Ornament of Eclipses (Grahanamanda of Parameshvara)

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THE ORNAMENT OF ECLIPSES Grahanamandana of Parameshvara

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

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

 

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.

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.

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

>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.

The ascending node of the moon is calculated by multiplying by 3 and dividing

the result by 20, 378.

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

The additive constant for the sun is 3r-18°-15'-7", correct to the seconds. For

the moon, it is 2r-0°-17'-1", correct to the seconds.

For the higher apsis of the moon, it is 2r-5°-23'-18"; and for the node it is

11r-1°-41'-14", correct to the seconds.

The additive constant to be added to the mean position in the first three cases

and in the case of the node to 12r-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.

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

A correction for difference of place has to be applied to the mean positions.

This I shall state next.

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.

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

To derive the Bhujântara and Carâdha corrections find the true sun separately:

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.

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

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.

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.

The carârdha-vinâdis decreased by their one-sixtieth part form the correction in

seconds for the sun.

For the moon, these vinâdis multiplied by 20 and divided by 91 give the correction in minutes.

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

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.

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

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.

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.

The conjunction of sun and moon takes place only at the moment of the syzygies (parva).

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.

This done, the sun and moon will be for the end of the syzygies and be of equal

minutes. Possibility of Eclipses

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

The true daily motion of the sun in minutes multiplied by 5 and divided by 9 is

its diameter in minutes.

The true daily motion of the moon in minutes divided by 25 is its diameter in minutes.

To the diameter of the sun should be added 8 seconds, and (8 seconds) should be

subtracted from the diameter of the moon.

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.

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.

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

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

I state now that which has to be done specially for the solar eclipse.

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.

The eastern ecliptic reduced by 3r (i.e., 90°) is called drk-ksepa-lagna (nonagesimal).

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).

If the lambana-nata-nâdikâs are more than 15, thes these subtracted from 30

should be taken as the lambana-nata-nâdikâs.

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.

The lambajyâ of the lambana-nata-nâdikâs divided by 60 are the lambana-nâdikâs

for the time taken. There is a correction for these.

The digits of the equinoctal shadow multiplied by 7 and divided by 9 are the

nâdikâs arising from the latitude. These are south.

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.

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.

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.

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.

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.

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.

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.

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

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.

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

>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.

 

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.

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.

>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.

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.

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.

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.

Thos non-differing half-durations are the true half-durations pertaining to the

first and last contacts. Half-duration of the Lunar Eclipse

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.

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.

The mid-eclipse of the moon is at the moment of the uncorrected time of

conjunction. Occurrence of an Eclipse

When the latitude is greater than half the sum of the diameters, there will be

no eclipse; otherise, there will be one.

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

I am stating the Valanas (deviations or changes in diurection) pertaining to the

different moments for drawing the diagram of eclipses. Latitudinal Deviation

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.

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

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

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.

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

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

For the graphical representation the minutes of diameters, deviations, etc.

should all be taken as so many digits.

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.

Measure off the deviation in the east-west line from the east and west sides.

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.

Southward deviations should be measured southwards and northward deviation

northward, in the same manner as sines are measured off on the circumference.

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.

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.

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

Again, with half the sum of the diameters describe a circle so that the eclipsed

body at its middle.

>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.

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.

>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.

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).

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.

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

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

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.

"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."

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.

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.

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.

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?

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

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.

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 12r for every 135 years. From the mean of the higher apsis should be

subtracted one minute for every 3 years.

With the application of these corrections, the mean positions of the sun and the

others will become accurate.

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