of the
Shadow section. SS′ is the Sun's radius, EE′ is the Earth's, and UU′ is that of the Shadow. SE is the
true distance of the earth from the Sun, and EU, that of the Moon from the earth. S′E′U′ is the
direct common tangent to the orbs of the Sun and the Moon. As the distance are very great when
compared with the radii, EE′, SS′, and UU′ are practically parallel. Draw ES″ parallel to E′S′, and
U′E″ parallel to U′E′. The triangles, ES″S and UE″E, are similar. Therefore,
E″E/EU = S″S/SE.

230 PAÑCASIDDHĀNTIKĀ X.4
∴ E"E = S"S × EU/SE = EU × (S'S - S' S")/SE
= EU × (S'S - E'E)/SE.
In order to get the angular diameter of the Shadow, we require the radius of the Shadow, UU',
= EE' - EE" = EE' - {EU (SS' - EE')/SE}
= 18 - Moon's true dist. × (Sun's radius - 18) ÷ Sun's true dist.
= 18 - Moon's true dist. × {18 × 5,14,787 ÷ (2 × 18 × 3438) - 18} ÷ Sun's true dist. (Here,
18 is the number obtained by reducing the earth's radius by 43, and given by the author in giving
the parallax, see IX.23.)
18 × 5,14,787 ÷ (2 × 18 × 3438) is the Sun's radius, since, of two orbs, the parallax as viewed
from one is the angular semi-diameter as viewed from the other. Therefore:
Sun's minutes of parallax: Sun's angular semi-diameter in minutes :: 18: Sun's reduced radius.
But the Sun's minutes of parallax = 18 × 3438 ÷ the Sun's true distance, and the Sun's semi-
diameter in minutes = 5,14,787 ÷ (2 ×the Sun's true distance) = 18 - Moon's true distance × 18
× 284.4 ÷ (90 × Sun's true dist.)
Since, sine angular diameter of the Shadow is got by multiplying the radius by 2, and the max.
tabular sine and dividing by the Moon's true distance, sine angular diameter of Shadow = {36 -
Moon's true dist. × 36 ÷ (90 × Sun's true dist. ÷ 284.4)} × 120 ÷ Moon's true distance, almost the
same as the author has given, but with 284.4 instead of 286.
If the number had been 17.9 instead of 18 taken as a whole number for convenience, then we
shall get 286 itself, as given by the author. We have already shown that the formula can be
simplified. The author must have given it in the involved form for indicating the geometrical
construction by way of proof.
When the numbers occurring are seen to be correct in the way shown by us, it is quite improper
for TS to read ṣaḍaṣṭadasra (278) as ṣaḍaśvadasra (276) and to agree with this, making the Sun's
reduced diameter as 146, in their proof, instead of the correct 149.73, got by dividing 5,14,787 by
3438.
[विमर्दकालः]
चन्द्रतमोव्यासयु(तिं) द्वाभ्यां हृत्वा ततो वर्गात् ॥ २ b ॥
विक्षेपवर्गहीनादासन्नपदे 'वियद्विद्विचन्द्र'घ्ने |
सूर्येन्दुभुक्तिविवरो(द्धृ)ते स्थिते(र्ना)डिका लब्धाः || ३ ||
प्रग्रहणे(न्दोः) कृत्वा विक्षेप [म] तोजनया स्थि(ति) र्भवति ।
एवं भूयो भूयः स्थित्य(वि) शेषः कृतो यावत् ॥ ४ ॥
Duration of the Eclipse
2b-3. Add the angular diameters of the Moon and the Shadow, divide by two,
and square it. Subtract the square of the Moon's latitude from this, and find

X.4 X. SAURA-SIDDHĀNTA — LUNAR ECLIPSE 231
the square root. Multiply this by 120 and divide by the difference of the
motions per day of the Sun and the Moon pertaining to the time of eclipse.
The duration of the eclipse is got in nāḍikās.
4. Find the Moon’s latitude at first contact and using this find a more correct
duration. Repeat this till there is no difference between the previous and the
next durations.
Note: Though the total duration alone is given here, we are expected to know how to find the
first and last contacts from this, from previous contexts. In the successive approximation, what is
said for the first contact must be taken for the last contact also. Therefore the following is asked to
be done:
(i) Rough duration = 120 √(half-sum of angular diameters)² – lat.² ÷ difference of instantane-
ous daily motions.
(ii) T ± half (i), are the rough first and last contacts.
(iii) Using the latitude of the rough first contact and repeating (i) gives successively better first
contacts.
(iv) Using the latitude of the rough last contact, and repeating (i), gives successively better last
contacts. (It should be noted that the shorter the duration the greater are the number of repetitions
required.)
Example 2. Continue Ex.1.
(i) Rough duration in nāḍis 120 √{ (76.9 + 31.77)/2 }² – 23.5² ÷ (780 – 60)
= 120 √ 54.34² – 23.5² ÷ 720 = 120 × 49 ÷ 720 = nāḍis 8-10.
(ii) Rough times first and last contacts = T – nā. 4-5: T + nā. 4-5.
(iii) The Moon at rough first contact = rā. 3-29-6.9,
Rāhu then = rā. 3-25-0.2.
Moon – Rāhu = 4° 6′.7.
From this the Moon’s latitude is 19′.35, north.
Using this, a more correct duration for first contact = √54.34² – 19.35² × 120 ÷ 720 = nā. 8-28.
Subtracting half this from the time of full moon, the first contact is at T – nā. 4-14. There is no
need to repeat, since the duration is long.
(iv) The Moon at rough last contact is rā. 4-0-53.1, and Rāhu then, rā. 3-24-59.8. From this the
Moon’s lat. is 27′.65, north. Using this, a more correct duration for last contact = √54.34² – 27.65²
× 120 ÷ 720 = nā. 7-48. Adding half this to full moon time, the last contact is at T + nā. 3-54. There
is no need to repeat.
The method has been explained several times before, which need not be repeated here. As for
understanding that the successive approximation is for getting the last contact also, though men-
2c. A.B. युतिः. B1.3. हत्या; B3. हत्या 4a. A.B.C. प्रग्रहणेन्दुः
3b. A.B. वियद्वि b. A.B. om. म. A.B. स्थितेः
d. A.B. धृते. A.B.C.D. स्थिते for स्थिते; A. लब्धा d. A.B. स्थित्यवशेषः