Chapter 4: Eclipses of the Sun

The diameter of the sun’s disk is six thousand five hundred yojanas ; of the moon’s, four hundred and eighty. We shall see, in connection with the next passage, that the diameters of the sun and moon, as thus stated, are subject to a curious modification, dependent upon and representing the greater or less distance of those bodies from the earth; so that, in a certain sense, we have here only their mean diameters. These represent, however, in the Hindu theory — which affects to reject the supposition of other orbits than such as are circular, and described at equal distances about the earth — the true absolute dimensions of the sun and moon. Of the two, only that for the moon is obtained by a legitimate process, or presents any near approximation to the truth. The diameter of the earth being, as stated above (i. 59), 1G00 yojanas, that of the moon, terms of the earth’s is .2716, or only about a tenth less. An estimate so nearly corfect supposes, of course, an equally correct determination of the moon’s horizontal parallax, distance from the earth, and mean apparent diameter. The Hindu valuation of the parallax may be deduced from the value given just below (v. 8), of a minute on the moon’s orbit, as 15 yojanas. Since the moon’s horizontal parallax is equal to the angle subtended at her centre by the earth’s radius, and since, at the moon's mean distance, 1' of arc equals 15 yojanas, and the earth’s radius, 800 yojanas, would accordingly subtend an angle of 53' 20" — the latter angle, 53' 20", is, according to the system of the Surya-Siddhanta, the moon’s parallax, when in the horizon and at her mean distance. This is consi- derably less than the actual value of the quantity, as determined by modem faience, namely 57' 1"; and it is practically, in the calculation of solar

These diameters, each multiplied by the true motion, and divided by the mean motion, of its own planet, give the corrected ( sphuta ) diameters. If that of the sun be multiplied by the number of the sun’s revolutions in an Age, and divided by that of the moon’s, < 3. Or if it be multiplied by the moon’s orbit ( kakshd ), and divided by the sun’s orbit, the result will be its diameter upon the moon’s orbit : all these, divided by fifteen, give the measures of the diameters in minutes. The absolute values of the diameters of the sun and moon being stated in yojanas, it is required to find their apparent values, in minutes of arc. In order to this, they are projected upon the moon’s orbit, or upon a circle described about the earth at the moon’s mean distance, of which circle — since 324,000-^-21,600=15 — one minute is equivalent to fifteen yojanas. The method of the process will be made clear by the annexed figure (Fig. 10). Let E be the earth’s place, EM or Em the mean distance of Fig. 19.

Calculate, by the equivalents in oblique ascension ( udayA - a avas) of the observer’s place, the orient ecliptic-point {lagna) for the moment of conjunction {parvavinddyas) : multiply the sine of its longitude by the sine of greatest declination, and divide by the sine of co-latitude {lamba) : the result is the quantity known as flic orient-sine {udaya). The object of this first step in the rather tedious operation of calculating the parallax is to find for a given moment — here the moment of true conjunction — the sine of amplitude of that point of the ecliptic which is then upon the eastern horizon. In the first place the longitude of that

Multiply the earth’s diameter by the true daily motion of the moon, and divide by her mean motion : the result is the earth’s corrected diameter ( suci ). The difference between the earth’s diameter and the corrected diameter of the sun

. Is to be multiplied by the moon's mean diameter, and divided by the sun’s mean diameter : subtract the result from the earth’s corrected diameter (suci), and the remainder is the diameter of the shadow ; which is reduced to minutes as before.

The ear 111’ a shadow is distant half the signs from the sun : when the longitude of the moon’s node is the same with that of the shadow, or with that of the sun, or when it is a few degrees greater or less, there will be an eclipse. To the specifications of this verse we need to add, of course, “ at the time of conjunction or of opposition.” It will he noticed that no attempt is made here to define the lunar and solar ecliptic limits, or the distances from the moon’s node within which eclipses are possible. Those limits are, for the moon, nearly 12° : for the sun, more than 17° . The word used to designate “ eclipse,” grahana, means literally “ seizure ”: it, with other kindred terms, to be noticed later, exhibits the influence of the primitive theory of eclipses, ns seizures of the heavenly bodies by the monster Halm. In verses 17 and 19, below, instead of grahana we have graha , another derivative from the same root grah or grabh , “ grasp, seize.” Elsewhere graha never occurs except as signifying “ planet,” and it is the only word which the Surya-Siddhanta employs with that signification : as so used, it is an active instead of a passive derivative

The longitudes of the sun and moon, at the moment of the end of the day of new moon (amdvdsya ) , are equal, in signs, etc. ; at the end of the day of full moon ( paurnamdsi ) they are equal in degrees, etc., at a distance of half the signs.

When diminished or increased by the proper equation of motion for the time, past or to come, of opposition or conjunc- tion, they are made to agree, to minutes : the place of the node at the same time is treated in the contrary manner. The very general directions and explanations contained in verses 6, 7, and 9 seem out of place here in the middle of the chapter, and would have more properly constituted its introduction. The process prescribed in verse 8, also, which has for its object the determination of the longitudes of the sun, moon, and moon’s node, at the moment of opposition or conjunction, ought no less, it would appear, to precede the ascertainment of the true motions, and of the measures of the disks and shadow, already explained. Verse 8, indeed, by the lack of connection in which it stands, and by the obscurity of its language, furnishes a striking instance of the want of precision and intelligibility so often characteristic of the treatise. The subject of the verse, which requires to be supplied, is, “ the longitudes of the sun and moon at the instant of midnight next preceding or following the given opposition or conjunction that being the time for which the true longitudes and motions are first calculated, in order to test the question of the probability of an eclipse. If there appears to be such a probability, the next step is to ascertain the interval between midnight and the moment of opposition or conjunction, past or to come : this is done by the method taught in ii. 66, or by some other analogous process : the instant of the occurrence of opposition or conjunction, in local time, counted from sunrise of the place of observation, must also be determined, by ascertaining the interval between mean and apparent midnight (ii. 46), the length of the complete day (ii. 69), and of its parts (ii. 60-68), etc.; the whole process is sufficiently illustrated by the two examples of the calculation of eclipses given in the Appendix. When we have thus found the interval between midnight and the moment of opposition or conjunction, verse 8 teaches us how to ascertain the true longitudes for that moment : it is by calculating — in the manner taught in i. 67, but with the true daily motions — the amount of motion of the sun, moon, and node during the interval, and applying it as a corrective equation to the longitude of each at midnight, subtracting in the case of the sun and moon, and adding in the case of the node, if the moment was then already past; and the contrary, if it was

The moon in the eclipser of the sun, coming to stand underneath it, like a cloud : the moon, moving eastward, enters the earth’s shadow, and the latter becomes its eclipser. The names given to the eclipsed and eclipsing bodies are either chadya and, as here, chadaka, “ the body to be obscured ” and “ the obscurer,” or grdhya and grab aha, “ the body to be seized ” and “ the seizer.” The latter terms are akin with graha.ua and graha, spoken of above (note to v. 6), and represent the ancient theory of the phenomena, while the others are derived from their modern and scientific explanation, as given in this verse.

Subtract the moon’s latitude at the time of opposition or conjunction from half the sum of the measures of the eclipsed and eclipsing bodies : whatever the remainder is, that is said to be the amount obscured.

When that remainder is greater than the eclipsed body, the eclipse is total ; when the contrary, it is partial ; when the latitude is greater than the half sum, there takes place no obscura- tion ( gr&sa)~ It is sufficiently evident that when, at the moment of opposition, the moon’s latitude — which is the distance of her centre from the ecliptic, where is the centre of the shadow — is equal to the suip of the radii of her disk and of the shadow, the disk and the shadow will just touch one another; and that, on the other hand, the moon will, at the moment of opposition, be so far immersed in the shadow as her latitude is less than the sum of the radii : and so in like manner for the sun, with due allowance for parallax. The Hindu mode of reckoning the amount eclipsed is not by digits, or twelfths of the diameter of the eclipsed body, which method we h&ye inherited from the Greeks, but by minutes.

Divide by two the sum and difference respectively of the eclipsed and eclipsing bodies : from the square of each of the resulting quantities subtract the square of the latitude, and take the square roots of the two remainders.

These, multiplied by sixty and divided by the difference of the daily motions ol‘ the sun and moon, give, in nadis, etc., half the duration ( sthiti ) of the eclipse, and half the time of total obscuration. These rules for finding the intervals of time between the moment of opposition or conjunction in longitude, which is regarded as the middle of the eclipse, and the moments of first and last contact, and, in a total eclipse, of the beginning and end of total obscuration, may be illustrated by help of the annexed figure (Fig. 21). Let ECL represent the ecliptic, the point 0 being the centre of the shadow, and let CD be the moon’s latitude at the moment of opposition; Fig. 21. L K H

Multiply the daily motions by the half-duration, in nadis, and divide by sixty : the result, in minutes, subtract for the time ol contact (pray rati a), and add for that of separation (moksha), respectively ;

By the latitudes thence derived, the half-duration, and likewise the half-time of total obscuration, are to be calculated anew, and the process repeated. In the case of the node, the ]) roper correction, in minutes, etc., is to be applied in the con- trary manner. This method of eliminating the error involved in the supposition of a constant latitude, and of obtaining another and more accurate deter- mination of the intervals between the moment of opposition and those of first and last contact, and of immersion and emergence, is by a series of successive approximations. For instance : AC, as already determined, being assumed 4 * as the interval between opposition and first contact, a new calculation of the moon’s longitude is made for the moment A, and, with this and the sum of the radii, a new value is found for AG. But now, as the position of A is changed, the former determination of its latitude is vitiated and must be made anew, and made to furnish .anew a corrected value of AC; and so on, until the position of A is fixed with the degree of accuracy required. The process must be conducted separately, of course, for each of the four quantities affected; since, where latitude is increasing, as in the case illustrated, the true values of AC and BG will be greater than their mean values, while GG and Ft 1 , the true intervals in the after part of the eclipse, will be less than AG and BG : and the contrary when latitude is decreasing.

The middle of the eclipse is to be regarded as occurring at the true close of the lunar day : if from that time the time of half-duration be subtracted, the moment of contact (grasa) is found ; if the same be added, the moment of separation.

In like manner also, if from and to it there be sub- tracted and added, in the case of a total eclipse, the half-time of total obscuration, the results will be the moments called those of immersion and emergence. The instant of true opposition, or of apparent conjunction (sec below, under ch. v. 9), in longitude, of the sun and moon, is to be taken as the middle of the eclipse, even though, owing to the motion of the moon in latitude, and also, in a solar eclipse, to parallax, that instant is not midway between those of contact and separation, or of immersion and emergence. To ascertain the moment of local time of each of these phases of the eclipse, we subtract and add, from and to the local time of opposition or conjunction, the true intervals found by the processes described in verses 12 to 15. The total disappearance of the eclipsed body within, or behind, the eclipsing body, is called nimilana , literally the ” closure of the eyelids, as in winking:” its first commencement of reappearance is styled unmilana, “ parting of the eyelids, peeping.” Wo translate the terms by “ immer- sion ” and ” emergence ” respectively.

If from half the duration of the eclipse any given interval be subtracted, and the remainder multiplied by the difference of the daily motions of the sun and moon, and divided by sixty, the result will be the perpendicular ( lcoti ) in minutes.

In the case of an eclipse ( graha ) of the sun, the perpen- dicular in minutes is to be multiplied by the mean half-duration, and divided by the true ( sphuta ) half-duration, to give the true perpendicular in minutes.

The latitude is the base ( bhuja ) : the square root of the sum of their squares is the hypothenuse (grava) : subtract this from half the sum of the measures, and the remainder is the amount of obscuration ( grdsa ) at the given time.

If that time be after the middle of the eclipse, subtract the interval from the half-duration on the side of separation, and treat the remainder as before : the result is the amount remaining obscured on the side of separation. The object of the process taught in this passage is to determine the amount of obscuration of the eclipsed body at any given moment during the continuance cf the eclipse. It, as well as that prescribed in the following passage, is a variation of that which forms the subject of verses the right-angled triangle formed by the line joining the centres of the eclipsed and eclipsing bodies as hypothenuse, the difference of their longi- tudes as perpendicular, and the moon’s latitude as base. And whereas, in the former problem, we had the base and hypothenuse given to find the perpendicular, here we have the base and perpendicular given to find the hypothenuse. The perpendicular is furnished us in time, and the rule supposes it to be stated in the form of the interval between the given moment and that of contact or of separation : a form to which, of course, it may readily be reduced from any other mode of statement. The interval of time is reduced to its equivalent as difference of longitude by a propor- tion the reverse of that given in verse 13, by which difference of longitude was converted mto time; the moon’s latitude is then calculated; from the two the hypothenuse is deduced ; and the comparison of this with the sum of the radii gives the measure of the amount of obscuration. Verse 21 seems altogether superfluous : it merely states the method of proceeding in case the time given falls anywhere between the middle and the end of the Eclipse, as if the specifications of the preceding verses applied only to a time occurring before the middle : whereas they are general in their character, and include the former case no less than the latter. When the eclipse is one of the sun, allowance needs to be made for the variation of parallax during its continuance; this is done by the process described in verse 19, of which the explanation will be given in the notes to the next chapter (vv. 14-17). In verse 20, for the first and only time, we have latitude called kshepa, instead of vikshepa , as elsewhere. In the same verse, the term employed for “ hypothenuse ” is grava, “ hearing, organ of hearing; ” this, as well as the kindred gravai^a, which is also once, or twice employed, is a synonym

From half the sum of the eclipsed and eclipsing bodies subtract any given amount of obscuration, in minutes : from the square of the remainder subtract the square of the latitude at the time, and take the square root of their difference.

Tlie result is the perpendicular ( hoti ) in minutes — which, in an eclipse of the sun, is to be multiplied by the true, and divided by the- mean, half-duration — and this, converted into time by the same manner as when finding the duration of the eclipse, gives the time of the given amount of obscuration ( grdsa ). The conditions of this problem arc precisely the same with those of the problem stated above, in verses 12-15, excepting that here, instead of requiring the instant of time when obscuration commences, or becomes total, we desire to know when it will be of a certain given amount. The solution must be, as before, by a succession of approximative steps, since, the time not being fixed, the corresponding latitude of the moon cannot be otherwise determined.

Multiply the sine of the hour-angle (nata) by the sine of the latitude ( ahsha ), and divide by radius : the arc corresponding to the result is the degrees of deflection (ralandn^ds ) , which are nortli and south in the eastern and western hemispheres ( kapdla ) respectively.

From the position of the eclipsed body increased by three signs calculate the degrees of declination : add them to the degrees of deflection, if of like direction ; take their difference, if of different direction : the corresponding sine is the deflection (■ valana ) — in digits, when divided by seventy. This process requires to be performed only when it is desired to project an eclipse. In making a projection according to the Hindu method, as will be seen in connection with the sixth chapter, the eclipsed body is represented as fixed in the centre of the figure, with a north and south line, and an east and west line, drawn through it. The absolute position of these lines upon the disk of the eclipsed body is, of course, all the time changing: but the change is, in the case of the sun, not observable, and

To the altitude in time (unnata) add a day and a half, and divide by a half-day ; by the quotient divide the latitudes and the disks ; the results are the measures of those quantities in digits ( angula ).