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.