Chapter 10: Phases of the Moon

The calculation of the heliacal rising ( udaya ) and setting (asta) of the moon, too, is to be made by the rules already given. At twelve degrees 1 distance from the sun she becomes visible in the west, or invisible in the east. In determining the time of the moon’s disappearance in the neighbour- hood of the sun, or of her emergence into visibility again beyond the sphere of his rays, no new rules are required; the same methods being employed as were made use of in ascertaining the time of heliacal setting and rising of the other planets : they were stated in the preceding chapter. The definition of the moon’s limit of visibility would have been equally in order in the other chapter, but is deferred to this in order that the several processes in which the moon is concerned may be brought together. The title of the chapter, grngonnatyadhihdra, “ chapter of the elevation of the moon's cusps ” grnga, literally “ horn ”), properly applied only to that part of it which follow^ the fifth verso, The degrees spoken of in this verse are, of course, “ degrees of time ” (kdldngds), or in oblique ascension.

Add six signs to the longitudes of the sun and moon respectively, and find, as in former processes, the ascensional equivalent, in respirations, of their interval (lagndntardsavas) : if the sun and moon be in the same sign, ascertain their interval in minutes.

Multiply the daily motions of the sun and moon by ' the result, in nadis, and divide by sixty ; add to the longitude of eacl*, the correction for its motion, thus found, and find anew their interval, in respirations ;

And so on, until the interval, in respirations, of the sun and moon is fixed : by so many respirations does the moon, in 4he light half-month (» ;uhla), go to her setting after the sun.

Add half a revolution to the sun’s longitude, and calculate the corresponding interval, in respirations : by so many respirajjons does the moon, in the dark half-month (krshnapalcsha ) , come to her rising after sunset. The question here sought to be solved is, how lorg after sunset upon any given day will take place the setting of the moon in the crescent half- month, or from new to full moon, and the rising of the moon in the waning half-month, or from full tc new moon. The general process is the same with that taught in the last chapter, for obtaining a like result as regards the other planets or fixed stcrs* we ascertain, by the rules of the seventh chapter — applying the correction for the latitude according to its value at the horizon, as determined by the first part of vii. 8 — the point of the ecliptic which sets with the moon; and then the distance in oblique ascen- sion between this and the point at which the sun set will measure the required interval of time. An additional correction, however, needs to be applied to the result of this process in the case of the moon, owing to her rapid motion, and her consequent perceptible change of place between the time of sunset and that of her own setting or rising: this is done by cal- culating the amount of her motion during the interval as first determined, and adding its equivalent in oblique ascension to that interval; then cal- culating her motion anew for the increased interval and adding its ascen- sional equivalent — and so on, until tho desired degree of accuracy is attained. The process thus explained, however, is not precisely that which is prescribed in the text. We are there directed to calculate the# amount of motion both of the sun and moon during the interval between the setting of the sun and that of the moon, and, having applied them to the longitudes of the two bodies, to take the ascensional equivalent of the distance -between them in longitude, as thus doubly corrected, for the precise time of the setting of the moon after sunset. In one point of view this is false and absurd; for when the sun has once passed the horizon, the interval to the setting of the moon will be affected only by her motion, and not at all by his. In another light, the process does not lack reason: the allow- ance for the sun’s motion is equivalent to a reduction of the interval from sidereal ( ndkshatra ) time to civil, or true solar (sdvana) time, or from respirations which are thirty-six-hundredths of the earth’s revolution on its ^axis to such as are like parts of the time from actual sunrise to actual sunrise. But such a mode of measuring time is unknown elsewhere in this treatise, which defines (i. 11-12) and employs sidereal time alone, adding

Of the declinations of the sun and moon, if their direc- tion be the same, take the difference ; in the contrary case, take the sum : the corresponding sine is to be regarded as south or north, according to the direction of the moon from the sun.

Multiply this by the hypothenuse of the moon’s mid-da^, shadow, and, when it is north, subtract it from the sine of latitude (aksha) multiplied by twelve ; when it is south, add it to the same.

The result, divided by the sine of co -latitude ( lamba ), gives the base ( bhuja ), in its own direction ; the gnomon is the perpendicular ( koti ) ; the square root of the sum of their squares '8 the hypothenuse. In explaining the method of this process, we shall follow the guidance of the commentator, pointing out afterwards wherein he varies from the strict letter of the text : for illustration w e refer to the accompanying figure 0 % 32 ). The figure represents the south-western quarter of the visible sphere, seen as projected upon the plane of the meridian; Z being the zenith, Y bhe south point, WY the intersection of the horizontal and meridian planes, and W the projection of the west point Let ZQ equal the latitude of the place of observation, and let QT and QO be the declinations of the sun and moon respectively, at the given time : then WQ, ST, and NO will be the projections of the equator and of the diurnal circles of the sun and moon Suppose, now, the sun to be upon the horizon, at S, and the moon to have a certain altitude, being at M: draw from M the perpendicular to the plane of the horizon ML, and join MS : it is required to know the relation to one another of the three sides of the triangle SLM, in order to the delineation of the moon’s appearance when at M, or at the moment of sunset Now ML is evidently the sine of the moon’s altitude at the given time, which may be found by methods already more than once described and illustrated. And SL is composed of the two parts SN and NL, of which the former depends upon the distance of the moon in declination from the sun, and the latter upon the moon’s altitude But SN is one of the sides of a right-angled triangle, in which the angle NSJ> is equal to the Observer’s co-latitude, and Nb to the sum of the sine of declination of the sun, cb or Wa, and that of the moon, Nc. Hence sin bSN : t>N : : R : SN er sin co-lat. : sum of Bines of decl • • R : SN HXrd 8N**(R x sum of sines of decl.) -esin eo-lat. In like manner, since, in the triangle MNL, the angles at M and N are respectively equal to the observer’s latitude and co-latitude, sin MNL : sin, LMN : : ML : NL 0, sfn co-lat. : sin. lat. : : sin alt. : NL and NL« (sin alt. x sin lat.)+»in oo-lat.

Tlic number oi minutes in the longitude of the moon diminished by that of the sun gives, when divided by nine hundred, her illumina od part (guhla) : this, multiplied by the number of digits (angula) of the moon’s disk, and divided by twelve, gives the same corrected ( sphuta ). The rule laid down in this verse, for determining the measure of the illuminated part of the moon, applies only to the time between new moon and full moon, when the moon is less than 180° from the sun : when her excess of longitude is more than 180°, the rule is to be applied as stated below, in verse 15. As tho whole diameter of the moon is illuminated when she is half a revolution from the sun, one half her diameter at a quarter of a revolution’s distance, and no part of it at the time of conjunc- tion, it is assumed that the illuminated portion of her diameter will vary as the pari of 180° by which she is distant from the sun; and hence fchat, assuming the measure of the diameter of her disk to be twelve digits, the number of digits illuminated may be found by the following proportion; as half a revolution, or 10,800', is to twelve digits, so is the moon’s distance from the sun in minutes to the corresponding part of the diameter illumi- nated : the substitution, in the first ratio, of 900: 1 for 10,800 : 12, gives the rule as stated in the text. Here, »t- will be noticed, we have for the

Fix a point, calling it the sun : from that lay off the base, in its own proper direction ; then the perpendicular, towards the west ; and also the hypothenuse, passing through the extremity of the perpendicular and the central point.

From the point of intersection of the perpendicular and the hypothenuse describe the moon’s disk, according to its dimen- sions at the given time. Then, by means of the hypothenuse, iirst make a determination of directions ;

And lay off upon the hypothenuse, from the point ol its intersection with the disk, in an inward direction, the measure of the illuminated part : between the limit of the illuminated part and the north and south points draw two fish-figures ( matsya ) ;

From the point of intersection of the lines passing through their midst describe an arc touching the three points : as the disk already drawn appears, such is the moon upon that day.

After making a determination of directions by means of the perpendicular, point out the elevated (unnata) cusp at the extremity of the cross-line : having made the perpendicular (fcofi) to be erect (unnata), that is the appearance of the moon. , 15. In the dark half-month subtract the longitude of the sun increased by six signs from that of the moon, and calculate, in the same manner as before, her dark part. In this case lay off the base in a reverse direction, and the circle of the moon on the west.

The corrected ( sphuta ) time of the aspect ( pata ) is the middle : if that be diminished by the half-duration, the result is the time of the commencement ; if increased by the same, it is the time of the end.