The following is a reproduction of a translation from an out-of-print book , originally published as "Grahanamandana of Paramesvara" by VVRI Press, Hoshiarpur, India in 1965. The original translator was K.V. Sarma, M.A., B.Sc. It deals with the calculation of eclipses. The author, Paramesvara, lived sometime between the years 1360 and 1455 CE. This text was itself written about July 15, 1410.

Invocation
1. Victory to the Lord of the Day, the Awakener of the universe, by coming into contact with whose rays the celestial bodies are seen illuminated.


Introduction

2. Just as the reflection of one's face can be seen clearly in a mirror, the eclipses of the sun and moon can be seen on the sphere.
3. Having learned the laghutantra and having observed on numerous occasions the positions of the celestial bodies in the celestial sphere, the Ornament of Eclipses is being composed by me to be a delight to astronomers.
4. Since the position of the moon obtained through the Parahita and other astronomical systems do not tally in eclipses, I set forth here the derivation of the celestial bodies enunciated by some of the earlier teachers.


Derivation of Mean Positions

5. From calculating the current Kali day (given here as July 15, 1410) by reducing 1,648,157 the sun's mean position is obtained by multiplying it by 58 and divinding the result by 21,185. The moon's mean position is obtained by multiplying by 143 and dividing by 3,907. That of the higher apsis of the moon is obtained by multiplying by 4 and dividing by 12,931.
6. The ascending node of the moon is calculated by multiplying by 3 and dividing the result by 20, 378.
7. Divide the revolutions of the sun by 283 for the sun, of the moon by 52, of the higher apsis by 11 and of the ascending node by 62. These are minutes. This is to be subtracted in the case of the sun and to be added in the case of the higher apsis and the moon.


Additive Constants

8. The additive constant for the sun is 3^r-18^°-15'-7", correct to the seconds. For the moon, it is 2^r-0^°-17'-1", correct to the seconds.
9. For the higher apsis of the moon, it is 2^r-5^°-23'-18"; and for the node it is 11^r-1^°-41'-14", correct to the seconds.
10. The additive constant to be added to the mean position in the first three cases and in the case of the node to 12^r-node). The planets are to be derived in the above-said manner by those who want conformity of observation with calculation in the computation of eclipses.
11. Some say that there is difference upon observation in the case of the node, the higher apsis and the moon, others in the the case of the moon, and still others in the case of the moon and the node. We now consider our view as correct.


Correction for Terrestrial Longitude

12. A correction for difference of place has to be applied to the mean positions. This I shall state next.
13. Multiply the daily motion in minutes by the difference in yojanas (of the place for which the eclipse is calculated from the central meridian) and divide by the circumference of the Earth at that place. The (resulting) minutes are to be added west of the central meridian, subtracted if east.
14. The central meridian is 18 yojanas east of the village of Ashvattha (modern Âlattûr in Central Kerala state). At this place, the circumference of the Earth is equal to 3,240 yojanas. The length of the equinoctal shadow at this place has been stated by ancient experts by means of the letters dus-tâ-strî.


Correction for the Equation of the Center

15. To derive the Bhujântara and Carâdha corrections find the true sun separately:
16. The mean sun reduced by its apogee is found (this is called Mandakendra). Its great sine (3438) multiplied by 3 and divided by 80 will be the Bhujâphala (equation of the center; lit. "earth result") which is additive or subtractive as the kendra is from Libra or Aries, respectively.
17. The Bhujâphala of the sun divided by 6 is for the sun, in seconds; for the moon by 27 in minutes. This correction to the respective mean positions is subtractive or additive according as the sun's bhujâphala is subtractive or additive.

Correction for Declinational Ascensional Difference
18. Take 20, 40, 57, 72, 82 and 85 as the jyâs for the half-signs (i.e., 15°, 30°, 45°, etc.) of the bhujâmsha (the angular distance covered in the first and third quadrants, and left to cover in the second and fourth quadrants is called bhujâ) of the true sun to which the precession of the equinoxes has been added. These jyâs multiplied by the length in angulas of the equinoctal shadow and then divided by 4 give the carârdha (declinational ascensional difference) in nâdikâs.
19. The mean daily motion of the planet in minutes multiplied by the sun's carârdha-vinâdis and divided by 3,600 gives the result in minutes. These are to be applied to the mean position.
20. The carârdha-vinâdis decreased by their one-sixtieth part form the correction in seconds for the sun.
21. For the moon, these vinâdis multiplied by 20 and divided by 91 give the correction in minutes.
22. At sunrise, the corrections beginning with Aries are subtractive and beginning with Libra they are additive. The reverse is the case at sunset. In the computation of the node, all are reversed.


Computation of True Positions

23. The mean sun and moon, calculated in the above manner, should be converted properly into their true positions. The great sine multiplied by 7 and divided by 80 will be the bhujâphala for the moon.
24. The bhujâphala of the moon is to be converted into arc. The number of degrees multiplied by itself and reduced by 3 are minutes. When this is added to the bhujâphala, the arc for the moon is obtained. This is not to be applied for Mars (and the other planets). Find the true positions of the sun and the moon for sunrise and sunset on the new or full moon day.


The 24 great sines from 0° to 90° at intervals of 3.75°:

3.75°: 225	33.75°: 1910	63.75°: 3084
7.5°: 449	37.5°: 2093	67.5°: 3177
11.25°: 671	41.75°: 2267	71.75°: 3256
15°: 890	45°: 2431	75°: 3321
18.75°: 1105	48.75°: 2585	78.75°: 3372
22.5°: 1315	52.5°: 2728	82.5°: 3409
26.25°: 1520	56.25°: 2859	86.25°: 3431
30°: 1719	60°: 2977	90°: 3438

Syzygy

25. Putting down the sun and the moon for sunrise on the new moon day, and the moon and the sun and six signs for sunset on the full moon day, consider their conjunction.
26. The difference between the sun and the moon, converted into minutes and again multiplied by 60 is divided by the differences of their true daily motions in minutes. The result withh be the nâdikâs to or from the conjunction.
27. The conjunction of sun and moon takes place only at the moment of the syzygies (parva).
28. The true daily motion in minutes is multiplied by the number of nâdikâs to the moment of conjunction and divided by 60 is added to the respective true positions when the conjunction has yet to take place, and subtracted when the conjunction is past.
29. This done, the sun and moon will be for the end of the syzygies and be of equal minutes.


Possibility of Eclipses

30. If the bhujâmsha of sun-node is less than 13°, an eclipse of the moon may be expected; and so for the sun if less than 11° plus the digits of the equinoctal shadow.


Measure of the Orbs

31. The true daily motion of the sun in minutes multiplied by 5 and divided by 9 is its diameter in minutes.
32. The true daily motion of the moon in minutes divided by 25 is its diameter in minutes.
33. To the diameter of the sun should be added 8 seconds, and (8 seconds) should be subtracted from the diameter of the moon.
34. The true motion of the moon in minutes divided by 10 and multiplied by the mean daily motion of the sun, the result divided by half the sum of the sun's true and mean daily motions with 50 seconds added will give the true diameter of the shadow.
35. The Samparka in the lunar eclipse is the sum of the diameters of the moon and the shadow. In the solar eclipse the sum of the diamaters of the sun and moon. Half the sum of the respective diameters is called the semi-sampraka in eclipses.
36. The sun is hidden by the moon even as a pot by another pot. The hiding of the moon by the shadow is like submergence in water.


Moon's Latitude

37. The great sine obtained from moon-node multiplied by 4 and divided by 51 is the moon's latitude. This, multiplied by its true motion and divided by its mean motion gives a more accurate latitude according to some. The latitude in minutes resulting from moon-node in Aries, etc., is north, and that resulting from the same in Libra etc. is south.


Special Work in the Solar Eclipse
Parallax in Longitude

38. I state now that which has to be done specially for the solar eclipse.
39. Using the sun at the time of conjunction and the measures of the rising of signs for the desired place, and the time of conjunction, calculate the rising point of the ecliptic in the east (lagna) at the moment of conjunction.
40. The eastern ecliptic reduced by 3^r (i.e., 90°) is called drk-ksepa-lagna (nonagesimal).
41. The degrees intervening between the drk-ksepa-lagna and the sun at that moment divided by 6 are the exact lambana-nata-nâdikâs (the nâdikâs from nonagesimal for parallax in longitude).
42. If the lambana-nata-nâdikâs are more than 15, thes these subtracted from 30 should be taken as the lambana-nata-nâdikâs.
43. 25, 50, 74, 97, 120, 140, 160, 177, 193, 207, 219, 227, 234, 238, 239. These are said to be the lambajyâs for the nâdikâs.
44. The lambajyâ of the lambana-nata-nâdikâs divided by 60 are the lambana-nâdikâs for the time taken. There is a correction for these.
45. The digits of the equinoctal shadow multiplied by 7 and divided by 9 are the nâdikâs arising from the latitude. These are south.
46. If the equinoctal shadow is more than 3 digits, it should be reduced by 3 and the square of the remainder divided by 45 should be subtracted from the latitudinal nâdikâs.
47. 20, 39, 56, 77, 80: These are the jyâs for the half-signs (15°, 30°, 45°, etc.) arising from the bhujâmsha of the drk-ksepa-lagna to which the precession of the equinoxes has been added.
48. These divided by 20 are the nâdikâs due to Apama (i.e., declinational nâdikâs). The nâdikâs arising from Aries etc. are north and those in Libra etc. are south.
49. The difference between the aksa and apama nâdikâs for opposite directions, and the sum for the same direction is the drk-ksepa-nata-nâdikâs; its direction should be taken as the resulting direction of the nâdikâs.
50. Find the lambajyâ of 15-minus-drk-ksepa and multiply by this the lambana calculated previously and divide by 239. The result will be the true lambana.
51. This lambana should be subtracted from or added to the nâdikâs elapsed to the time of conjunction. It should be added when the sun is less than the drk-ksepa-lagna, and subtracted if otherwise.
52. Calculate again as before the drk-ksepa-lagna and the sun for the time of conjunction corrected for lambana and find the lambana for that. Add this lambana to or subtract it from the nâdikâs elapsed to the time of non-corrected conjunction.
53. For this time again, find sine drk-ksepa-lagna and the lambana. Apply this to the non-corrected time of conjunction. Repeat this until the value of the lambana obtained does not differ from that of the previous. The mid-eclipse of the sun will be at the time of conjunction corrected by the lambana by successive approximation.


Moon's Parallax in Latitude

54. Now take the nâdikâs of the zenith distance of the nonagesimal obtained by successive approximation during the work, and then its corresponding sine lambajyâ. That multiplied by the difference in degrees of the true daily motions of the sun and the moon, and divided by 60, will normally give the true nati (parallax in latitude) in minutes.
55. Calculate as specified before the latitude of the moon at that moment. Sum of the parallax and the latitude in the same direction and difference in opposite directions. The result obtained will be the more accurate latitude of the moon for computing the sun's eclipse.


Half-duration of the Solar Eclipse

56. From the square of half the sum of the diameters of the sun and moon subtract the square of the corrected latitude. Find the root of the remainder. Multiply it by 60 and divide the result by the difference between the true daily motions of the sun and the moon. The result will be the nâdikâs of the half-duration of the eclipse. This is the usual method of finding the half-duration of eclipses.
57. Subtract from or add to the time of conjunction corrected for parallax in longitude the half-duration of the eclipse and find, respectively, the times of the first contact and the last contact.
58. Calculate for these particular moments the nonagesimal, sun and moon. Using the values obtained, calculate, once for all, the true parallax in longitude and the more accurate latitude, as before.
59. From the square of half the sum of the diamaters subtract the square of the latitude. Add the result to the square of the difference between the latitudes at mid-eclipse and at the chosen time, if in the same direction; and their sum if in opposite directions. Find the root. The root multiplied by 60 and divided by the difference in minutes between the daily motions will give half-duration. The half-duration should always be calculated in this manner.
60. Take the lambana for the chosen time and that for the mid-eclipse. Find their difference if both are positive or both are negative. This added to the half-duration will be the true half-duration in the case of the sun's eclipse.
61. When, however, one of the two lambanas is negative and the other positive, the half-duration added to the sum of the two lambanas will be the true half-duration.
62. To the time of conjunction corrected for parallax add or subtract, as directed above, the two half-durations. Find again the lambanas and the half-durations. Do this again until the respective half-durations become non-differing.
63. Thos non-differing half-durations are the true half-durations pertaining to the first and last contacts.


Half-duration of the Lunar Eclipse

64. Calculate for the moon the two half-durations in the same manner as above, but without the calculation for parallax. Herein, the true latitude is only that derived from moon-node.
65. The half-duration calculated using the moment of first contact and that calculated using the moment of last contact are respectively subtracted from or added to the time of conjunction. The moments will be the nâdikâs of first and last contacts.
66. The mid-eclipse of the moon is at the moment of the uncorrected time of conjunction.


Occurrence of an Eclipse

67. When the latitude is greater than half the sum of the diameters, there will be no eclipse; otherise, there will be one.
68. When the latitude is less than half the eclipsing body minus the eclipsed body, there will be a total eclipse; if it is greater, the eclipse will not be total.


Graphical Representation of Eclipses

69. I am stating the Valanas (deviations or changes in diurection) pertaining to the different moments for drawing the diagram of eclipses.


Latitudinal Deviation

70. The equinoctal shadow in terms of digits multiplied by hour angle (nata) and divided by 12 give the minutes of the deviation due to latitude.
71. In the moon's case these are northwards when the first contact takes place before noon, and southwards after noon. For the sun, the opposite is the case. For both the direction of the deviation for the last contact is the opposite.


Deviation Due to the North or South Course of the Moon

72. 1, 3 and 6 are the sines of deviation pertaining to the northward and southward courses of the moon in minutes for the koti-râsi (in odd quadrants, the degrees of the râshi required to complete the quadrant is called Koti, and in the even, those gone is Koti) of the moon corrected for precession. The direction of deviation is the same as that of the ayana for first contact in the case of the moon. It is the opposite for the sun. For both at the last contact will be opposite to that at their first contacts.


Deviation Due to Celestial Latitude

73. For the moon the deviation due to celestial latitude is given by the latitude multiplied by 2 and divided by 7. Its direction will be opposite to that of the latitude, both for first and last contacts.
74. For the sun, the deviation due to celestial latitude is given by the latitude in minutes divided by 2. Its direction will be that of the latitude, both for first and last contacts.


Total Deviation

75. The celestial-latitudinal, equinoctal and terrestrial-latitudinal deviations are to be multiplied individually by the actual diamater of the eclipsed body and divided by 32. The sum of the three results is the three are in the same direction and difference if in different directions will be the true total deviation. When this is more than half the diameter of the eclipsed body, it should be subtracted from the diameter of the eclipsed body and the remainder taken as the true deviation; in this case, however, east and west should be interchanged.


The Eclipse Diagram

76. For the graphical representation the minutes of diameters, deviations, etc. should all be taken as so many digits.
77. First, draw the orb of the eclipsed body using a string of length equal to half its diameter. Across the circle draw the east-west and north-south lines.
78. Measure off the deviation in the east-west line from the east and west sides.
79. The deviation for the first contact should be measured from the east side for the moon, and for the sun from the west side. The deviation for the last contact should be measured from the west for the moon, and for the sun from the east.
80. Southward deviations should be measured southwards and northward deviation northward, in the same manner as sines are measured off on the circumference.
81. At the intersections of the circumference and the respective deviations, mark the two points representing the first and last contacts. As these points occur the first and last contacts of the eclipsed body.
82. Take the mid-points on the circumference of the arcs formed by the points as the south and north points. On the line passing through these, mark off from the center the celestial latitude at mid-eclipse, in the direction of the latitude in the case of solar eclipse and in the opposite direction in the lunar eclipse.
83. Taking the tip of this latitude-line as the center describe the eclipsing orbit using string measuring half the diameter of the eclipsing body. The eclipsing body will hide that portion of the eclipsed body which lies within this circle and not which is outside it.


Eclipse at Any Moment

84. Again, with half the sum of the diameters describe a circle so that the eclipsed body at its middle.
85. From the center draw two lines passing through the points of the first and last contacts and extending up to the outer circle. Mark on the circumference the two points at the ends of these two lines. Call these Âdya (first) and Antya (last); Madhya (middle) will be the point at the tip of the latitude at mid-eclipse.
86. Construct the arc of the circle passing through the three points. That will represent the path of the eclipsing planet, since the planet moves along that circle.
87. From the square of the sum of the semi-diameters subtract the squares of the latitudes at the first and last contacts; the roots of the remainders will be, respectively, the bases pertaining to the first and last contacts.
88. The difference between the mid-eclipse and the chosen time, multiplied by the respective bases and divided by the respective half-durations is termed here as the ista-bâhu (base pertaining to the chosen moment).
89. The root of the sum of the squares of the ista-bâhu and of the moon's latitude at the chosen time is the ista-shalâkâ (the distance between the centers of the bodies at the chosen moment). It should be measured off from the center in the direction obtained for the time.
90. Describe the orb of the eclipsed body with its center at the point of intersection of the path of th eclipser and the ista-shalâkâ. The eclipsed portion at the desired time of the sun or the moon will be seen in that.


Eclipses Not To Be Indicated

91. An eclipse of the sun if less than the eighth part of its diameter will not be visible due to its brilliance, and is not to be indicated. Similarly, for the moon, less than a sixteenth part of its diameter will not be distinguishable on account of its great brightness and thus is not to be indicated.


Conclusion

92. Thus has been enunciated the computation of eclipses according to principles derived from the ancient texts. The times as obtained from this may, at times, differ slightly from observation.
93. "Predictions of the effects occurring earlier or later than the times due are given on the authority of ancient texts on the subject." - so says Varâhamihira in his Samhita in the section entitled "Prediction of Effects of Eclipses."
94. This being the case, it is to be postulated by learned astronomers well versed in theory that in the computation of the eclipses of the sun and the moon a correction not stated in old texts must exist.
95. Such a correction has to be postulated by astronomers after observing a large number of eclipses and with due consideration to the principles of spherics, or in light of instructions of masters.
96. It is not possible to measure off on its orb the eclipsed part of the sun, on account of its brilliance. Hence find that portion from circles of sunlight falling in residence.
97. When it is not possible to measure off on its orb even the dark portion of the cresent moon, how then will it be possible on the sun's orb, bright with countless dazzling rays?
98. May this Ornament of Eclipses, composed in a hundred verses by the twice-born named Parameshvara, endure for long in the minds of astronomers.


Appendix: Additional Corrections to the Grahanamandana, from Parameshvara's Drgganita

99. A further correction has to be applied to the mean position of the planets enunciated in the Ornament of Eclipses. That correction, too, I shall state, since that has not been specified by me there.
100. One second should be subtracted for every 200 years from the mean position of the sun derived according to the Ornament of Eclipses to get its correct mean position. In the case of the moon, however, one second should be added to its mean position for every 41 years. In the case of the node, one second should be added to 12^r for every 135 years. From the mean of the higher apsis should be subtracted one minute for every 3 years.
101. With the application of these corrections, the mean positions of the sun and the others will become accurate.