Chapter 6: Projection of Eclipses

Since, without a projection ( chedyaka ), the precise (sphuta) differences of the two eclipses are not understood, I shall proceed to explain the exalted doctrine of the projection. The term chedyaka is from the root chid , “ split, divide, sunder,” and indicates, as here applied, the instrumentality by which distinctive differences are rendered evident. The name of the chapter, parilekhddhi- Jcdra, is not taken from this word, but from parilekha , ” delineation, figure,” which occurs once below, in the eighth verse.

Having fixed, upon a well prepared surface, a point, des- cribe from it, in the first place, with a radius of forty-nine digits ( angula ), a circle for the deflection ( valana ) : 3 Then a second circle, with a radius equal to half the sum of the eclipsed and eclipsing bodies ; this is called the aggregate- circle ( samasa ); then a third, with a radius equal to half the eclipsed body.

The determination of the directions, north, south, east, and west, is as formerly. In a lunar eclipse, contact ( graham ) takes place on the east, and separation ( moksha ) on the west; in a solar eclipse, the contrary. The larger circle, drawn with a radius of about three feet, is used solely in laying off the deflection ( valana ) of the ecliptic from an east and west circle. We have seen above (iv. 24, 25) that the sine of this deflection was reduced to its value in a circle of forty-nine digits’ radius by dividing

In a lunar eclipse, the deflection (valana) for the con- tact is to be laid off in its own proper direction, but that for the separation in reverse; in an eclipse of the sun, the contrary is the case.

From the extremity of either deflection draw a line to the centre : from the point where that cuts the aggregate-circle ( samasa ) are to be laid off the latitudes of contact and of separa- tion.

From the extremity of the latitude, again, draw a line to the central point : where that, in eithercase, touches the eclipsed body, there point out the contact and separation.

Always, in a solar eclipse, the latitudes are to be drawn in the figure ( parilekha ) in their proper direction; in a lunar ec- lipse, in the opposite direction.... The lines uM and drawn from v and v', the extremities of the rines or arcs which measure the deflection, to the centre of the figure, re- present, as already noticed, the direction of the ecliptic with reference to an east and west line at the moments of contact and separation. Erom them, accordingly, and at right angles to them, are to be laid off the values of the moon’s latitude at those moments. Owing, however, to the principle adopted in the projection, of regarding the eclipsed body as fixed in the centre of the figure, and the eclipsing body as passing over it, the lines uM and -a'M do not, in the case of a lunar eclipse, represent the ecliptic it- self, in which is the centre of the shadow, but the small circle of latitude, in which is the moon’s centre : hence, in laying off the moon’s latitude to determine the centre of the shadow, we reverse its direction. Thus, in the case illustrated, the moon’s latitude is always south: we lay off, then, the lines kl and k'V, representing its value at the moments of contact and separation, northward : they are, like the deflection, drawn as sines, and in such manner that their extremities, l and l\ are in the aggregate-circle : then since IM and VM are each equal to the sum of the two semi -diameters, and Ik and Vkf to the latitudes, kM and k f M will represent, the distances of the centres in longitude, and l and V the places of the centre of the shadow, at contact and separation : and upon describing circles from l and V , with radii equal to the semi-diameter of the shadow, the points c and s, where these touch the disk of the moon, will be the points of first and last contact : c and s being also, as stated in the text, the points where IM and VM meet the circumference of the disk of the eclipsed body.

The deflection is to be laid off — eastward, when it and the latitude are of the same direction; when they are of different direc- tions, it is to be laid off westward : this is for a lunar eclipse; in a solas*, the contrary is the case.

From the end of the deflection, again, draw a line to the centra] point, and upon this line of the middle lay off the latitude, in the direction of the deflection.

From the extremity of the latitude describe a circle with a radius equal to half the measure of the eclipsing body : what- ever of the disk of the eclipsed body is enclosed within that circle, so much is swallowed up by the darkness ( tamas ). The phraseology of the text in this passage is somewhat intricate and obscure; it is fully explained by the commentary, as, indeed, its meaning is also deducible with sufficient clearness from the conditions of the prob- lem sought to be solved. It is required to represent the deflection of the ecliptic from an east and west line at the moment of greatest obscuration, and to fix the position of the centre of the eclipsing body at that moment. The deflection is this time to be determined by a secondary to the ecliptic, drawn from near the north or south point of the. figure. The first question is, from which of these two points shall the deflection be laid off, and the line to the centre drawn. Now since, according to verse 10, the latitude itself is to be measured upon the line of deflection, the latter must be drawn southward or northward according to the direction in which the latitude is to be laid off. And this is the meaning of the last part of verse 8; “ in accordance,” namely, with the direction in which, according to the previous part of the verse, the latitude is to be drawn. But again, in which direction from the north or south point, as thus determined, shall the deflection be measured? This must, of course, be determined by the direction of the deflection itself : if south, it must obviously be measured east from the north point and west from the south point; if north, the contrary. The rules of the text are in accordance with this, although the determining circumstance is made to be the agreement or non- agreement, in respect to direction, of the deflection with the moon’s latitude — the latter being this time reckoned in its own proper direction, and not, in a lunar eclipse, reversed. Thus, in the case for which the figure is drawn, as the moon’s latitude is south, and must be laid off northward from M, the deflection, v f, w ff , is measured from the north point; as deflection and latitude are both south, it is measured east from N. In an eclipse of the

By the wise man who draws the projection ( chedyaha ), upon the ground or upon a board, a reversal of directions is to be made in the eastern and western hemispheres. This verse is inserted here in order to remove the objection that, in the eastern hemisphere, indeed, all takes place as stated, but, if the eclipse oc- curs west of the meridian, the stated directions require to be all of them reversed. In order to understand this objection, we must take notice of the origin and literal meaning of the Sanskrit words which designate the cardinal directions. The face of the observer is supposed always to be east- ward: then “ east ” is praflc , “ forward, toward the front ” ; “ west ” is pageat , “ backward, toward the rear “ south ” is dahshina, “ on the right ** north ” is uttara , “ upward ” ( i.e ., probably, toward the moun- tains, or up the course of the rivers in north-western India). Thege words apply, then, in etymological strictness, only when one is looking eastward — and so, in the present case, only when the eclipse is taking place in the eastern hemisphere, and the projector is watching it from the west side of his projection, with the latter before him: if, on the other hand, he re- moves to E, turning his face westward, and comparing the phenomena as they occur in the western hemisphere with his delineation of them, then " forward ” ( prdflc ) is no longer east, but west; *' right ” {dahshina) is no longer south, but north, etc. It is unnecessary to point out that this objection is one of the most frivolous and hair-splitting character, and its removal by the text a waste of trouble : the terms in question have fully acquired in the language an absoluto meaning, as indicating directions in space, without regard to the position of the observer.

Owing 7 to her clearness, even the twelfth part of the moon, when eclipsed ( gmsta ), is observable; but, owing to his piercing brilliancy, even three minutes of the sun, when eclipsed, are not observable. The commentator regards the negative which is expressed in the latter half of this verse as also implied in the former, the meaning being that an obscuration of the moon’s disk extending over only the twelfth part of it does not make itself apparent. We have preferred the interpre- tation given above, as being better accordant both with the plain and simple construction of the text and with fact.

At the extremities of the latitudes make three points of corresponding names; then, between that of the contact and that of the middle, and likewise between that of the separation and that of the middle,

Describe two fish-figures ( matsya ) : from the middle of these having drawn out two lines projecting through the mouth and tail, wherever their intersection takes place, 1G. There, with a line touching the three points, describe an arc : that is called the path of the eclipsing body, upon which the latter will move forward. The deflection and the latitude of three points in the continuance of the eclipse having been determined and laid down upon the projection, it is deemed unnecessary to take the same trouble with regard to any other points, these three being sufficient to determine the path of the eclipsing body : accordingly, an arc of a> circle is drawn through them, and is regard- ed as representing that path. The method of describing the arc is the same with that which has already been more than once employed (see above, iii. T-4, 41-42) : it is explained here with somewhat more fullness than before. Thus, in the figure, l, l ", and V are the three extremities of the moon’s latitude, at the moments of contact, opposition, and separation, respectively : we join W 1 , V'V , and upon these lines describe fish-figures (see note to iii. 1-5); their two extremities (“ mouth ” and “ tail ”) are indi- cated by the intersecting dotted lines in the figure : then, at the point, not included in the figure, where the lines drawn through them meet one another, is the centre of a circle passing through l, V f , and V.

From half the sum of the eclipsed and eclipsing bodies subtract the amount of obscuration, as calculated for any given time : take a little stick equal to the remainder, in digits, and, from the central point,

Lay it off toward the patli upon either side — when the time is before that of greatest obscuration, toward the side of contact; when the obscuration is decreasing, in the direction of separation — and where the stick and the path of the eclipsing body

Meet one another, from that point describe a circle with a radius equal to half the eclipsing body : whatever of the eclipsed body is included within it, that points out as swallowed up by the darkness (tamas).

Take a little stick equal to half the difference of the measures (mdna), and lay it off in the direction of contact, calling it the stick of immersion ( nil nil ana ) : where it touches the path,

From that point, with a radius equal to half the eclipsing body, draw a circle, as in the former case : where this meets the circle of the eclipsed body, there immersion takes place.

So also for the emergence (unmilana), lay it off in the direction of separation, and describe a circle, as before : it will show the point of emergence in the manner explained. The method of these processes is so clear as to call for no detailed explanation. The centre of the eclipsing body being supposed to be always in the arc W'V . drawn as directed in the last passage, we have only to fix a point in this arc which shall be at a distance from M corresponding to the calculated distance of the centres at the given time, and from that point to describe a circle of the dimensions of the eclipsed body, and the result will be a representation of the then phase of the eclipse. If the point thus fixed be distant from M by the difference of the two semi- diameters, as MT, Me', the circles described will touch the disk of the eclipsed body at the points of immersion and emergence, i and e.

This mystery of the gods is not to be imparted indiscrimi- nately : it is to be made known to the well-tried pupil, who remains a year under instruction. The commentary understands by this mystery, which is to be kept wjth so jealous care, the knowledge of the subject of this chapter, the deli- neation of an eclipse, and not the general subject of eclipses, as treated in the past three chapters. It seems a little curious to find a matter of so subordinate consequence heralded so pompously in the first verse of the chapter, and guarded so cautiously at its close.