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

IX.9 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 205 Example 6. Find the true Moon at true noon at Ujjain, the Days being 5,28,931. From example 2 (a), the mean Moon for 5,28,931 days gone is rā. 4-11-25.7. From ex. 3 (a) the Moon’s apogee for the given days gone is, rā. 5-5-33.2. From these, (b) (i) The Moon’s anomaly = rā. 4-11-25.7 – rā. 5-5-33.2 = rā. 11-5-52.5. (ii) Sine anomaly = sine rā. 11-5-52.5 = sine rā. 0-24-7.5 = 49' 2". Sine equation of the centre = 49' 2" × 31 ÷ 360 = 4' 13".3. Equation of the centre = arc 49' 2" = 2° 1'. (iii) True Moon at mean noon = rā. 4-11-25.7 + 2° 1' = rā. 4-13-26.7. (iv) True Moon at true noon = rā. 4-13-26.7 + 1° 34'.1 × 729.1 ÷ 21,600 = rā. 4-13-26.7 + 3'.2 = rā. 4-13-29.9 (That the Moon’s daily motion is 729'.1 will be seen from example under verse 13 below. The addition is as the Sun’s Eq.C.) It should be noted here that the apogee of the Sun, given as 80° is too far from the correct apogee for the time of the work viz. 77° 19'. There is no doubt about the reading here, since the Ārdharātrika and the Khaṇḍakhādyaka too give 80°. So much error is unbelievable in the Saura, and must be explained thus: At first the practice might have been to get the mean longitude of the Sun for the days from the commencement of the true solar year and 80° deducted to get the anomaly, for this would be equivalent to deducting about 77° 50', (since the Eq.C at this time is about 2° 10'), from the correct mean Sun, not much different from the correct 77° 19' to be deducted. Later, by some mistake, the deduction of 80° was instructed to be done from the correct mean Sun itself. The apogee for the time computed by the Modern Sūrya Siddhānta is 77° 15'. From the instruction to multiply the sine of the Sun and Moon’s anomalies by 14 and 31, respec- tively, and divide by 360, to get the sine of the respective equation of the centre, we see that this Siddhānta actually uses epicycles like the Āryabhaṭīya etc, though not mentioning the word, and we can say that epicycles appear in the Hindu Siddhāntas for the first time in the Saura, and the others following using epicycles and excentries. The Modern Sūrya Siddhānta gives the same degrees of epicycle for the Sun, but 32° for the Moon instead of 31°. Further, in the Saura, the epicycle is uniform, while in many Siddhāntas like the Āryabhaṭīya there is difference between the degrees at the ends of odd and even quadrants. For instance, the degree of epicycle mentioned above for the Sun and the Moon in the Sūrya Siddhānta is for odd quadrants, being less by 20 minutes at even quadrants. The computations mentioned above can be simplified, since the multiplier and the divisor are constants and small arcs are proportionate to the sines. Thus, we can get the Sun’s Eq.C. in minutes by multiplying its sine anomaly by 1.114. In the example, multiplying 84' 27" by 1.114 we get 94' 6'', the equation of the centre. We can get the Moon’s Eq.C. by multiplying its sine anomaly by 2.467, and if the result is in excess of 225 minutes, adding 1/235 of the excess to the result. In the example, multiplying 49' 2" by 2.467, we get the equation of the centre, 121' 3". In the same way, we find the Sun’s maximum equation of the centre to be , 120' × 1.114 = 133'.7. The correct maximum for the period of our author is 119'.5. The large difference is due to the Moon’s Annual Equation being wrongly applied to the Sun with its sign changed, in Hindu astronomy, as already alluded to, since by doing so the tithi is not affected, the constants having been derived by the analysis of the syzygies, which are, in essence, ends of particular tithis. Adding the maximum Annual equation to the correct equation of the centre of the period, we have 131'.5. See how close this is to the value, 133.7 of the Saura, and how far from the 140' of the Pauliśa, and the 143' of the Romaka and of Ptolemy.

206 PAÑCASIDDHĀNTIKĀ IX.9 In the same way, the maximum of the Moon’s equation of the centre is 120' × 2.467 + (120' × 2.467 − 250) ÷ 235 = 296' + .3' = 296' .3. This too was determined by analysis of syzygies at the occurrence of eclipses. According to modern astronomy, the mean of the maximum equation of the centre of the Moon at true syzygies is 297'.3, a difference of only one minute! (At mean syzygies it is 303'.5). Epicyclic theory We shall now proceed to explain the epicyclic theory of planetary motion, used by this Siddhānta, and show how it works, by relating it to the modern theory, which latter is as follows: The earth and the other planets like Mercury etc. move round the Sun in eclipses, with the Sun at one of the two foci. The point nearest to the Sun on the ellipse is the perihelion, and the most distant, aphelion, which, from the point of view of the earth, are called the perigee and apogee, respectively. In the same manner, the Moon moves in an eclipse round the earth at one focus. This fact relating to the planets was first discovered by the European astronomer Kepler and is called Kepler’s First Law of planetary motion. The line joining the Sun and the planet (or the earth and the Moon), called the radius vector, sweeps equal areas in equal time. This is Kepler’s Second Law of planetary motion. From this it can be readily inferred that the motion of the planet is swiftest at perihelion (or perigee for the Moon) and lowest at aphelion. Kepler’s Third Law, that the square of the periodic time of the planets round the Sun is proportionate to the cube of the distance, is not wanted for our purpose here. The celebrated astronomer, Newton, showed that all the three laws follow from his Theory of Universal Gravitation, that all bodies attract one another with a force proportionate to their masses, and inversely proportionate to the square of the distance between them. But ancient Indian astronomers held the view that the earth is the centre round which the Moon, Sun and planets move. All the motion is in circles, and uniform. To explain the non-uniformity of the apparent motion caused by the equation of the centre, it was assumed that these bodies moved in circles called epicycles, (manda-vṛttas), the centres of which moved in circles round the earth as centre. In the case of the star-planets, another set of epicycles called epicycles of conjunction or śīghra-vṛttas were assumed, the effect of which is to convert helio-centric positions into geocentric. As the observers are on the earth, it is geocentric positions that are wanted, and therefore given, whether by Hindu astronomy or by modern western astronomy. The inaccuracy in the positions given by the former is due to unawareness of the elliptic motion and the inability to observe accu- rately for want of adequate instruments, with the result that small errors in the constants accumu- lated in course or time, to give large errors. It has been said that whether the heliocentric theory is adopted or the geocentric theory, the result in so far as this goes, is the same. Then, it may be asked, is it not better to adopt the geocentric theory which agrees with our perception? No. There is a clinching proof for the motion of the earth round the Sun, in the phenomenon of aberration which makes the Sun and star planets appear to be a little in advance of their real positions, the quantity being so small that very accurate measure- ment is required to find it, capable only by modern instruments. The heliocentric theory is also simpler, and satisfies the requirement of least assumption. Adhering to only circular motion, so satisfying to their minds, the ancients had to get the equa- tion of the centre, caused by the motion on the ellipse. They sought to achieve it in two ways, by using epicycles, as indicated already, or excentric circles, or both. How the ex-centric is used for the purpose is explained as follows:

IX.9 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 207

Fig. IX. 1-a.

The earth E, is the centre of the Orbit-circle, of 'radius' equal to the sine of Three rāśis (120' in this work). O indicates the first point of Meṣa, in the direction EO. EA is the direction of apogee, A, O EA being the longitude of apogee (P is the perigee). X is the centre of the ex-centric circle on which the Sun, Moon or star-planets move uniformly according to the mean motion. X is on EA, at a distance, towards A, equal to the sine of the maxi-minimum Eq. C. S is the position of the body, angle SXO' being the mean longitude of the body. XO' also is directed to the first point of Mesha. EO and XO being practically parallel. ASX = mean longitude – longitude of apogee = mean ano- maly.

SEO is the true longitude, which has got to be found. Since XO' and EO are parallel, SEO = SXO' – XSE, where XSE is the Eq. C. Since XSE changes sign on the right hand side of PA, SEO = SXO' + XSE on that side. Thus we have that in the first case, when the body is from apogee to perigee, i.e. when the anomaly is less than six rāśis, the equation of the centre is subtractive. In the second case, where the mean anomaly is more than six rāśis, is it additive.

Next, for the equation of the centre. If the maximum Eq.C, represented by EX, is small, as in general, then taking SE and SX to be practically equal, sine Eq. C = sine XSE = XE. sine SXE ÷ SE = XE. sine SXA ÷ SX = max Eq.C × sine mean anomaly ÷ 120' (120' being the radius).

In this computation, the astronomers belonging to the school of Āryabhaṭa find the Eq.C using the actual radius vector, SE, in accordance with the geometric representation. But Bhāskaracārya in his Siddhānta Śiromaṇi does not use it, and gives reasons for not using it. Now, if degrees of epicycle are given, as in this Siddhānta, instead of sine maximum Eq. C, these degrees are multiplied by 120' and divided by 360° to get sine maximum Eq. C. (i.e. EX). Therefore, sine Eq. C = (degrees of epi- cycle × 120' ÷ 360°) × sine anomaly ÷ 120'

208 PAÑCASIDDHĀNTIKĀ IX.9 = degrees of epicycle × sine anomaly ÷ 360°, as in the text. We shall now see how the use of the epicycle gives the Eq.C. [Diagram showing epicycle model with centre E (Earth), deferent circle, epicycle centred at C, body S, apogee A, A', direction to first point of Meṣa, and point O] Fig. IX. 1-b. Here too, E, the centre of the earth is the centre of the orbit circle of radius 120'. On the orbit circle, the centre C of the epicycle on which the body S is situated, moves according to the mean motion of the body. The radius of the epicycle is the degrees of epicycle given in the text × 120' ÷ 360°, which is the maximum Eq.C, as already seen. At the two points of intersection of the epicycle with the line of apogee, EA, arc A the apogee, and P the perigee. The body moves on the circumference of the epicycle, with its mean motion in the direction opposite to the motion of C. Angle ACS is the anomaly. Now, draw EA' parallel to CS. Then angle AEA' also is the anomaly. Since CEO is the mean body, A'EO is equal to the longitude of apogee. Since SEO is the true longitude and CEO is the mean longitude, angle CES is the Eq.C. Therefore, mean longitude minus Eq.C. = true long, (the anomaly being less than 6 rāśis in the figure. If the anomaly is more than 6 rāśis, S is greater than P, and the Eq. C. becomes additive). The Eq. C. is got in the same way as in the excentric method. We have now to show that in either of the methods, the Eq. C. is the same, i.e. angle XSE in fig 1a = angle CES in fig 1b. In the two triangles XSE and CES in the respective figures, it has been

IX. 10 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 209 said that EX is equal to CS. XS is equal to CE, both being equal to 120'. Angles SXE and SCE are also equal, since they are 180° minus the equal mean anomalies, SXA and SCA. Therefore, the two triangles are congruent, and so angles XSE and CES are equal, as required to be shown. In the matter of the correctness of the degrees of epicycle we have to take the authority of the work. That it agrees with the Original Saura can be seen, the same being found also in the Ārdharātrika system and Khaṇḍakhādyaka. Bhujāntara correction We shall now show why the correction called Bhujāntara is done. The Days used in the formulae are mean solar days. (That is why the mean longitudes are taken to be proportional to them.) There- fore the true longitudes got are for mean noon. But we want the longitudes for true noon. So we have to apply a correction which is the motion during the interval between mean and true noons. If the Sun's Eq.C. is positive it is east of its mean position, and reaches the meridian later than mean noon by a certain number of prāṇas (prāṇa = 4 seconds of time, one sixth of a vināḍī) equal to the number of minutes of arc. Being later, the Sun and Moon's motion during the interval has to be added. If the Eq.C. is negative, then the true Sun is west of its mean position and true noon is earlier. Therefore the motion is to be subtracted. The motion in the interval is found by the proportion: (Since there are 21600 prāṇas in a day) 21600 : daily motion :: Minutes of Sun's equation of the centre: motion during the interval. From this it can be seen that this correction has got to be done not only to the Sun, but also to the Moon and the star-planets as well. Udayāntara correction We must add that this correction for equation of the centre is not sufficient. Another correction has got to be made for what is called Udayāntara or reduction to the equator, i.e. reducing the motion on the ecliptic to motion on the celestial equator which is the circle on which time has to be measured. Both these corrections form the equation of time, being the interval between true and mean noons. But Hindu astronomers prior Śrīpati were unaware of this correction. [देशान्तरसंस्कारः] पञ्चाशता त्रिभिस् त्र्यंशसंयुतैर्योजनैश्च नाड्येका । समपूर्वपश्चिमस्थैर्नित्यं शोध्या च देया च ॥ १० ॥ Deśāntara correction 10. One nāḍī for every 53 1/3 yojanas has to be deducted or added (to Ujjain noon) by people in places east and west, respectively, of the Ujjain meridian, (to get their own noon.) Ujjain was the Greenwich of the Hindus, and the line of longitude passing through Laṅkā, Ujjain and the North pole was taken as the prime longitude. It is well-known that noon occurs earlier and earlier as the longitude of a place is more and more east, and vice versa. The author says that for every 53 1/3 yojanas of distance east or west, there is one nāḍī earlier or later. The idea is that there- fore the daily motion of the body should be multiplied by the nāḍīs got, divided by 60, and the resulting minutes of arc should be subtracted or added to the true longitude, according as the place is east or west, to get the longitude for local noon. 10a. A. पञ्चांशताः

210 PAÑCASIDDHĀNTIKĀ IX.11 Example 7. Days, 52931. Benaras is east of Ujjain longitude by 68 yojanas. Find the true Moon at noon at that place. From example 6, the true Moon at Ujjain noon is rā.4-13-29.9. The difference in time for Benaras = nāḍikās (68 ÷ 53 1/3), earlier. From this, the correction for local noon = 729'.1 × 68 ÷ (53 1/3 × 60) = 15'.5, subtractive. (The Moon's daily motion for the day will be shown to be 729'.1 under verse 13.) ∴ The true Moon required = rā.4-13-29.9 − 15′.5 = rā.4-13-14.4. The instruction is explained thus: The author takes it that for the region of Ujjain the length of the latitude circle is 3200 yojanas. The Sun, in its apparent diurnal motion westward goes once round the circle in 60 nāḍikās, crossing all the meridians on the earth. There are thus 3200 ÷ 60 = 53 1/3 yojanas for one nāḍī, and a place east by this distance has its meridian crossed by the Sun, i.e. its noon earlier by one nāḍī, and west, later. The motion for this time is calculated by the proportion, 60 nāḍīs: daily motion :: the nāḍīs got: the motion for the same. That the motion is deductive if the noon is earlier, and vice versa is plain. (This is for direct motion, the Sun and the Moon alone being considered here. If the daily motion is retrograde, as is possible in the case of the star-planets, it is obvious that the subtraction and addition have to be reversed.) But it is to be noted that in the Ārdharātrika of Āryabhaṭa and in the Khaṇḍakhādyaka, the diameter of the earth is given as 1600 yojanas from which the equatorial circumference got is 5027 yojanas. Therefore the Original Saura must have given the same values. The Modern Sūrya Siddhānta, and the Siddhāntas that follow it also give the same. From this the latitude circle at or near Ujjain should be given according to them as 5027 cos 24° = 4600 yojanas. According to the Āryabhaṭīya, which uses a yojana measure one and a half times that of Saura etc., the equatorial circumference would be 3300 yojanas. From this, it is 14° latitude circle that would be 3200 yojanas, and not Ujjain latitude circle. Why should the author use the yojana measure of the Āryabhaṭīya instead of that of the Saura, and in that why should he use the yojanas of the 14° latitude circle instead of the Ujjain (24°) latitude circle seems inexplicable. [रविशशिनोर्मध्यभुक्तिः] नव(तिः) सप्तशतीन्दोः सचतुस्त्रिंशद्विलिप्तिका भुक्तिः । षष्ट्यैका विकलाऽष्टकं च मध्या सहस्त्रांशोः ॥ ११ ॥ Mean motion of the Sun and the Moon 11. The mean daily motion of the Moon is 790′ 34″, and that of the Sun is 59′ 8″. These can be derived from the mean motions of the Sun and the Moon given in the first and second verses. In the former it is said that in 2,92,207 days there are 800 revolutions of the Sun. ∴ 800 revolutions ÷ 2,92,207 = 59′ 8″, the motion for one day. In the latter, we have that there are 9,00,000 revolutions of the Moon in 2,45,89,506 days. ∴ the motion per day = 9,00,000 revolu- tions ÷ 2,45,89,506 = 790′ 34″. The correction in verse 4 is too small per day to consider. 11a. A.B.नव (A.नच) सप्तसर्तींदोः b. B.त्रिंशद्विलिप्तिकासु भुक्तिः d. B.मध्याग्रहस्त्रांशोः

IX.14 IX. SAURA-SIDDHĀNTA — SOLAR ECLIPSE 211 [चन्द्रकेन्द्रभुक्तिः] सप्तकला वित्र्यंशाश्चन्द्रोच्चस्येन्दुभुक्तिरनयोना | केन्द्रस्य परिज्ञेया स्फुटभुक्तिश्चा(न)या कार्या || १२ || Motion of Moon's anomaly 12. The daily motion of the Moon's apogee is 6 2/3 minutes. The Moon's mean daily motion less the motion of the apogee is the daily motion of the Moon's (mean) anomaly. The true daily motion is to be found using this motion of anomaly. We get that the daily motion of the Moon's anomaly is 790' 34" - 6' 40" = 783'54". The Sun's mean daily motion itself is the motion of its anomaly, since its apogee has no motion according to this Saura, as we have already said. Even if motion is taken into account, it is so small that it is prac- tically nothing per day. The rule to get the daily motion of Moon's anomaly is explained thus: From verse 3 above, we see that there are 900 revolutions of Moon's apogee in 29,08,789 days. ∴ in one day, the motion is 900 revolutions ÷ 29,08,789 = 6' 41". The correction per day is practically nothing and so left out. The author gives it as 6' 40", for convenience of expression, since the 1' left out will not affect the result materially. Since anomaly is mean longitude minus longitude of apogee, the motion of anomaly is mean motion minus motion of apogee, for it is the daily motions that add up to form the longitude. [रविशशिनोः स्फुटभुक्तिः] केन्द्रान्तरज्या गुणिता 'तिथिवर्गे'णोद्धृता च परिणा(म्या) | तत्कार्मुकं क्षयचयौ भुक्तौ मृगकर्कटाद्येषु || १३ || तत्कालभुक्ति(रेषा)ऽऽहोरात्रिकी शशिविशेषात् | व्यासार्धहता भुक्तिः स्फुटभुक्तिहता स्फुटः कर्णः || १४ || True motion of Sun and Moon 13. The daily motion of anomaly should be multiplied by the current sine- interval and divided by 225. This should be reduced to the epicycle, i.e. multiplied by the degrees of epicycle and divided by 360°. The change in sine Eq.C, is got. Its arc should be subtracted from the mean daily motion, if the anomaly falls within rāśis 9 to 3, and added if it falls within rāśis 3 to 9. 14. This is the true motion per day, for the moment (for which the anomaly is taken.) The true daily motion in the case of the Moon is got by subtracting 12a. B.विचित्र्यंशा 13a. C.केन्द्रज्यान्तरगुणिताः; D. ॰ज्यागुणिता b. A.रतयोना b. B. वर्गेणोधृता A. परिणाम्यः; B.C.D. परिणाम्य d. B. स्फुटभक्तिश्चात् या; A. श्रातया d. B1.3. भुक्तो मृगकर्कराद्येषु

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