Chapter 9: Heliacal Risings and Settings

Now is set forth the knowledge of the risings ( ndaya ) and settings (astamaya) of the heavenly bodies of inferior brilliancy, whose orbs are overwhelmed by the rays of the sun. The terms used for the heliacal settings and risings of the heavenly bodies, or their disappearance in the sun’s neighbourhood and their return to visibility, are precisely the same with those employed to denote their rising (ndaya) and setting (asta, a*iamaya, astamdna) above and below the horizon. The title of the chapter, udaydst ddhihdra , is literally trans- lated in our heading.

Jupiter, Mars, and Saturn, when their longitude is greater than that of the sun, go to their setting in the west ; when it is less, to their rising in the east : so likewise Venus and Mercury, when retrograding.

The moon, Mercury, and Venus, having a swifter motion, go to their setting in the east when of less longitude than the sun ; when of greater, to their rising in the west. These specifications are of obvious meaning and evident correctness. The planets which have a slower motion than the sun, and so are overtaken by him, make their last appearance in the west, after sunset, and emerge again into visibility in the east, before sunrise : of those which move mere rapidly than the sun, the contrary is true : Venus and Mercury belong to either class, according as their apparent motion is retrograde or direct.

Calculate the longitudes of the sun and of the planet — in the west, for the time of sunset ; in the east, for that of sunrise —

Then the ascensional equivalent, in respirations, of the interval between the two (lagndntaraprdnds) will give, when divided by sixty, the degrees of time ( kdldnqds ) ; or, in the west, the c ascensional equivalent, in respirations, of the interval between the two when increased each by six signs. Whether a planet will or will not be visible in the west after sunset, or in the east before sunrise, is in this treatise made to depend solely upon the interval of time by ( which its setting follows, or its rising precedes, that of the sun, or upon its distance from the sun in oblique ascension; to the neglect of those other cwbumstances — as the declination of the two bodies, and the distance and direction of the planet from the ecliptic — which variously modify the limit of visibility as thus defined. The ascertainment of the distance in oblique ascension, then, is the object of the rules given in these verses. In explaining ths method of the process, we will consider first the case of a calculation made for the eastern horizon. The time of sunrise having been determined, the true longitudes and rates of motion of the sun and the planet m question are found for that moment, as also the latitude of the planet. Owing to the latter’s removal in latitude from the ecliptic, it will not pass the horizon at the same moment with the point of the ecliptic which determines its longitude, and the point with which it does actually rise must be found by a separate process. This is accomplished by calculating the apparent longitude of the planet, according to the method taught in the seventh chapter There is nothing in the language of the text winch indicates that the calculation is not to be made in full, as there prescribed, and for the given moment of sunrise: as so conducted, however, it would evidently yield an erroneous result; for, the planet being above the horizon, the point of the ecliptic to which it is then referred by a circle through the north and south points of the horizon is not the one to which it was referred by the horizon itself at the moment of its own rising. The commentary removes this difficulty, by specifying that the akshadrkkarman , or that part of the process which gives the correction for latitude, is to be performed’ ‘ ‘ only as taught in the first half- verse ” — that is, according to the former part of vii, 8, which contains the rule for determining the amount of the correction at the horizon — omitting the after process, by which its value is made to correspond to the altitude of the planet at the given time. Having thus ascertained the points of ' the ecliptic which rise with the sun and with the planet respectively, the corresponding equatorial interval, or the distance of the planets in oblique ascension, is found by a rule already given (iii. 50). The result is expressed in respirations of sidereal time, which are equivalent to minutes of tbs-

The degrees of setting ( astdngds ) are, for Jupiter eleven; for Saturn, fifteen ; for Mars, moreover, they are seventeen :

Of Venus, the setting in the west and the rising in the east take place, by reason of her greatness, at eight degrees ; the setting in the east and the rising in the west occur, owing to her inferior size, at ten degrees :

So also Mercury makes his setting and rising at a distance from the sun of twelve or fourteen degrees, according as he is retrograding or rapidly advancing.

At distances, in degrees of time (kdlabhdgds) , greater than these, the planets become visible to men ; at less distances they become invisible, their forms being swallowed up ( grasta ) by the brightness of the sun. The moon, it will be noticed, is omitted here; her heliacal rising and setting are treated of at the beginning of the next following chapter. In the case of Mercury and Venus, the limit of visibility is at a greater or less distance from the sun according as the planet is approaching its inferior or superior conjunction, the diminution of the illuminated portion of the disk being more than compensated by the enlargement of the disk itself when seen so much nearer to the earth. ?$• / 33

The difference, in minutes, between the numbers thus stated and the planet’s degrees of time (kdldng&s), when divided by the difference of daily motions- — or, if the planet be retrograding, bv the sum of daily motions — gives a result winch is the time, in days, etc. 11 . The daily motions, multiplied by the corresponding as- censional equivalents (tallagndsavas) , and divided by eighteen hundred, give the daily motions in time ( k&lagati ) ; by means of these is found the distance, in days etc., of the time past or to come. Of these two verses, the second prescribes so essential a modification of the process taught in the first, that their arrangement might have been more properly reversed. If we have ascertained, by the previous rules, th§ distance of a planet in oblique ascension from the sun, and if we know the distance in oblique ascension at which it will disappear or re-appear, the interval between the given moment and that at which disappearance or re-appearance will take place may be readily found by dividing by the rate of approach or separation of the two bodies the difference between their actual distance and that of apparition and disparition * but the divisor must, of course, be the rate of approach in oblique ascension, and not in longitude. The former is derived from the latter by the following propor- tion: as a sign of the ecliptic, or 1800 / , is to its equivalent in oblique as- cension, as found by iii. 42-45, so is the arc of the ecliptic traversed by each planet in a day to the equatorial equivalent of that arc. The daily rates of motion in oblique ascension thus ascertained are styled the “ time- motions ” (kdlagati), as being commensurate with the “ tfme-degrees (kdldngds)

Svati, Agastya, Mrgavyadha, Citra, Jyeshtha, Punar- vasu, Abhijit, and Brahmahrdaya rise and set at thirteen degrees.

Hasta, Qravana, the Phalgunis, Qravishtha, Rohini, and Magha become visible at fourteen degrees ; also Vi 9 ftkh.fl and Ayvinls

Bharara. Push) a, and Mrga^rsha, owing to their faint- ness, are seen at twenty-one degrees; the rest of the asterisks become visible and invisible at seventeen degrees. These are specifications of the distances from the sun in oblique ascension (kdldngds) at which the asterisms, and those other of the fixed stars whose positions were defined in the preceding chapter, make their heliacal risings and settings. The asterisms we are doubtless to regard as represented by their junction-stars ( yogatdrd ). The classification here made of the stars in question, according to their comparative magnitude and brilliancy, is in many points a very strange and unaccountable one, and by no means calculated to give us a high idea of the intelligence and care of those by whom it was drawn up. The first class, comprising such as are visible at a distance of 13° from the sun, is, indeed, almost wholly composed of stars of the first magnitude; one only, Punarvasu (/? Geminorum), being of the first to second, and having for its fellow one of the first (a Geminorum). But the second class, that of the stars visible at 14°, also contains four which are of the first magnitude, or the first to second, namely, Aldebaran (ttohini), Regulus (Magha), Deneb or f3 Leonis (Uttara-Phalguni), and Atair or a Aquilee ((^Jravana); and, along with these one of the second to third magnitude, 8 Leonis (Purva-Phalguni), three of the third, and one, t Libras (Vi<jakM), of the fourth. In this last case, however, it might be possible to regard a Librse, of the second magnitude, as the star which is made to determine the visibility of the astensm. Among the stars of the third class, again, which are visible at 15°, is one, a Ononis (Ardra), which, though a variable star, does not fall below the first to second magnitude; while with it are found ranked six stars of the third magnitude, or of the third to fourth. The class of those which are visible at 17°, and which are left unspecified, contains two stars of the fourth magnitude, but also two pf the second, one of which, a Andromed© or y Pegasi (Uttara-BMdrapad&), : is mentioned below (v. 18) among those which are never obscured by the too near approach of the sun. The stars forming the class which are not to he seen within 21° of the sun are all of the fourth magnitude, but they are no less distinctly visible than two of those in the preceding class; and indeed, Rharani is palpably more so, since it contains a star of the third magnitude, which is perhaps (see above) to be regarded as its junction-star. $ince Agni, Brahma, Apamvatsa, and Apas are not specially mentioned, it is to be assumed that they all belong in the class of those visible at 1/7°, and they are so treated by the commentator: the first pf them (fi Tauri) is

The degrees of visibility (drgydngas), if multiplied by eighteen hundred and divided by the corresponding ascensional equivalent ( udaydsavas ), give, as a result, the corresponding degrees on the ecliptic ( kshetrdngas ) ; by means of them, likewise, the time of visibility and of invisibility may be ascertained. % r t This verse belongs, in the natural order of sequence, not after the passage next preceding, with which it has no special connection, but after verse 11. Instead of reducing, as taught in that verse, the motions upon the ecliptic to motions in oblique ascension, the “ degrees of time ” (lcdldngds) may themselves be reduced to their equivalent upon the corres- ponding part of the ecliptic, and then the time of disappearance or of re- appearance calculated as before, using as a divisor the sum or difference of daily motions along the ecliptic. The proportion by which the reduction is made is the converse of that before given; namely, as the ascensional equivalent of the sign in which are the sun and the planet is to that sign itself, or 180<y, so are the “ degrees of visibility ” ( drgydngds , or ledldngds) of the planet to the equivalent distance upon that part of the ecliptic in which it is then situated. The technical name given to the result of the proportion is kshetrdngds: kshetra is literally “field, territory,” and the meaning of the compound may be thus paraphrased : “ the limit of visibility, m degrees, measured upon that part of the ecliptic which is, at the time, the territory occupied by the planets in question, or their proper sphere.” 17 . Their rising takes place in the east, and their setting ih. the west ; the calculation of their apparent longitude ( drkkarman ) is to be made according to previous rules ; the ascertainment of the time, in days etc., is always by the daily motion of the sun alone.

Abhijit, Bra hm ah relay a, Svati. Qravana ( vdishnava ), Qravishtha ( v&sava ), and TJttara-Bhadrapada (ahirbudhnya ) , owing to their northern situation, are not extinguished by the sun’s rays. It may seem that it would have been a more orderly proceeding to omit the stars here mentioned from the specifications of verses 12-15 above; but there is, at least, no inconsistency or inaccuracy in the double statement of the text, since some of the stars may never attain that distance in oblique ascension from the sun which is there pointed out as their limit of visibility. We have not thought it worth the trouble to go through with the calcula- tions, and ascertain whether, according to the data and methods of this treatise, these six stars, and these alone, of those which the treatise notices, would never become invisible at Ujjayini. It is evident, however, as has already been noticed above (viii. 20-21), that the star called Brahma or Prajapati (S Aurigae) is not here taken into account, since it is 8° north of GBrahinahrdaya, and consequently cannot become invisible where the latter does not.