सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

467 planet at the moment are to be computed. The time in between the two lagnas, which will be in Sāvana measure pertaining to the planet gives the time that has elapsed after the rise of the planet. Comm. According to computation, the difference of times of the current lagna and the Udayalagna as com- puted taking the present position of the planet will be the time measured along the diurnal path of the planet and as such is in Sānava measure. Bhāskara mentions here a very subtle point. In the analysis the latitude of the planet is taken into account while finding the Udayalagna. Time measured in this case is called Kṣetrātmika and not Kālātmika for the following reason. Suppose A is the Udayalagna of the planet and B the current lagna both points being on the ecliptic. Let us say that the current lagna is posterior to the Udayalagna (there is no loss of generality in supposing this). By the time of the current lagna, the planet is no more at A but will have moved a little towards B. Let the present position of the planet on the ecliptic be A'. The corresponding arc of AB on the equator gives the sidereal measure of the time that has elapsed after the rise of the planet whereas the corresponding arc of A'B on the equator gives the Sāvana measure of the same time. Here this Sāvana is that pertaining to the planet and not that pertaining to the Sun which is Saura Sāvana. In other words it is the Sāvana pertaining to the particular planet, because the arc AA' is traversed by the planet in question as per its own velocity. The stress is here on the word Tat-Kāla-graha, ie. the longitude of the planet at the moment at which the shadow or the zenith-distance of the planet is sought. Taking this as the position of the planet and taking the latitude of the planet also into account compute the Udayalagna. This will be A' cited above. To obtain the

468 sidereal measure of the same time we have to take A and not A'. Verse 12. Sāvana measure alone is to be employed while finding the gnomonic shadow ie. while the zenith-distance of the planet is to be computed, because the arc of the diurnal circle of the planet indicates only Sāvana measure. Suppose the Udayalagna falls short of the current lagna, then the Bhōgyakāla ie. the remaining rising time of the Rāśi in which the planet is situated added to the elapsed time of the Rāśi of the current lagna together with the sum of the rising times of the Rāśis in between gives the difference of the Udayalagna of the planet and the current lagna. Comm. (Ref. fig. 111). Let p be the planet's place on the Ecliptic at the moment in question when the lagna is L (or the foot of the latitude of the planet in case it is not on the ecliptic). Let PA be the remainder of the Rāśi in which the planet is situated. Then the time of rising of the arc PA is here termed Bhōgyakāla. Let DL be the arc of the Rāśi in which the current lagna L is situated, which has arisen. The time of rising of this arc DL is called Bhuktakāla. The times of risings of the Rāśis in between namely AB, BC, CD are called Madhyōdayās. The sum of the rising times of PA, AB, BC, CD, DL gives the time in between the rise of P and that of L which is the time that has elapsed after the planet has arisen upto the current Lagna ie. the rising time of L. Verse 13. The method of computing the gnomonic shadow of the planet or what is the same the zenith- distance of the planet, as per the method of computing that of the Sun (extended to the case of the planet) after

469 finding the Sphutakrānti (ie. modern declination) of the planet. The Krānti of the planet or the declination of the foot of the latitude of the planet added to the latitude rectified called Sphuṭasāra, gives what is called Spaṣṭakrānti of the planet and its Hsine is called Spaṣṭa-Krāntijyā. From this H sin δ, H cos δ etc. is to be computed (as mentioned in Triprasnādhikāra in the context of finding the zenith- distance of the Sun). From the time that has elapsed after the rise of the planet called the Unnata, the shadow is to be computed as in the case of the Sun’s shadow. Having thus computed the shadow or what is the same the zenith-distance of the Moon or that of the stars, the instru- ment called Nalaka could be pointed to the spot where that celestial body is situated. Comm. The words Krānti, Spaṣṭakrānti were explained before. Also the method of calculating the zenith-distance from a knowledge of the declination was described before in Triprasnādhikāra. Verses 14, 15. Consideration of horizontal parallax in the case of visibility of the Moon or planets. H cos z or what is called the Mahāśaṅku of the Moon or planet is to be reduced by 1/15th of the respective daily motion gives the visible Sanku when the radius is taken to be 3438. If the radius is taken as 120, 1/450th of the daily motion is to be subtracted. If H cos z is less than 1/15th (or 1/450th) of the daily motion, then the Moon is not visible. This applies to the other planets as well, but as this is a neglible matter, [the earlier Āchāryas did not suggest this. Comm. In the context of the subject of parallax, we mentioned that in the case of the Moon, the horizontal parallax happens to be 1/15 of the Moon’s daily motion

470 approximately. The same is extended to the case of the planets as well. To get the proportion in the case of taking the radius to be 120 only instead of 3438, rule of three is applied as follows. “If the radius be 3438, 𝑣/15 is parallax, what will it be if the radius be 120?” The answer is 𝑣/15 × 120/3438 = 𝑣 / (3438/8) = 𝑣/430 very approximately as stated. If H cos z be less than this 1/15 th or 1/430 th of the daily motion, the parallax does not allow the Moon to be seen. Bhāskara specifically talks about the Moon only because, the earlier Āchāryas did not apply the correction of parallax to the case of the planets, because it is not appreciable. Verse 16. If an operation is neglected because its effect is not appreciable, or is not of much use, or because it is apparent, or if it implies great labour, or again if it implies a lot of exposition that would make the text unduly voluminous, ignoring the necessity of that oper- ation should not be treated as wrong. Comm. Here Bhāskara upholds the earlier Āchāryas not stipulating parallax in the case of planets, because it is not appreciable. Here ends Graha-cchāyādhikāra.

GRAHŌDAYĀSTĀDHIKĀRA Verse 1 and first half of verse 2. The Udayalagna of a planet is termed Prāk-Dṛk-graha and the Astalagna is termed the Paschima-Dṛk-graha. If the Prāk-Dṛk-graha happens to be less than the current lagna the planet had already risen. If it be greater, the planet is still to rise. Similarly if the Paschima-Dṛk-graha is less than the current lagna, the planet had already set ; otherwise is yet to set. Comm. The words Prāk-Dṛk-graha and Paschima- Dṛk-graha are coined to indicate the points of intersection of the ecliptic with the eastern and western horizon respectively when the planet is rising or setting, when the planet has latitude. When the planet has no latitude Prāk-Dṛk-graha coincides with the rising planet and the Paschima-Dṛk-graha with the setting planet. When the planet has a latitude, the foot of the latitude, which signifies the planet’s position on the ecliptic differs from the two Dṛk-grahas. In this case ie. when the planet has a latitude the Prāk-Dṛk-graha’s longitude will be less than that of the planet’s longitude whereas the Paschima-Dṛk- graha’s longitude will be greater than that of the planet. If x and y be the differences of the longitudes of the planet and those of the Dṛk-grahas respectively, it is clear from a figure that tan x = (tan β) / (tan θ) and tan y = (tan β) / (tan ϕ) where β is the latitude of the planet and θ and ϕ the angles which the ecliptic makes with the horizon at the planet’s rising and setting respectively. Since θ and ϕ are then respectively the zenith- distances of the pole of the ecliptic in each case, and since

472 in the course of half a sidereal day, these zenith-distances could not be equal, x and y cannot be equal. Second half of verse 2 and verse 3. To find the side- real time in between the given time and the time of the planet's rising. Find the time that has elapsed after the planet's rise, from a knowledge of the Iṣṭalagna at a given time and Prāk-Dṛk-graha ie. the longitude of the rising point of the ecliptic at the time when the planet rises. This time will be in Sāvana measure, because we have taken the planet's position at the given moment and not the position of the rising planet. From this time, know- ing the daily motion of the planet of the day in question, obtain the arc; that would have been traversed in between the moments. Subtracting this arc from the planet's position at the moment, the planet's position at rising would be obtained approximately; approximately, because as per the procedure enunciated in verse eleven, Graha- cchāyādhikāra, the Prāk-Dṛk-graha of the rising planet, and that obtained from the position of the planet at the moment differ. From this approximate time again com- pute the arc that would have been traversed by the planet during that time. Subtracting this arc which is nearer the truth from the planet's present position, we obtain a more approximate position of the rising planet. Again computing the Prāk-Dṛk-graha from this position, calcu- late the time from this Prāk-Dṛk-graha and the lagna of the moment. Repeating the process we will obtain the actual sidereal time in between the given time and the actual rising time of the planet. It is sidereal because, we have found the time as per the procedure of verse (11) referred to, where we have used तत्कालखेटोदय and not the actual ,खेटोदय, i.e the rising lagna of the planet obtained from the present position and not that of the rising planet.

473 This difference between the nature of the times was already dealt with. The matter of this verse appears to have been unnecessarily complicated by Bhāskara but on careful scrutiny it is not so ; for, the problem is to find the time that has elapsed after the rise of the planet upto a given time. Here the data are the given time and the corres- ponding position of the planet not on the horizon but elsewhere. We could not know immediately the time at which the planet rose. To find it alone, this method of successive approximation has to be used and there is no other go. Had we known the position of the planet while rising, the corresponding Prāk-Dṛk-graha could be found exactly and as this position of the Prāk-Dṛk-graha is a point of the ecliptic and the Iṣṭa-lagna ie. the rising point of the ecliptic at the given moment is also a point on the ecliptic, the time between the moments of rising of these two points of the ecliptic could be found in sidereal measure directly by noting the rising times of the Rāśis in between, without an appeal to the method of successive approximation. Verse 4. The computation of the times of heliacal rising and setting of a planet in contradistinction to its rising time during a day on account of earth’s diurnal rotation. The times of rising and setting of a planet have been dealt with. Now I shall tell the procedure to be adopted in computing the times of heliacal rising and setting of a planet. If a planet has a daily motion less than that of the Sun, it rises heliacally in the east, and sets in the west ; otherwise the reverse. Comm. Take the example of the superior planets, say that of Jupiter. Since Jupiter moves slower than the Sun, 60

474 the Sun will have to overtake Jupiter and not vice versa. When it is the Sun that is to overtake, taking a position of Jupiter near the western horizon gradually the planet gets fainter and fainter as the Sun approaches him from west to east. Ultimately one evening the planet ceases to be perceptible within a particular distance of the Sun. This we call heliacal setting of the planet. Having thus set in the west, after a few days the planet rises in the east. This is so because, after the planet's heliacal setting, the Sun gradually approaches the planet from west to east; at a particular moment will have a longitude equal to that of the planet and then gradually gains in longitude over the planet. After a few days, in the eastern horizon, the planet will emerge from the rays of the Sun before Sun- rise; it is not the planet moving west from east but it is the Sun moving from west to east so that the Sun recedes away from the planet towards east. The planet also will be having a motion from west to east but it being slower than that of the Sun, the latter gains over the former in its motion from west to east. In the case of the inferior planets, say for example, Mercury, the velocity of Mercury is greater than that of the Sun; as such, Mercury will be overtaking the Sun and not vice versa. Thus in the eastern horizon, it is Mercury that enters the rays of the Sun, going from west to east, so that he sets in the east. After a few days of this heliacal setting, Mercury acquires a longitude equal to that of the Sun, and while overtaking the Sun from west to east, he emerges out of the rays of the Sun in the west, which is therefore heliacal rising. But, when Mercury is retrograde, the case is different. Some days after heliacal rising in the west, Mercury attains his maximum elonga- tion and then begins to retrograde. The elongation then gradually decreases, and he will set again in the rays of the Sun in the west itself. A few days thereafter, still continuing retrograde, he emerges out of the Sun's rays

475 in the east, thus rising heliacally. After attaining the maximum elongation in his retrograde motion, his motion will then become direct, so that he again approaches the Sun, going from west to east. Next a little before overtaking the Sun, he sets heliacally in the east, and thereafter, having overtaking the Sun, he rises heliacally in the west. Here, one point is to be mentioned. The time between Mercury's maximum elongation in the west and again the maximum elongation in the east (which are termed maximum eastern elongation and maximum western elongation with respect to the Sun) will be far less than the time between the maximum elongation in the east (ie. maximum western elongation with respect to the Sun) and that in the west (ie. maximum western elongation with respect to the Sun) in as much as when Mercury is retrograde, and the Sun having always direct motion, the relative velocity will be the sum of the retrograde velocity of Mercury and the direct velocity of the Sun. In the case of the superior planets, as mentioned above, the superior planets set heliacally in the west and rise heliacally in the east. They will not rise heliacally in the west and will not set heliacally in the east which happens only if either the superior planet has a greater velocity than the Sun or the Sun has a retrograde motion which is never the case. Verse 5. Speciality with respect to Mercury and Venus. Mercury and Venus rise heliacally in the west in their direct motion, (attain maximum elongation before they) become retrograde, set heliacally there itself, then rise heliacally in the east continuing to be retrograde, (attain maximum elongation there before they) next become direct

476 and gradually set there (to rise again in the west) as before. Comm. Explained above. Verse 6. Kālāmśas or distance in degrees from the Sun within which the planets rise or set heliacally. The Kālāmśas with respect to the Moon, Mars, Mer- cury, Jupiter, Venus and Saturn, or the degrees of distance from the Sun within which they rise or set heliacally are 12, 17, 14, 11, 10, 15 respectively. In the case of Mercury and Venus when they are retrograde the Kālāmśas are 12, and 8 respectively. Comm. The Kālāmśas given above depend upon the luminosity of the respective planets. When Mercury and Venus happen to be retrograde, the Kālāmśas happen to be 2° less in each case because they are then nearest to the earth and as such being most luminous as seen by us will not set heliacally till they are very near the Sun. Verse 7. To compute the moment when a planet rises or sets heliacally. If it is to be known when a planet rises or sets helia- cally the position of the Prāk-Dṛk-graha or the Paschima- Dṛk-graha as the case may be (Prāk-Dṛk-graha in case the rising or setting takes place in the east or the other in the other case) and that of the Sun also are to be computed on a day a little before the day of rising or setting as prognosticated by the Śīghra anomaly. In case the planet rises or sets in the west, to obtain the lagna the position of the Sun is to be increased by 180°. Comm. Clear.

477 First half of verse 8. To obtain what are called Iṣṭa-Kālāṁśas. The time between the rising of the planet or of the Dṛk-graha and that of the Sun measured in ghaṭīs multi- plied by six gives what are called Iṣṭa-Kālāṁśas. Comm. Having found the approximate position of the Dṛk-graha as mentioned above when the planet is likely to rise or set heliacally, let the time in between the rising of this Dṛk-graha and that of the Sun (if it be the case of setting or rising in the west the position of the Sun is to be increased by 180° because the astalagna directed to be found in verse (1) is the point of intersection of the ecliptic with the eastern horizon, which is removed 180° from the setting point of the Sun) be multiplied by six. Since both the Dṛk-graha and the Sun's position are points on the Ecliptic and since we have considered the time in between their rising moments, which is measured on the equator, and again since this time is measured in ghaṭīs, the number of ghaṭīs multiplied by six give the degrees, for, each ghaṭī corresponds to six degrees of the equator (sixty ghaṭīs corresponding to one sidereal day). These degrees are said to be Iṣṭa-Kālāṁśas, which means that it gives the arc in between the feet of the declination circles of the Dṛk-graha and the Sun at the Iṣṭa-kāla ie. that particular time considered. Second half of verse 8 and verses (9) and (10). If the number of degrees so found ie. the Iṣṭa- Kālāṁśas fall short of or exceed the number of Kālāṁśas postulated for the rising or setting of the planet, then the planet's rising has to take place or has already taken place respectively and vice versa in the case of setting. The number of minutes of the difference of the prescribed and Iṣṭa-Kālāṁśas multiplied by 1800 and divided by the rising time of the Rāśi expressed in Kalās and again

478 divided by the difference of the daily motions of the planet and the Sun expressed in minutes of arc if the planet is direct in motion or divided by the sum of those daily motions if the planet be retrograde gives the days elapsed after rising or to elapse for the rising to take place. Again, compute the positions of the Dṛk-graha and the Sun for the moment thus obtained and repeat the process till the actual moment is obtained. Comm. Suppose the prescribed Kālāṁśas for rising be x and suppose the Iṣṭa-Kālāṁśas are y such that y<x; then for the planet to rise, the arc of the equator in between the feet of the declination circles of the Sun and the Dṛk-graha has to increase for the planet to rise which means that this takes some more time to happen. Similarly if y>x, the planet has already risen. The question of the arc decreasing does not arise in this case of rising, because the planet's position in the east is behind that of the Sun, and in the case of a superior planet the Sun has to advance further for the planet to rise ie. the arc has to increase and in the case of a retrograde inferior planet also, the arc will be increasing. In the case of an inferior planet being direct, the arc will be decreasing no doubt, but we have to remember that this is a case of an inferior planet setting and not rising which we are considering. In the case of setting, in the east, however, of the inferior planet the moment of setting has already elapsed and thus the condi- tion is reverse to that of rising. The case of setting of a superior planet in the east never happens. In the case of setting of a superior planet in the west, if the Iṣṭa- Kālāṁśas be less than the prescribed Kālāṁśas, ie. y<x, setting must have already taken place, the Sun having approached the planet from behind and already effected heliacal setting. Thus this is also reverse to the condition of rising as stated. In the case of an inferior planet setting in the west if y>x, the planet should have set already since the inferior planet is retrograde while setting in the west.

479 This condition is also reverse to what has been stated in the case of rising. The case of a superior planet rising in the west also never happens, because it is the Sun that overtakes the planet and also the superior planet cannot be retrograde while near the Sun. The case of rising which has already elapsed the Iṣṭa- Kālāṁśas y will be evidently greater than x ie. y > x. To obtain the time by which the rising will take place after the moment in question, ie. the rising of a superior planet which is direct and an inferior planet which is retrograde, we have to take the difference of the prescribed and Iṣṭa- Kālāṁśas and find out the time by rule of three as follows. (1) Since we have to find the corresponding arc of the ecliptic in minutes of arc, from (y - x) × 60' the difference of the Kālāṁśas of the equator converted into minutes ie. asus we have first to use the proportion. "If by the rising time of the Rāśi t in asus in which the planet is situated we have 1800' of the ecliptic, what will we have by (y - x) × 60? The answer is (y - x) × 60 × 1800 / t . Next we have to find the days when this arc of the ecliptic is covered by the proportion. "If the Sun overtakes the planet by (u - v) minutes of the ecliptic per day, how many days are taken to cover the above arc?" The answer is as stated. The answer gives in days together with fraction of a day when the planet is likely to rise. We have to repeat the process because the motions of the planet as well as the Sun differ from moment to moment and the time calcu- lated above by rule of three taking into account their motions for the entire day will be only approximate. The computation in the case of setting of a planet may be similarly considered, the setting of a superior planet in the west or that of a direct inferior planet in the east.

480 Verses 11 and 12. If the Prāk-Dṛk-graha has a longitude greater than that of the Sun or the Paschima- Dṛk-graha has one less than that of the Sun, the sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas converted into minutes of arc will have to be used to compute the elapsed days or the days after which the respective phenomenon is going to occur. If the Iṣṭa-Kālāṁśas y be greater than the prescribed namely x, the reverse to what has been stated in the second half of verse (8) and in the first line of verse (9) happens ie. a phenomenon which has elapsed when y < x will happen in the future and vice versa. Comm. If the Prāk-Dṛk-graha has a longitude greater than that of the Sun, (since the prescribed Kālāṁśas are between the planet and the Sun, the planet being behind the Sun, and the Iṣṭa-Kālāṁśas are now on the other side of the Sun) the planet has advanced over its position of heliacal setting be it a superior planet or inferior by the sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence the days that have elapsed after the heliacal setting have to be calculated with the sum of the two Kālāṁśas. If it be also a case of a retrograde inferior Prāk-Dṛk-graha rising, having a longitude greater than that of the Sun, even then the prescribed Kālāṁśas are behind the Sun and the Iṣṭa-Kālāṁśas ahead of the Sun. So from the position of the planet’s heliacal rising, the Sun and the planet have advanced in opposite direction to a distance which is the sum of the prescribed Kālāṁśas and Iṣṭa-Kālāṁśas. Hence computation has to proceed with the sum of the two Kālāṁśas to get the time that has elapsed after the planet’s heliacal rising. Similar is the argument for the Paśchima- Dṛk-graha. We have commented on verses from the latter half of (8) upto (10) where the Iṣṭa-Kālāṁśas y happen to be less

481 than x. If y>x, or again the Prāk-Dṛk-graha has a longi- tude greater that of the Sun, the phenomenon of rising or setting which happened when y<x, will have to be taken as going to happen and that which was going to happen when y<x, must have happened already. It is enough to consider just one case instead of all the four cases since the argument is similar as before. Let us take y>x and it is an inferior planet in the east behind the Sun. According to the latter half verse (8) the rising had to take place y being less than x, whereas, now, y being greater than x, rising had already taken place, in contra- distinction to what happens when y<x. Here ends the Udayāstādhikāra.

SṚṄGONNATYADHIKĀRA Verse 1. Either in the last quarter of a lunation, or in the first quarter, on the day when the elevation of the cusps of the crescent Moon is to be determined, then either at the moment of Moon-rise or Moon-set or (for the matter of that during any part of the night) the Hsine of the altitude of the Moon is to be computed by noting the time from the moment of Moon-rise. Comm. Either in the first quarter or the last quarter of the lunation, ie. when the phase of the Moon according to the definition of phase in modern terms is less than half, the Moon will be a crescent. Also, generally, on the back-ground of the horizon, we notice that one of the cusps is more elevated than the other. This elevation goes by the name Sṛṅgonnati. Even in the middle half of the lunation, Bhāskara mentions that Brahmagupta and some others (meaning Sripati whom he closely follows) attempted at finding the elevation (strictly speaking elevation during the second quarter and depression during the third quarter) of the dark horns. Bhāskara does not appreciate this since, nobody would think about this as it does not appeal to the eye at all. Verse 2. To find the Hsine of the altitude of the Sun. The Hsine of the altitude of the Sun is to be com-puted, assuming the rising Sun to be in the opposite hemisphere, south or north (ie. if he be originally in the northern, assume him to be in the south) and using the formula given in verse 54 of Tripras'nādhyāya "अथोन्नता- दूनयुताच्चरेण" given the time measured in asus that has elapsed after Sunset.

483 Fig. 112 Comm. During the early part of the night or during the latter part thereof, when the Sun is below the horizon, the Sun will be occupying symmetrical positions with respect to the horizon at times which are equally removed from Sunset and the next Sun-rise. This is clear from the figure 112. Let A and B be two such symmetrical positions where AM and BN are the Hsines of the altitudes. Evidently AM = BN, C P̂ A = C P̂ B. Since the rising eastern hour-angle of the Sun equals he setting western hour- angle, and since C P̂ A = C P̂ B, the time elapsed after Sunset when the Sun is at B, will be equal to the time before Sun-rise in the position A. Hence by congruence AM = BN = Hsine of the altitudes in the two symmetric positions. Using the modern formula from the triangle PZS, cos z = sin ϕ sin δ + cos ϕ cos δ cos h, when C P̂ A = C P̂ B = h cos z will be the same in the two positions A and B. Putting z = 90 + θ, cos z = - sin θ will be the same ie. H sin θ will be the same in the two positions ie. the Śaṅkus will be the same. In the formula

484 cos z = sin ϕ sin δ + cos ϕ cos δ cos h Put z = 90 + θ, h = 90 + H, δ = δ in the positions A, B and z = 90−θ, h=90−H. δ=−δ in the positions A', B' We have then − sin θ = sin ϕ sin δ − cos ϕ cos δ sin H and sin θ = − sin ϕ sin δ + cos ϕ cos δ sin H which are identical. This means that the altitude θ below the horizon with +δ in the positions A, B is computable with −δ in the positions A', B'. This accounts for the statement made ‘गोलविपर्ययेण’. The second statement of the latter half of verse (2) says Śankutala = Śanku × ˢ/₁₂ which we have proved in Tripraśnādhyāya. Verse (3) and first half of (4). To obtain the Bhuja of the Sun. The Śankutala of the inverse altitude or the altitude below the horizon, is north (in contradistinction to what it is above the horizon). The sum or difference of the Agrā and Śankutala according as they are of the same direction or of opposite directions, is the Bhuja. The sum or difference of the Bhujas of the Sun and Moon, according as they are of opposite or the same directions is what is called the Spaṣṭa Bhuja whose direction is to be construed as that of the Moon. If the Bhuja of the Moon falls short of that of the Sun, then the direction of the Spaṣṭa Bhuja is that opposite to that of the Moon. Comm. In fig. 113, we have shown five different positions of the Sun S₁ to S₅ the feet of the Śankus being B₁ to B. From these feet of the Śankus draw perpendiculars on the Udayāstasutras as well as on the East-west line their points of intersection being respectively A₁ to A₅ and C₁ to C₅. The perpendiculars from the feet of the Śankus on the East-west line go by the name Bhujas, the perpendi-

485 Fig. 113 cular distances between the Udayāstasutras and the East- west line are called Agrās and the perpendiculars from the feet of the S'ankus on the Udayāstasutras are called S'ankutalas which were all defined in the course of the Tripras'nādhyāya. We have from the spherical triangle PZS, sin δ = sin ϕ cos z + cos ϕ sin z sin a so that sin δ / cos ϕ = tan ϕ cos z + sin z sin a which was shown as A=S+B in the Tripras'nādhyāya ie. Agrā = Sankutala + Bhuja. Changing δ into —δ and a into —a we have different formulae, the standard form being A=S+B. We don't propose to change ϕ into —ϕ, because in India this case does not arise. Thus our latitude being north, the S'anku- talas defined above as the distances from the feet of the Sankus from the Udayāstasutras are always deemed as south. It need not be reiterated that the Udayāstasutras

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are the straight lines joining the rising and setting points of the celestial body. As such, these Udayāstasutras are all parallel to the East-west line. In fig. 114, in the horizontal plane containing the points A's, B's and C's in the respective cases, according to the Hindu convention, we talk of Agrā and Bhuja as being Uttara Agrā, Dakṣiṇāgrā, Uttara Bhuja and Dakṣiṇa Bhuja. Śaṅkutalam is always taken to be south except in the fifth case shown in figs. 113 and 114 when the altitude happens to be below the horizon. In this case the Śaṅkutala is spoken as north. Thus the general formula A=S+B assumes the following various forms, in the respective cases. (1) Uttarāgrā—Dakṣiṇa Śaṅkutala = Uttara Bhuja. This corresponds to A, B, C of fig. 114 where AC, is Uttarāgrā, BA, Dakshina Śaṅkutala and BC, Uttara Bhuja. (2) In the case of A₃ B₃ C₃, Śaṅkutala—Agrā = Dakṣiṇa Bhuja since B₃ A₃—C₃ A₃=B₃ C₃. This occurs after the Sun's diurnal path has crossed the prime-vertical and the Sun has a position on the south of the prime- vertical, having a northern declination. In the case of A₄ B₄ C₄, B₄ C₄ = B₄ A₄ + A₄ C₄ ie. Dakṣiṇa Bhuja = Dakṣiṇa Śaṅkutala + Dakṣiṇāgrā. In the fifth case when the Sun's altitude is below the horizon, B₅ C₅ = B₅ A₅ + C₅ A₅ ie. Uttara Bhuja = Uttara Śaṅkutala + Uttarāgrā. In the sixth case when the Sun has a southern declination and an altitude below the horizon, B₆ C₆ = A₆ C₆ + A₆ B₆ ie. Dakṣiṇa Bhuja = Dakṣiṇāgrā minus Uttara Śaṅkutala. In the case of A₂ B₂ C₂, Uttarāgrā =