Chapter 2: True Places of the Planets

Forms of Time, of invisible shape, stationed in the zodiac ( hhagana ), called the conjunction (qighrocca), apsis (man- docca), and node (pata), are causes of the motion of the planets.

The planets, attached to these beings by cords of air, are drawn away by them, with the right and left hand, forward or backwajd, according to nearness, toward their own place*

The so-called apex (ucca), when in the half-orbit in front of the planet, draws the planet forward; in like manner, when in the half-orbit behind the planet, it draws it backward.

In like manner, also, the node, Rahu, by its proper force, causes the deviation in latitude ( vikshepa ) of the moon and the other planets, northward and southward, from their point of dec! ination (apakrama ) .

When in the half-orbit behind the planet, the node causes it to deviate northward; when in the half-orbit in front, it draws it away southward.

In the case of Mercury and Venus, however, when the node is thus situated with regard to the conjunction (gighra), these two planets are caused to deviate in latitude, in the manner stated, by the attraction exercised by the node upon the conjunc- tion. c The name Rahu, by which the ascending node is here designated, is properly mythological, and belongs to the monster in the heavens, which, by the ancient Hindus, as by more than one other people, was believed to occasion the eclipses of the sun and moon by attempting to devour them. The word which we have translated “ force ” is rnnhas , more properly “ rapidity, violent motion:” in employing it here, the text evi- dently intends to suggest an etymology for rahu , as coming from the root rah or raruh, “ to rush on ” : with this same root Weber (Ind. Stud. i. 272) has connected the group of words in which rahu seems to belong. For the Hindu fable respecting Rahu , see Wilson’s Vishnu Purana, p. 78. The moon’s descending node was also personified in a similar way, under the name of Ketu, but to this no reference is made in the present treatise.

Mars and the rest, on account of their small size, are, by the super natural beings (ddivata) called conjunction ( Qighrocca ) and apsis ( mandoeca ), drawn away very far, being caused to vacillate exceedingly.

Hence the excess (dhana) and deficiency ( rna ) of these latter is very great, according to their rate of motion. Thus do the planets, attracted by those beings, move in the firmament, carried on by the wind., The dimensions of the, sun and moon are stated below, in iv. 1 ; tfiose of the other planets, in vii. 13. We have ventured to translate aiivegila , at the. end of the tenth verse, as it is given above, because that translation seemed so much bettor to suit the requirements of the sense than the better-supported rendering caused to move with exceeding velocity.” In so doing, "we have assumed Unit the noun vega , of which the word in question is a denominative, re- tains something of the proper meaning of the root vij, “ to tremble,” from which it comes.

The motion of the planets is of eight kinds : retrograde ( rahra ) , somewhat retrograde (anunakra), transverse (hi til a ) ,

By reason of this and that rate of motion, from day to day, the planets thus come to an accordance with their observed places (efrf) — this, their correction (sphutikarana), I shall care- fully explain. Having now disposed of matters of general theory and preliminary explanation, the proper subject of this chapter, the calculation of the true (spfyuta) from the mean places of the different planets, is ready to be taken up. And the first thing in order is the table of sines, by means of which all the after calculations are performed.

The eighth part of the minutes of a sign is called the first sine ( jydrdlw ); that, increased by the remainder left after sub- tracting from it the quotient arising from dividing it by itself, is the second sine.

Three thousand four hundred and thirty-one; three thousand four hundred and thirty-eight. Subtracting these, in re- versed order, from the half-diameter, gives the tabular versed- sines ( utkramajyardhapindaka ) :

The quotient thus obtained add to the tabular sine called the preceding; the result is the required sine. The same method is prescribed also with respect to the versed sines.

The degrees of the sun's epicycle of the apsis (manda- paridhi) are fourteen, of that of the moon, thirty-two, at the end of the even quadrants; and at the end of the odd quadrants, they are twenty minutes less for both.

At the end of the even quadrants, they are seventy- five, thirty, thirty-three, twelve, forty-nine; at the odd ( oja ) they are seventy-two, twenty-eight, thirty-two, eleven, forty-eight,

By the corrected epicycle multiply the base-sine ( bhujajyd ) and perpendicular-sine ( kotijyd ) respectively, and divide by the number of degrees in a circle : then, the arc corres- ponding to the result from the base-sine ( bhujajydphala ) is Ihe equation of the apsis (mdnda phala), in minutes, etc. All the preliminary operations having been already performed, this is the final process by which is ascertained the equation of the apsis, or the amount by which a planet is, at any point in its revolution, drawn away from its mean place by the disturbing influence of the apsis. In modern phraseology, it is called the first inequality, due to the elliptieity of the orbit ; or, the equation of the centre. Figure 3, upon the next page, will serve to illustrate the method of the process. Let AMM'P represent a part of the orbit of any planet, which is supposed to be a true circle, having E, the earth, for its centre. Along this orbit the planet would move, in the direction indicated by the arrow, from A through M and M' to P, and so on, with an equable motion, were it not for the attraction of the beings situated at the apsis (mandocca) and conjunction ( gighrocca ) respectively. The general mode of action of these beings has been explained above, under verses 1-5 of this chapter : we have now to ascertain the amount of the disturb- ance produced by them at any given point in the planet’s revolution. The method devised is that of an epicycle, upon the circumference of which the planet revolves with an equable motion, while the centre of the epicycle traverses the orbit with a velocity equal to that of the planet’s mean motion, having always a position coincident with the mean place of the planet. At present, we have to do only with the epicycle which represents the disturbing effect of the apsis (mandocca). * The period of the planet’s revolution about the centre of the epicycle is the time which it takes the latter to make the circuit of the orbit from the apsis around to the apsis again, or the period of its anomalistic revolution. This is almost precisely equal to the period of sidereal revolution in the case of all the planets excepting the moon, since their* apsides are re- garded by the Hindus are stationary (see above, under i. 41-44) : the moon’s apsis, however, has a forward motion of more than 40° in a year; henc?e the moon’s anomalistic revolution is very perceptibly longer than its sidereal, being 27 d 13 h 18 m . The arc of the epicycle traversed by the planet at any mean point in its revolution is accord- ingly always equal to the arc of the orbit intercepted between that

To the square of this sum or difference add the square of the result from the base-sine ( bahuphala ); the square root of their sum is the hypothenuse (harm) called variable ( cala ). Multiply the result from the base-sine by radius, and divide by the variable hypothenuse :

The arc corresponding to the quotient is, in minutes, etc., the equation of the conjunction ( gdighrya phala); it is em-

In the case of all the planets, and both in the process of correction for the conjunction and in that for the apsis, the equa- tion is additive ( dhana ) when the distance ( kendra ) is in the half- orbit beginning with Aries; subtractive (ma), when in the half- orbit beginning with Libra. The rule contained in the last verse is a general one, applying to ,all the processes of calculation of the equations of place, and has already been anticipated by us above. Its meaning is, that when the anomaly (mandakendra), or commutation (y ighrakendra ), reckoned always forward from the planet to the apsis or conjunction, is less than six signs, the equation of place is additive; when the former is more than six signs, equation is subtractive. The reason is made clear by the figures given above, and by the explanations under verses 1-5 of this chapter. It should have been mentioned above, under verse 29, where the word kendra was first introduced, that, as employed in this sense by the Hin- dus, it properly signifies the position (see note to i. 53) of the “ centre ” of the epicycle — which coincides with the mean place of the planet itself — relative to the apsis or conjunction respectively. In the text of the Rurya-Siddhanta it is used only with this signification : the commentary employs it also to designate the centre of any circle. Sinco the sun and moon have but a r’ngle inequality, according to the Hindu system, the calculation of their true places is simple and easy. With the other planets the case is different, on account of the existence of two causes of disturbance in their orbits, and the consequent necessity both of applying two equations, and also of allowing for the effect of each cause in determining the equation due to the other. For, to the appre- hension of the Hindu astronomer, it would not be proper to calculate the two equations from the mean place of the planet; nor, again, to calculate either of the two from the mean place, and, having applied it, to take the new position thus found as a basis from which to calculate the other; since the planet is virtually drawn away from its mean place by the divinity at either apex ( ucca ) before it is submitted to the action of the other. The method adopted in this Siddhanta of balancing the two influences, and arriving at their joint effect upon the planet, is stated in verses 43 and 44. The phraseology of the text is not .entirely explicit, and would bear, if taken alone, a different interpretation from that which the commentary puts upon it, and which the rules to be given later show to be its true meaning; this is as follows: first calculate from the mean place of the planet the equation of the conjunction, and apply the half of it to the mean place; from the position thus obtained calculate the equation of the apsis, and apply half of it to the longitude as already XI

Multiply the daily motion ( bhukti ) of a planet by the sun’s result from the base-sine ( bahuphala ), and divide by the number of minutes in a circle ( bhacahra ) ; the result, * in minutes, apply to the planet’s true place, in the same direction as the equation was applied to the sun. By this rule, allowance is made for that part of the equation of time, or of the difference between mean and apparent solar time, which is due to the difference between the sun’s mean and true places. The instru- ments employed by the Hindus in measuring time are described, very briefly and insufficiently, in the thirteenth chapter of this work; in all

Multiply the result by the corresponding epicycle • of the apsis (mandaparidhi ) , and divide by the number of degrees in

Subtract the daily motion of a planet, thus corrected for the apsis ( manda ), from the daily motion of its conjunction U'ighra) ; then multiply the remainder by the difference between the last bypothenusc and radius, ,

And divide by the variable hypothenuse ( cala karna) : the result is additive to the daily motion when the hypothenuse is greater than radius, and subtractive when this is less ; if, when subtractive, the equation is greater than the daily motion, deduct the latter from it, and the remainder is the daily motion in a retrograde ( vakra ) direction.

When at a great distance from its conjunction (ftghrocca), a planet, having its substance drawn to the left and right by slack cords, comes then to have a retrograde motion.

To the nodes of Mars, Saturn, and Jupiter, the equa- tion of the conjunction is to be applied, as to the planets them- selves respectively ; to those of Mercury and Venus, the equation of the apsis, as found by the third process, in the contrary direction.

The sine of the arc found by subtracting the place of the node from that of the planet- — -or, in the case of Venus and Mercury, irom that oi the conjunction — being multiplied by the extreme latitude, and divided by the last hypotlienuse — or, in the case of the moon, by radius — gives the latitude ( vihshepa ),

When latitude and declination ( apakrama ) are of like direction, the declination (krdnti) is increased by the latitude ; when of different direction, it is diminished by it, to find the true ( spashta ) declination : that of the sun remains as already determined. How to find the declination of a planet at any given point in the ecliptic, or circle of declination ( krdntivrtia ), was taught us in verse 28 above, taken in connection with verses 9 and 10 of the next chapter: here we have stated the method of finding the actual declination of any planet, as modified by its deviation in latitude from the ecliptic. The process by which the amount of a planet’s deviation in latitude from the ecliptic is here directed to be found is more correct than might have been expected, considering how far the Hindus were from compre- hending the true relations of the solar system. The three quantities employed as data in the process are, first, the angular distance of the planet from its node; second, the apparent value, as latitude, of its greatest removal from the ecliptic, when seen from the earth at a mean distance, equal to the radius of its mean orbit; and lastly, its actual distance from the earth. Of these quantities, the second is stated for each planet in the concluding verses of the first chapter; the third is correctly represented by the variable hypofchenuse ( cala karna) found in the fourth process for determining the planet's true place (see abofe,

Multiply the daily motion of a planet by the time of rising of the sign in which it is, and divide by eighteen hundred ; the quotient add to, or subtract from, the number of respirations in a revolution : the result is the number of respirations in the day and night of that planet. In the first half of this verse is taught the method of finding the increment or decrement of right ascension corresponding to the increment or decrement of longitude made by any planet during one day. For the “ time of rising ” (vdayaprdnds, or, more commonly, udaydsavas , liter- ally “ respirations of rising ”) of the different signs, or the time in respira- tions (see i. 11), occupied by the successive signs of the ecliptic in passing the meridian — or, at the equator, in rising above the horizon — see verses 42-44 of the next chapter. The statement upon which the rule is founded is as fol low's : if the given sign, containing 1800' of arc (each minute of arc corresponding, as remarked above, under i. 11-12, to a respiration of sidereal time), occupies the stated number of respirations in passing the meridian, what number of respirations will be occupied by the arc traversed by the planet on a given day? The result gives the amount by which the day of each planet, reckoned in the manner of this Siddhanta, or from transit to transit across the inferior meridian, differs from a sidereal day : the difference is additive when the motion of the planet is direct; subtractive, when this is retrograde. Thus, to find the length of the sun’s day, or the interval between tw'o successive apparent transits, at the time for which his true longitude and rate of motion have already been ascertained. The sun’s longitude, as corrected by the precession, is 9 9 8° 40'; he is accordingly in the tenth sign, of which the time of rising (udaydsavas), or the equivalent in right ascension, is 1935 7 '. His rate of daily motion in longitude is 61' 26". Hence the proportion 1800' : 1935^ : : 61'26" : 66^.04 show's that his day differs from the true sidereal day by 11 v 0 ?J .04. As his motion is direct, the difference is additive : the length of the apparent day is therefore 60 w 11* 0"-t>4, which is equivalent to 24 h O'” 27 *.5, mean solar time. According to the Nautical Almanac, it is 24 /J 0 W 28*. 6. By a similar process, the lpngth of Jupiter’s day at the same time is found to

Multiply the sine of declination by the equinoctial shadow, and divide by twelve ; the result is the earth-sine ( kshitijyd ) ; this, multiplied by radius and divided by the day- radius, gives the sine of the ascensional difference (earn) : the number of respirations due to the ascensional difference

Is shown by the corresponding arc. Add these to, and subtract them from, the fourth part of the corresponding day and night, and the sum and remainder are, when declination is north, the half-day and half-night;

When declination is south, the reverse; these, multi- plied by two, are the day and the night. The day and the night of the asterisms (bha) may be found in like manner, by means of their declination, increased or diminished by their latitude. We wore taught in verse 59 how to find the length of the entire day of a planet at any given time; this passage gives us the method of ascertaining the length of its day and of its night, or of that part of the day during which the planet is above, and that during which it is below, the horizon. In order to this, it is necessary to ascertain, for the planet in question, its ascensional difference (cum), or the difference between its right and oblique ascension, the amount of which varies with the declination of the pjanct and the latitude of the observer. The method of doing this is stated in verse 61 : it may be explained by means of the last figure (Fig. 8). First, the value of the line AB, which is called the “ earth- sine ” ( kshitijyd ), is found, by comparing the two triangles ABC and CHE, which are similar; since the angles ACB and CEH are each equal to the latitude of the observer. The triangle CHE is represented here by a triangle of which a gnomon of twelve digits is the perpendicular, and its equinoctial shadow, cast when the sun is in the equator and on the meridian (see the next chapter, verse 7, etc.), is the base. Hence the proportion EH : HC : : BC : AB is equivalent — since BC equals DF, the sine of declination — to gnom. : eq. shad. : : sin decl. : earth-sine^ But the arc of which AB is sine is the same part of the circle of diurnal

The portion ( bhoga ) of an asterism ( bha ) is eight hundred minutes; of a lunar day ( tithi ), in like manner, seven hundred and twenty. If the longitude of a planet, in minutes, be divided by the portion of an asterism, the result is its position in asterisms : by means of the daily motion are found the days, etc. The ecliptic is divided (see ch. viii) into 27 lunar mansions or aster- isms, of equal amount; hence the portion of the ecliptic occupied by each asterism is 13° 20 7 , or 800 / . In order to find, accordingly, in which asterism at a given time, the moon or any other of the planets is, we have only to reduce its longitude, not corrected by the precession, to minutes, and divide by 8(X) : the quotient is the number of asterisms traversed, and the remainder the part traversed of the asterism in which the planet is. The last clause of the verse is very elliptical and obscure; according to the commentary, it is to be understood thus : divide by the planet’s true daily motion the part past and the part to come of the current asterism, and the quotients are the days and fractions of a day which the planet has passed, and is to pass, in that asterism. This inter- pretation is supported by the analogy of the following verses, and is doubt- less correct. The true longitude of the moon was f und above (under v. 39) to be 11 • 17° 39', or 20,859'. Dividing by 800, we find that, at the given time, the moon is in the 27th, or last, asterism, named Revati, of which it has traversed 59', and has 741' still to pass Over. Its daily motion being 737', it has spent 28*’ 4 ; \ and has yet to continue l rf 0 W 19* 3*, in the asterism. r The latter part of this process proceeds upon the assumption that the planet’s rate of motion remains the same during its whole continuance in the asterism. A similar assumption, it will be noticed, is made in all the processes from verse 59 onward; its inaccuracy is greatest, of course, where the moon’s motion is concerned. Respecting the lunar day (tithi) see below, under verse 66.

From the number of minutes in the sum of the longi- tudes of the sun and moon are found the yogas, by dividing that sum by the portion (bhoga) of an asterism. Multiply the minutes past and to come of the current yoga by sixty, and divide by the

Prom the number of minutes in the longitude of the moon diminished by that of the sun are found the lunar days ( tithi ), by dividing the difference by the portion ( hhoga ) of a lunar day. Multiply the minutes past and to come of the current lunar day by sixty, and divide by the difference of the daily motions of the two planets ; the result is the time in uadis. The iithif or lunar day, is (see i. IB) one thirtieth of a lunar month, or of the time during which the moon gains in longitude upon the sun a whole revolution, or 360° : it is, therefore, the period during which the difference of the increment of longitude of the two planets amounts to 12°, or 720 7 , which arc, as stated in verse 64, is its portion (bhoga). To find the current lunar day, we divide by this amount the whole excess of the longitude of the moon over that of the sun at the given time; and to find the part past and to come of the current day, we convert longitude into time in a manner analogous to that employed in the case of the yoga. Thus, to find the date in lunar time of the midnight preceding the first of January, 1860, we first deduct the longitude of the sun from that of the moon; the remainder is 2* 29° 24', or 5364': dividing by 720, it appears that the current lunar day is the eighth, and that 324' of its portion arc traversed, leaving 396' to ho traversed. Multiplying these numbers respectively by 60, and dividing by 675' 38", the difference of the daily motions at the time, we find that 28 n 46 r 2 p have passed since the beginning of the lunar day, and that it still has 35 n 10” 8*' to run. The lunar days have, for the most part, no distinctive names, but those of each half month (paksha — sec above, under i. 48-51) are called first, second, third, fourth, etc., up to fourteenth. The last, or fifteenth, of each half has, however, a special appellation : that which concludes the first, the light half, ending at the moment of opposition, is called paurnamasi , purnimd, pdrnnmd , “ day of full moon; ” that which closes the month, and ends with the conjunction of the two planets, is styled amdtidsyd, “ the day of dwelling together.” Each lunar day is farther divided into two halves, called karana , as. appears from the next following passage,

The fixed ( dhruva ) karanas , namely gahuni, ndga , catushpada the third, and kinstughna , are counted from the latter half of the fourteenth day of the dark half-month.

After these, the karanas called movable (earn), namely bava, etc., seven of them : each of these karanas occurs eight times in a month.

Half tiie portion ( bhoga ) of a lunar day is established as that of the karanas. . . . Of the eleven karanas, four occur only once in the lunar month, while the other seven are repeated each of them eight times to fill out the remainder of the month. Their names, and the numbers of the half Lunar diys to winch each is applied, are presented below: