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

rate of 15° per hour or 6° per nadi or 10 Vinādis per degree or 60 asus per 60 minutes of arc or as many asus as there are minutes of arc in the right ascension of the mean Sun. So, what is stated by Bhāskara is the difference of the minutes of arc in the mean longitude of the Sun and the minutes of arc in his right ascension ie. (l—a) expressed in minutes. We know that the total equation of time arising out of unequal motion in the true longitude of the Sun ie. ☉ in comparision with the equal motion in his right ascension ie. a is measured by ☉—a which may be expressed as (☉—l) + (l—a) where l is his mean longi- tude, ☉—l = Equation of centre and so the time expressed by ☉—l is the equation of time due to obliquity. The difference l—a arises out of the obliquity of the ecliptic and so the time expressed by l—a is the equation of time due to obliquity. We have the modern formula Cos ω = tan a / tan l so that (1 — cos ω) / (1 + cos ω) = (tan a — tan l) / (tan a + tan l) = (sin a — l) / (sin a + l) ∴ Sin (a — l) = tan² ω/2 sin (a + l). As a is very nearly equal to l, we could write, when expressed in radiaus a—l = tan² ω/2 sin 2l. Thus the maximum difference between a and l arises when 2l = 90° ie. l = 45 degrees ie. at the mid-point of the first quadrant; the mini- mum difference is when 2l = 270 ie. l = 135 ie. at the middle point of the 2nd quadrant. Also the numerical magnitudes of the max. as well as the minimum value are each tan² ω/2 ie. they are equal. Since this is expressed in radiaus, converting into time the numerical value of the max. and minimum equation of time due to obliquity is 9.87′. Thus we can write l—a = 9.87′ sin 2l. Again where l = 225, 2l = 450 so that l—a will have a positive max.; and again when l = 315, 2l = 630 so that l—a will have a negative maximum value. Also when l has values 0, 90, 180, 270 it is zero. Thus the equation of time due to obliquity is zero at ♈, ie. the vernal equinox; +ve in the first quadrant

. 207 increasing from zero to 45 degrees and then decreasing from 45° to 90°; assuming the value zero at 90°, then negative in the second quadrant negatively increasing from zero to a maximum as l increases from 90° to 135°, and then negatively decreasing from the maximum value to zero at the end of the second quadrant ; again behaving in the 3rd quadrant as in the first and in the fourth as in the second. (6) It was stated by Mr. Mazumdar in his introduction to the Siddhānta S'ekhara of S'ripati published by the Calcutta University as well as by pandit Babuaji Misra, the editor thereof that this Udayāntara correction was first mentioned by S'ripati, and it meant equation of time due to obliquity. In fact S'ripati states (Verse I ch. eleven) “ अन्त्यभ्रमेण गुणिता रविबाहुजीवाऽभीष्टभ्रमेण विहृता, फलकार्मुकेण, बाहोः कलासु रहितास्ववशेषकं ते, यातासवो युग्मयुजोः पदयोः धनर्णम् ” l expressed in minutes — H sin⁻¹ (H sin ☉ H cos ω / H cos δ) express- ed in asus = what are called elapsed asus and are +ve, +ve, +ve and —ve in the successive quadrants. We saw before that H sin α = (H sin ☉ H cos ω / H cos δ) so that S'ripati meant l - α, the former expressed in minutes of arc and the latter in asus, or what is the same (l—α) both expressed in minutes or asus. These give the gain of l over α. Immediately after this verse S'ripati goes to a different topic, and never mentions (as understood from the printed text) any further details as to what is to be done with these Yātāsus. Since Bhāskara says explicitly that for these Yātāsus, the planets are to be corrected, we may surmise that there should have been in S'ripati's text also another verse detailing the usage of those asus. (7) We shall now attend to what Bhāskara gives by way of explanation of the verses in question, The Ahar-

208 gaṇa found by us is what is called Madhyama Sāvana- ahargaṇa or number of days according to mean solar reckoning, a reckoning made on the basis of taking uniform motion of the Sun in right ascension. The planets computed for the Sun-rise on this basis, are to be corrected for the true Sun-rise which goes according to the measure of ☉ ie. the true longitude of the Sun. But in the name of Bhujāntara we have-effected the correction for ☉ - l ; so what remains is to effect correction for l - a. This correction is known as Udayāntara. In other words, since we could not compute the Sphuṭa Sāvana-ahargaṇa, when we deem that x days have elapsed according to the mean solar reckoning from, the epoch upto the mean Sun-rise x ± f might have elapsed according to Sphuṭa Sāvana-ahargaṇa where f is a fraction. A correction is to be effected for this fraction f of a day and it is of the form ☉ - a = ☉ - l + l - a. In the beginning Bhāskara said that mean planets are had being computed out of the mean solar ahargaṇa, at the time when the mean Sun is about the eastern horizon of the equator. Why did he say ‘about the horizon’? It is because the mean solar ahargaṇa indicates a Sun-rise on the basis of equal motion in right ascension whereas on the basis of equal motion in l, he would be about the horizon. Had we been able to compute Sphuṭa-Sāvana-ahargaṇa, we could have got direct the planetary position when the true Sun is on the horizon. The correction for the difference in ☉ and l having been attended to through Bhujāntara, we are now to attend to the correction for the difference l - a. Verse 64. Another way of looking at the same. Had we obtained the Ahargaṇa in terms of the local risings of the Rasis in the place of the equatorial, and computed the planetary positions for the local true Sun- rise obtained that way, we would have done the three

209 corrections namely Bhujāntara, Chara and the Udayāntara as well. Comm. The correction of chara arises out of the difference between the equatorial risings and local risings of the mean Sun. The local mean Sun pertaing again to uniform motion along the celestial equator Udayāntara has to be effected for l—a, ie. for the local mean Sun on the ecliptic. Thus we have obtained the planetary posi- tions for the local mean Sun-rise and to obtain them for the True Sun-rise, Bhujāntara is to be effected. Verse 65. An alternative method of effecting the Udayāntara correction. Double the H sine of the Sāyana longitude of the Sun derived out of the smaller H sin-table, being multiplied by the daily motion of the planet and divided by 270 and the result in seconds of arc is to be corrected in the plane- tary position positive or negative according as the Sun is in the even or odd quadrants. Comm. (1) In symbols the correction indicated is (m H sin 2l) / 270 where m is the planet's mean daily motion in minutes of arc, l the longitude of the mean Sun, and H sine is taken where the radius = 120. In the commen- tary under the verse, Bhāskara adds 'Each quadrant of the eclipse rises (at the equator) in ¼th of a day but each Rāśi does not rise in 1/12 of a day. Since this Udayāntara correction vanishes when the Sun is at the ends of quadrants it must be construed that this correction increases positively or negatively from the beginning of the quadrant to the middle and decreases from the middle to the end of a quadrant. Saying that a particular Rāśi rises at the equator in n nādikās is only an approximate statement since the Rāśi does not rise uniformly. That is why astronomers like Aryabhata 27

stipulated finding the risings of smaller arcs like horas and Dṛkkāṇas. Find (H sin l × H cos ω) / (H cos δ) ie. H sin α. Take the asus in the arc of this H sine ie. find α in minutes. Then l - α gives the number of asus for which the correct- ion in the planetary position is to be effected, for, by these asus the True Sun-rise is accelerated or belated. In the middle of a quadrant these asus will be a little above 26 Vinadis. To obtain them at any point of the quadrant, take H sin 2l as the argument so that the maximum correct- ion will be had at the middle of the quadrant by this argument. Then the rule of three is ‘ If by 120 as radius, we have 26 Vinadis, what shall we have for H sin 2l ?’ The result is (H sin 2l × 26) / 120 = (H sin 2l) / 4½ approximately. Then another rule of three. “ If by 60 Vinadis we have m' of the planetary motion, where the daily motion of the planet is m°, what shall we have for (H sin 2l) / 4½ ?” The result is (H sin 2l × m) / (60 × 4½) minutes = (H sin 2l × m) / 270 . Then the sign of the correction is clear. (2) We shall now prove Bhāskara’s statement that at the middle of the quadrant, the correction is 26 Vinadis. We saw above that l - α = tan² ω/2 H sin 2l. It is really creditable on the part of Bhāskara to have seen by intuit- ion that the argument is H sin 2l. The maximum correction is therefore tan² ω/2 expressed in asus or minu- tes of arc. Let x = tan² ω/2 log x = 2 log tan ω/2. Take ω = 24 Then log x = 2 × 1̄.3275 = 2̄.6550 ∴ x = ·04519 radian = 155 minutes of arc ie. 155 asus = 155 / 6 = 26 apply.

211 (3) Taking the case of the Moon, the max correct- ion amounts to (790 × 120) / 270 = 5' — 51''. Verses 66, 67. The computation of Tithi, Nakṣatra and Yoga. The elongation of the Moon ie. the excess of the Moon's longitude over that of the Sun in degrees. (In case the former is smaller, add 360° to it and subtract Sun's longitude) being divided by 12 and 6, the quotients represent the elapsed tithis and Karaṇas. If the elapsed Karanas are K, count K — 1 beginning from Bava to get the current Karaṇa and count from Śakuni to get the current one beginning from the mid-moment of Kṛṣṇachaturdasī. Take again the planetary position or that of the Moon in particular as well as the sum of the longitudes of the Sun and the Moon both expressed in minutes of arc and divided by 800. The first quotient gives the elapsed stars ie. the Stars covered by the planet or the Moon; whereas the second quotient gives the elapsed Karaṇas. Then take the remainders in seconds and divide by the respective daily motions in minutes i.e. in the case of the planets or the Moon divide by their respective motions and in the case of yogas divide by the sum of the motions of the Sun and the Moon. Then the results give the times in nādīs as to how much the next nakṣatra or yoga have elapsed. If it be required to find as to how long the next nakṣatra or yoga will last, substract the remainder from 800, and divide by the daily motions in minutes as mentioned above. The results give as to how many nādīs beginning from the morning concerned, the next nakṣatra or yoga will last. Comm. (1) A lunation i.e. the time from the moment of New Moon to the next New Moon is divided into 30 parts called Tithis. Their names are pratipat, Dwitiyā etc. upto the 15th purnima or full Moon and again

212 pratipat, Dwitīyā etc. upto the 30th i.e. Amāvāsyā or New Moon. Thus pratipat starts when ☾ = ☉ i.e. when the longitude of the Moon is equal to that of the Sun i.e. from the moment of New Moon when the Moon is in conjunction with the Sun. अमा सह = वसतः सूर्य्याचन्द्रमसावस्या मित्यमावस्या i.e. Amāvāsyā is that point of time when the Sun and the Moon are together. Pratipat lasts till ☾ = ☉ + 12°; then Dwitīyā begins and lasts till ☾ = ☉ + 24; Thus purnima begins from the moment when ☾ = ☉ + 168 and lasts till ☾ = ☉ + 180° and Amāvasyā begins when ☾ = ☉ + 348 and lasts till ☾ is again equal to ☉ Thus a tithi is the time that is taken by the Moon to overcome the Sun by 12° beginning from the moment of New Moon. In other words the tithi is a measure of the phase of the Moon with a particular convention. (2) In modern astronomy the phase of the Moon or that of a planet is measured by the formula (1 + cos EPS) / 2 where P is the planet or the Moon. Since ES is almost Fig. 29 parallel to PS (Fig. 29) ÊPS may be taken to be nearly equal to P̂ES which is the elogation of the planet or the Moon so that phase = (1 - cos EPS) / 2 approximately. So

213 in modern astronomy also the phase is measured through the elongation. But the maximum phase in modern astronomy is taken to be unity as could be seen from the formula, for, when EPŜ = 180°, the phase = 1. Thus the phase multiplied by 15 gives approximately the tithi. We shall have occasion to deal with this topic later. (3) Suppose ☾ − ☉ = E°. The number of the tithis elapsed is equal to the quotient in E / 12. Take the remainder r°. It means during the current tithi which has a duration of 12°, r° have elapsed. Let the daily motions of the Moon and the Sun be m and s so that the Moon overtakes the Sun by m − s (expressed in minutes for convenience) during 60 nādīs. The elapsed time of the current tithi in nādīs is given by (r × 60 × 60) / (m − s). But r × 60 × 60 = Seconds of arc of the remainder so that it is said “गतैष्यविलिप्तिकाः”. If it be required to find how many nādīs the next tithi lasts, (12−r)° or (12−r) × 60′ is to be gained by the Moon over the Sun. So the time in Nādīs = ((12 − r) × 60 × 60) / (m − s) = एष्यविलिप्तिकाः / (m − s) . (4) A lunation is again divided into 60 Karaṇas and so 6° of increase in elongation correspond to a Karaṇa. These Karaṇas are eleven in number, out of which 4 are fixed to occur during the latter half of Kṛṣṇa Chautùrdasī, during the two halves of Amāvāsyā and the first half of the Sukla pratipat. Their names are Sakuni, Chatuṣpāt, Nāgava, and Kimstughna. So there remain 56 halves of tithis during the lunation during which the remaining seven Karaṇas, Known as Bava, Bālava etc. rotate eight times. Thus the first half of the duration of any tithi is covered by one Karaṇa and the latter by another. The computations of Karaṇas proceeds as with respect to tithis

214 but dividing the elongation by 6, we are asked to count K — 1, (where K is the quotient) beginning with Bava because the first half of Sukla pratipat is covered by the fixed Karaṇa Kimstughna. The computation of the elapsed as well as the remaining nādīkas of a particular Karaṇa, it proceeds on the same lines as that of a tithi. (5) The Nakṣatra in which a planet or the Moon is situated is then found. The zodiac is divided into 27 equal divisions beginning with its zero point and each division is named after the brilliant star of that division. Those stars are Aświnī, Bharaṇī etc. The star occupied by the Moon has a special significance in the Hindu calendar, it being spoken of that every day is presided over by a nakṣatra. This happens so because the Moon’s sidereal period is approximately 27 days. (6) To compute the Nakṣatra in which the planet or Moon is, take its longitude in minutes of arc λ; divide by 800, since each star division consists of 360° / 27 = (360 × 60) / 27 minutes = 800'. The quotient gives the elapsed nakṣa- tras. To get the elapsed nādikās of the current nakṣatra or the remaining, let the remainder be r'. Then the proportion is ‘If during the day of 60 nādīs, the planet or the Moon goes p minutes what time does it take to cover r' or 800 — r'’. The answer is (r' × 60) / p' or ((800 — r') × 60) / p' = गतैष्यविलिप्तिकाः / p' . (7) Regarding yogas, let the longitudes of the Sun and the Moon be ☉ and λ in minutes of arc; let their daily motions be s and m in minutes. If the sum of the longi- tudes is 800', we say the first yoga named Viṣkambha is over, if the sum is 1600', the second Yoga, named prīti has elapsed. Thus going on if the sum is 360°, we say the 27 yogas have elapsed and the first again begins. Since the sum of the daily motions of the Sun and the Moon

215 are on the average 59'—8''+790'—35''=850', to cover one Yoga, they take roughly one day since the duration of a yoga is of 800'. To get the number of elapsed yogas, divide (⊙+M)' where M is the longitude of the Moon by (s+m). To get the elapsed nādīs which are given by the remainder r' or the remaining nādīs of the current yoga which are given by 800; the proportion is ‘If by s+m gain in the sum of the longitudes we have 60 nādīs, what time is indicated by r' or 800—r'?’. The answer is (r' × 60)/(m + s) or ((800 — r)' × 60)/(m + s) = गतैष्यविलिप्तिकाः/(m + s) . By the word गतैष्यविलिप्तिकाः is meant therefore the number of seconds of arc as many as the remainder r or (800—r) is in minutes or what is the same the remainder or 800—r converted into seconds. (8) The tithi, karaṇa, nakṣatra of the Moon and yoga constitute four of the Angas of the panchānga or the Hindu Calendar the fifth being the week-day. All these five are supposed to have their effects good or bad on living beings. Verses 68, 69. The correction what is called Nata- karma. The Zenith distances of the Sun and the Moon at the end of Purṇimā or Amāvāsyā at the time of lunar or solar eclipse, being expressed in nādīs, are multiplied by 6 to get the degrees. Let their H sine be got from the short table of H sines. Multiply it by the equations of centre of the Sun and the Moon. Divide by 4920, and 4361 respectively. If the Sun be in the Eastern hemisphere, let the result pertaining to the Sun be subtracted from his position ; if in the Western, let it be added to his position. If the Moon be in the Eastern hemisphere and if his equation of centre be negative let the result be added to his position; if the equation of centre be positive, let the result be sub- tracted from his position in either of the hemispheres.

216 Again from these positions, let the tithi be computed and again let the above process be carried out until a constant time is arrived at for conjunction or opposition. Comm. (1) The above procedure is accepted by Bhaskara as Āgama enunciated by Brahmagupta and reiterated by Chaturveda as giving results that accorded with observation. We shall see that the correction known in modern astronomy as 'correction due to astronomical refraction' is indicated here, though it was not stated explicitly. In fact what was stated by Brahmagupta was that the periphery of the Manda epicycle given as 13⅔° for the Sun and as 31-36 for the Moon hold good only on the meridian but the periphery of the Sun is to be increased or decreased by 20' according as the equation of centre is negative and the Sun is the Eastern equatorial horizon, or western equatorial horizon. If the equation of centre be positive the reverse correction is to be effected in the periphery i.e. for negative equation of centre. Periphery on the Eastern equatorial horizon = 14°-0 On the meridian = 13°-40' On the Western equatorial horizon = 13°-20' For positive equation of centre Periphery on the Eastern unmandala i.e. equatorial horizon = 13°-20' On the meridian = 13°-40' On the Western unmandala = 14°-0 In the case of the Moon, for negative equation of centre. On the Eastern unmandala = 30 - 44 On the meridian = 31 - 36 On the Western unmandala = 32 - 28

." "Include all Sanskrit shlokas, sutras, mathematical derivations, tables, numerical examples, Hindi translation, commentary, and footnotes. Prefix each line with line marker like , ." Let's represent the fraction cleanly using standard text notation, e.g., 1/3 × 1/120 = H sin Z / 360 or similar clean text. Wait, let's see how lines correspond to the printed page. Let's trace every printed line:

Line 1: `217` (page number at the top)
Line 2: `For positive Equation of centre.`
Line 3: `On the Eastern unmandala = 30 - 44`
Line 4: `On the meridian = 31 - 36`
Line 5: `On the Western unmandala = 30 - 44`
Line 6: `In between the meridian and the unmandala, pro-`
Line 7: `portion is to be used. If by H sine Z equal to R, there`
Line 8: `is a difference of 20' in the periphery of the Sun, what`
Line 9: `will it be for an arbitrary H sin Z? The result is`
Line 10: `H sin Z × 1/3 × 1/120

218 (3) Consider for the Sun first for negative equation of centre. (a) On the east, the negative equation being increased, elevation is effected. (b) On the west, the negative equation of centre being rendered less, elevation is effected. (c) For positive equation of centre, on the east it being lessened, again elevation is effected. (d) For +ve equation of centre on the Western side it being increased again it is elevated. Thus with respect to the Sun, the stipulated correction effects elevation in all the cases which accords with the effect of the modern refraction. Then let us consider the case of the Moon. (e) In the case of negative equation of centre, it being lessened in the East, the effect is depression. (f) And being increased in the west, the effect is depression. (g) In the case of positive equation of centre, it being reduced in the east, elevation is effected. (h) And in the west, it being reduced, elevation is effected. Thus in the case of the Moon, the phenomenon is recorded as depression in the case of negative equation of centre. Out of these two cases again the equation of centre being lessened is truly desirable as the Hindu equation of centre is in excess of the true value. So it need not be interpreted as depression. Regarding the other case, the error might have been due to the fact that a lesser parallax being taken, whose effect is to depress

219 the celestial body, depression might have been noticed during the course of an eclipse wherein conjunction or opposition had to be belated; or again the greater equation of centre as was postulated for the Sun, than what it should be when his equation of centre was negative might have depressed the Sun, so that the Moon had to be depressed to arrive at the correct moment of conjunction or opposition. Thus this correction of Natakarma which was accepted by Bhāskara on the reported Āgama of Brahmagupta and also on the right endorsement of Chaturveda, must have been in fact no other than the effect of the phenomenon of astronomical refraction, and what further strengthens this observation is the prescri- ption of H sin Z which is proportional tan Z. Also the modern formula being A tan Z where A = 58″ approxi- mately, when Z is sufficiently large, the magnitudes given by Brahmagupta are of the same order as that of the moderns. This correction of Natakarma really reflects much credit on the ancient Hindu observations. Verse 70. Computing the planetary position for a given moment. The daily motion of the planet being multiplied by the time that has elapsed or that is to elapse at which the planetary position is to be found, and divided by 60, and the result being substracted from or added to the planetary position found, will render the position hold good for the moment in question. The Sun and the Moon will become by this process of what is called तात्कालिकीकरण equal up to minutes for the moment of conjunction or opposition. For opposition only the Rāśis differ whereas the degrees, minutes and seconds in their positions will be equal whereas for opposition, the positions are equal in all respects ie. Rāśis, degrees, minutes and seconds too. Comm. The meaning is clear.

220 Verses 71 to 75. Obtaining what are called Sukṣma- nakṣatras. The computation of the nakṣatras done as pres- cribed before, is only approximate. Now I shall give the method of obtaining what are called Sukṣma- nakṣatras as prescribed by the Ṛiṣis that are required to note auspicious occasions regarding marriages, journeys etc. people who knew about it, told that the six stars Viśākha, Punarvasu, Rohiṇī, and the three Uttaras or Uttaraphālgunī, Uttarāṣādhā, Uttarabhadrā have the duration of one and half stars ie. 3/2 × 790′ — 35″ = 1185′ — 52″. The six stars Āslesha, Ārdrā, Swāti, Bharaṇī, Jyeshṭhā and Śatabhishak have half the duration of a star ie. 395′ — 17″. The remaining 15 alone have one nakṣatra duration ie. 790′ — 35″. A star's duration is the mean daily motion of the Moon ie. 790′ — 35″. The sum total of all the above 27 stars being subtracted from 360°, give the duration of the star what is called Abhijit which occurs after Uttarāṣādha and before Śravaṇa. To obtain the star in which a planet is situated, convert its longitude in minutes of arc and substract the durations of the stars from Aświni as many as could be substracted . The number of stars whose durations are thus substracted are deemed to have elapsed. The remainder is called the gata or elapsed portion of the current star and the difference of this gata and the duration of the current nakṣatra is called the Ēṣya, ie. unelapsed portion. To obtain the elapsed time or the unelapsed time of the current star the gata or the ēṣya is to be multiplied by 60, and divided by the daily motion of the planet concerned the result being in nādīs. Comm. One line in the verse 72 is evidently missing which should name Rohiṇī and the three Uttaras. We are able to know them from Bhāskara's commentary as well as from Brahmagupta and Śrīpati. In the course of the commentary Bhāskara reiterates what was stated

221 by Brahmagupta, that Ṛiṣis like Puliśa, Vasiṣṭha and Garga spoke about these Sukṣmanakṣatras. The duration of Abhijit calculated as directed is 254' — 18''. The computation as directed is easy for understanding. The reason for the durations indicated is not clear but is to be taken as based on Astrology. Verses 76, 77. The duration of the planets’ transit into new Rāśis and the duration of the interval between successive stars, tithis, Karaṇas and yogas. The disc of the planet multiplied by 60, and divided by its daily motion gives the nāḍīs of transit of the planet from Rāśi to Rāśi. This duration is considered to be holy for performing Vedic rites. It is the holiest with respect to the Sun’s transit in particular. A planet in its transit gives partly holy results not so much as the Sun, depending upon the nature of the previous and succeeding Rāśis. The duration of Sandhi for tithis is got by dividing the disc of the Moon expressed in seconds by the difference of the daily motions of the Moon and the Sun; so also with respect to Karaṇas. The Sandhi between two Nakṣatras is obtained by the same measure of the disc of the Moon expressed in seconds of arc being divided by the Moon’s daily motion. The Sandhi between two yogas is got by dividing the same numerator by the sum of the daily motions of the Sun and the Moon. Comm. (1) The Sandhi is the period that elapses during the transit of the disc concerned between the Rāśis and nakṣatra divisions. With respect to tithi, Karaṇa and yoga, the divisions are imaginary not being seen in the Sky and the disc concerned is that of the Moon, and not that of the Sun though the Sun’s motion is also taken into account. We say a transit from a division to another is current so long as the disc lies partly in the previous and partly in the latter. So the Sandhi begins when the disc touches the next division and ends when its hind part

222 touches the previous division. In other words it is the interval between the first contact of the disc with the succeeding division and the last contact with the previous division. (2) In the case of the Sun the duration of the Sans- krānti is equal to (32' ½) / (59 — 2) of a day taking the maximum magnitude of the disc and the minimum daily motion approximately = 65 / (2 × 57) × 60 nāḍīs = 650 / 19 = 34 — 12 nāḍīs approximately or more accurately (32 ²¹/₄₀) / (56 ¹¹/₁₂) = 1301 / 40 × 12 / 683 = 3903 / 6830 of a day = 3903 / 683 × 6 nāḍīs = 23418 / 683 = 34 — 17 nāḍikās. Taking liberal boundaries, the वृद्धकारिका or the elders saying is विंशतिः पूर्वे, विंशतिः परे ie. 40 nāḍikās on the whole. Here ends the Spaṣṭāshikāra.

ॐ THE TRIPRAŚNĀDHIKĀRA Verse 1. The purport of this chapter. Pandits say that this is the science of time in as much as, herein there is described the method of knowing the direction and the point of space (where a celestial body is situated) given the time. Hence I expound that chapter, which gives that knowledge and which abounds in very important statements, which forms the quintessence of the science of astronomy. Comm. This chapter is called Tripraśnādhikāra, since this deals with the three questions pertaining to the direction and the point of space of a celestial body for a given time ie. dealing with Deśa, Dik and Kāla. In this chapter we come across the Hindu methods of spherical trigonometry, and gnomonics or Śaṅkuvedha or observations with the help of a gnomon. Also we find herein a usage of what is called ‘Golayantra’ or the armillary sphere, which helped the Hindu astronomers to solve all diurnal problems. We find herein Bhāskara excelling himself. This chapter abounds in a good number of technical terms and without a knowledge of this chapter, no one could call himself a Hindu astronomer. Verses 2-4. To compute what is called lagna given the time. The lagna or the ascendant as it is called or the point of intersection of the ecliptic with the horizon at a given point of time is obtained as follows. Obtain the Sāyana longitude of the Sun at the point of time at which it is required to find the lagna. Supposing the Sun is in the rth degree of a particular Rāśi, the number of asus which

224 give the rising time of the arc of (30 — r)° of that Rāśi are called the Bhogyāsus, or the asus which are taken by the remainder of the Rāśi to rise at the place. They are equal to. ((30 — r) × T) / 30 where T gives the asus of the rising time of that Rāśi. The Bhuktāsus on the other hand are the asus which pertain to the rising time of the arc of the first r° of that Rāśis, which are therefore equal to (r × 𝒯) / 30 ; from the given time substract the Bhogyāsus formulated above; then substract also the rising times of as many subsequent Rāśis as could be substracted. Let 'R' be the remainder of the time given. If t be the rising time in asus of the next Rāśi, (R × 30°) / t gives the number of degrees by which the lagna has advanced in the next Rāśi. These degrees added to the previous Rāśis beginning from ♈ the equinoctial point give the Sāyana or the modern longitude of the lagna. From this Sāyana longitude if we substract the Ayanamsa, we have the Nirayana or the Hindu longitude measured from the Hindu Zero-point of the ecliptic. If, however, the given time after Sun-rise expressed in asus say 'a' falls short of the Bhogyāsus defined above, then, (a × 30) / 𝒯 where 𝒯 is the rising time in asus of the Rāśi in which the Sun is situated, added to the longitude of the Sun, gives the Sāyana longitude of the lagna. Comm. The substance of these verses, though appears to be simple, yet is complicated which can be better under- stood with the help of a figure (Ref. fig. 30). Let SEN be the horizon, ♈ER the celestial equator and ♈AL the ecliptic where L is the point of lagna. Required to find ♈L the Sāyana longitude of L from which if Ayanamsa be substracted we get the Hindu

225 longitude. Let rA, AB, BC, CD, DF be the successive Sāyana Rāśis called Sāyana Meṣa, Sāyana Vṛṣabha etc. (The Nirayana Rāśis also starting from the Hindu zero-point are called Nirayana Meṣa, Nirayana Vṛṣabha etc. and if in Hindu Astronomy we use the words simply as Meṣa, Vṛṣabha etc. especially in panch- āngas ie. the Hindu almanacs, we have to construe them as belong- ing to the Nirayana or the Hindu system whose zero-point is called Fig. 30 the first point of the constellation Aswini and not r). rL = rD + DL = an integral number of Rāśis, say, ‘n’ of them ie. n × 30° + DL, where DL is the arc of the Rāśi carrying the Lagna L. The question then resolves itself into knowing how many Rāśis precede ‘D’ from r and what the measure of DL is. The data are (1) the Nirayana longitude of the Sun as computed by the methods of Hindu astronomy (2) The Ayanamsa of the year ie. ‘ro’ where o is the Hindu zero-point of the ecliptic. (3) The time after Sun-rise at the place, given in Sāvana units at which we are required to find the Lagna. Finding the Lagna at a given point of time at a given place is not only necessary in astronomy but its importance is more in what is called horary astrology or Muhūrta Sāstra, whose purpose is to fix an auspicious moment for the performance of marriages etc. as well as in astrology in casting a horoscope. The finding of the rising times of the various Rāśis at the equator, as well as at a given place was dealt with in the previous chapter. Those rising times are found in sidereal units, a sidereal day being divided into 60 nādis, or 60 × 60 Vinādis or 60 × 60 × 6 asus. These units are of constant magnitude since a sidereal day, which is 29