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

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

226 the period of diurnal rotation of the earth is of constant magnitude. In the data given above, the time is normally given in Sāvana units, ie. mean solar units. A Sāvana day is of 60 × 60 × 6 + 59 asus, which is greater than a sidereal day by 59 asus because the mean Sun advance by 59' — 8'' per day among stars and as such the Sun-rise the next day is belated by 59 asus approximately. The given time in Sāvana units could be converted into sidereal units as per the approximate ratio 21600 : 21660 which means for every Sāvana nādi we have to add one asu or for every hour four seconds. Then the procedure of finding the Lagna will be a little different from what it would be if we proceed with the Sāvana units. The complex- ity mentioned before, arises out of the Sāvana units, and this has been explained by Bhāskara in Golādhyāya under the title तात्कालिकीकरणवासना in the beginning of the chapter called त्रिप्रश्नवासना. Now the procedure will be explained. Let (Fig. 30) S' be the position of the Sun at the Sun-rise and S his position at the time at which the Lagna is to be found so thst the Sun has advanced by S'S ie. which measured in minutes of arc is called gati- kalās. In a mean solar day these gatis-kalas would be 59, and in the given time after Sun-rise they will be proportional. The time given after Sun-rise pertains to the rising time of the arc S'L which we have in sidereal units. We shall first find the Lagna using sidereal units by measuring S'L, so that we may latter understand Bhāskara's reasoning for his stipulation of तात्कालिकीकरण on which basis he finds the Lagna with the Sāvana units taking the position of the Sun at S instead of S'. We know the position of the sun at sun-rise ie. S', so that the rising time of As' ie. the previous arc in the Rāsi in which the Sun is situated is given by Bhuktasus defined above and the rising time of S'B the remaining arc of the Rāsi is given by Bhogyāsus. Subtract from the given time converted into sidereal units if they are not

, CD.->BC, CD.- no periods between B and C! In line 10:s'B, BC, CD and T the rising time of the Rāsi ... Wait, what is that symbol? Wait, look at&c.again: &has a loop at the top, a loop at the bottom, and a stroke to the right. Thenc. Look at the image: it is literally&c.! Wait, let's look at: the Rāsi &c. Wait, does it saythe Rāsi &c.? Wait, "T the rising time of the Rāsi &c." -> why "&c."? Because T is the rising time of the Rāśi in which DL lies! (or the Rāśi containing DL, etc.?) Wait, look at formula on line 3: formula is [S'B (indegrees) × T] / 30 where T is the rising time of that Rāsi expressed in asus. In formula on line 8: the magnitude of DL in degrees by the formula (R × 30) / T where ‘R’ is the remainder in time after subtracting the rising times of s'B, BC, CD and T the rising time of the Rāsi ...` Wait, in line 3: "T is the rising time of that Rāsi

228 the argument of an imaginary opponent namely “ Is the time given measured after Sun-rise Sāvana (mean solar) or Nākṣatra (ie. sidereal)? If it be the former, how is it you are subtracting the rising times of sA, AB etc. which are sidereal from your Sāvana units? Further, should you not take the position of the Sun at Sun-rise namely S', because the time given is what has elapsed after Sun-rise ? Also, why should you complicate matters by accepting Sāvana units when the question is simple if dealt with sidereal units ? To this Bhāskara answers as follows — “ True it is, what you say. Generally in day to day life, time is given only in Sāvana units and not side- real. Further you cannot avoid Sāvana measure, for, in the case of an arc moved by a planet, in its diurnal circle time is measured in the Sāvana units pertaining to the planet. (The Sāvana units of a planet are different from what they are for the Sun depending upon the arc moved by the planet in question during a day). These Sāvana units are what are termed Kṣetra-Vibhā- gātmika or what depend upon the arc covered in the diurnal path in contradistinction to the Kāla-Vibhāgat- mika units or sidereal units. (In other words pure time is what is measured in sidereal units which is a standard measure, whereas time which has the bias of the motion of the planet also ie. which we seek to measure by the arc moved by a planet in its diurnal path, is Kṣetra-Vibhāgāt- mika). Thus having had to accept the Sāvana measure also, we seek to proceed on that basis, though we could convert the Sāvana measure into the sidereal and proceed without complication ”. The argument which Bhāskara gives for तात्कालिकीकरण is further as follows. The measure of the arc SL using the rising times in asus ie. sidereal units, is the Sāvana measure of the arc S'L done in sidereal units. Instead of subtracting the rising time of S'A from the given time converted into

229 sidereal, we subtract the rising time of SA from the given Sāvana time and the result will be the same, for, — SA = — (S′A — S′S) = S′S — S′A. which means subtracting SA from the time tantamounts to increasing by S′S and subtracting S′A. This increase by S′S is add- ing what have been defined as gati-kalas so that automati- cally the Sāvana units got converted into sidereal units, by taking the position S and subtracting SA instead of taking the position S′ and subtracting S′A. The result is the same. So, it is said ‘तात्कालिकार्कस्य’ in the beginning of the verse 2. In the second case mentioned in verse 4, if the time given, falls short of the Bhogyāsus, then simply (x × 30) / T where x is the time given in asus and T the rising time of the Rāsi in which both the Sun and the lagna are then situated gives the arc in degrees which if added to the longitude of the Sun gives that of the lagna. Here also the position ‘S’ counts. Verses 5 to 6½. To find conversely the time that has elapsed after Sun-rise given the lagna. The Bhogyasus of the Sun and the Bhuktāsus of the Lagna together with the rising times of intermediate Rasis gives the time required. If the Sun and the Lagna both be in the same Rāsi, then the arc in between them, multiplied by T and divide by 30, gives the time required. If, however, the longitude of the Lagna falls short of that of the Sun, ie. if the Sun be below the horizon, (in this case SL > 180°) then finding the time of rising of SL and subtracting from a day, we have the time of the Lagna before Sun-rise.

230 However, here. there is one complication if we consider the तात्कालिकार्क ie. s the Sun at the given time. This position of s cannot be had unless we know the time, which is itself required. So we get in the first place the time pertaining to S'L, which is the time measured in sidereal units the position S' being that of the Sun at Sun- rise. If we don't take recourse to convert this time in sidereal units to Sāvana units using the proportion between them, then the alternative is to obtain the position S using the time obtained and then calculate the time again in Sāvana units. If the time given to find the Lagna, be sidereal, it goes without saying that we find it from S'. Also S' being given and if the time after sun-rise is required in sidereal units for a given Lagna, the method of succes- sive approximation is unnecessary. Verse 7. To find the Lagna before Sun-rise called Vilōmalagna. Suppose it be required to find the lagna before Sun- rise, given the time before Sun-rise. Obtain the then posi- tion of the Sun and find his Bhuktāsus; subtract them from the given time; from the remainder, subtract the rising times of as many Rāsis as could be, rāsis behind the Sun's position. If R be the remainder in the time after these subtractions, then R × 30/T where T is the rising time of is the next preceding Rāśi, together with n × 30, where n the number of integral Rāśis subtracted and the arc of the next Rāśi by which the Sun has advanced in his Rāśi at the time of the Lagna (known from the position of S found) the sum total of these three items being subtracted from the position of S gives the longitude of L. Comm. Easy. Verse 8. To obtain the East-West line.

231 The East-West line is roughly the join of the extremities of the morning shadow as well as that in the after-noon of a gnomon placed at the centre of a circle drawn on a plane with any arbitrary radius, when those shadows equal the radius of the circle. But this line is to be deflected keeping its western point ie. the extremity of the morning shadow fixed through a distance K (sin δ₁ ~ sin δ₂) / cos ϕ at the eastern end perpendi- cular to it, where the above distance is measured in units, which measure K. Comm. The east and west points are where the celestial equator cuts the horizon. The east point is thus the point where the Sun-rises when he is exactly at the vernal equinox. The question is how to draw the east-west line on a plane. For this we are asked to draw a circle with any radius on that plane. The plane is described here as अम्भःसुसमीकृतक्षिति that kind of sur- face as is determined by the surface of water there. Such a kind of surface forms approximately a horizontal plane not of course exactly because such a surface is really spherical, the earth being a sphere. But because the radius of the earth is sufficiently large, we can take such a surface to be a horizontal plane for all practical purposes. Having drawn a circle place the gnomon vertical at the centre. In the morning note the extremity of the shadow when it equals the radius. In the afternoon also mark the point when the shadow equals the radius. Join those two points. It represents roughly the east-west line, roughly because the Sun's declination changes in between the two moments however small the change might be. Ignoring the change in the declination, this line will be east-west because of the following reason. The length of the shadow is 12 tan z where 12 units are the measure of the gnomon. But since the shadow on both the occasions is equal to the radius of the circle z, the zenith-distance

232 will be the same on both the occasions. Then the spherical triangles PZS₁ and PZS₂ where P is the celestial pole, Z the Zenith and S₁ and S₂ are the positions of the Sun on the two occasion, are congruent their three sides being respe- ctively equal, provided we take PS₁ = PS₂ ie. 90 — δ₁ = 90 — δ₂ ie. δ₁ = δ₂ on both the occasions. When the two triangles are thus congruent, PẐS₁ = PẐS₂ ie. S₁ and S₂ are equidistant from the plane of the Prime-Vertical. Hence the extremities of the shadows will be equidistant from the East-West line ; or this may be seen in another way ; S₁ S₂ will be perpendicular to the meridian plane and as such parallel to the plane of the Prime-Vertical. The correction mentioned in the verse is known as the Agrāntara correction which was originally given by Chaturvedā chārya and then accepted by Sripati. Why it is called Agrāntara is because it is a change in what are called Karnavrittāgras of the two occasions where we shall see in due course that the formula for Karnavrittāgra is (K sin δ) / (cos φ) where K is hypotense of the gnomonic triangle formed by the gnomon and its shadow S at any place and time. This correction is a very minute correction and as a matter of fact could be ignored. But the fact that the correction was cognized and correctly formulated testifies to the knowledge of the sphere which the above acharyas had. Assuming the formula of the Karnāgra here (it will be proved by us later in this chapter) if δ₁ and δ₂ be the declinations on the two occasions respectively the Karnāgras will be (K sin δ₁) / (cos φ) and (K sin δ₂) / (cos φ) , K being equal on the two occasions because the shadows are equal. Hence the correction being the difference of the Agrās, it is [K (sin δ₁ ~ sin δ₂)] / (cos φ) as stated by Bhās- kara,

233 We shall now prove it in modern terms from the spherical triangle PZS. We have the formula sin δ = sin ϕ cos z + cos ϕ sin z sin a where PZS = 90 — a, a being the Hindu azimuth measured from the East point. Multiply the above equation by K and divide throughout by cos ϕ where K is called the Chāyākarṇa equal to √(12² + S²), S being the gnomonic shadow at the moment. Hence K sin δ / cos ϕ = K cos z tan ϕ + K sin z sin a (1) Fig. 31 Fig. 32 But from Fig. 31, K cos z = 12, K sin z = S, (2) and from Fig. 32, S sin a = b where b is called the Chāyābhuja ie. Chāyābhuja = K sin z sin a (3). Thus K sin δ / cos ϕ = 12 tan ϕ + b. But again from Fig. 33, when ☉ the Sun at vernal equinox is on the meridian and as such has a meridian zenith-distance equal to ϕ, 12 tan ϕ = s where s is called the Vishuvat-chāyā or equinoctial shadow. Thus we have K sin δ / cos ϕ = s + b (4). Again if the Sun be on the horizon, from Fig. 34, E ☉ is called the Agrā, A, so that from the triangle P ☉ N, 30

234 Fig. 33 Fig. 34 cos (90 — δ) = cos ϕ cos (90 — A) ie. sin δ / cos ϕ = sin A or in the Hindu form H sin A = R (sin δ / cos ϕ) = Agrajyā (5). This Agrajyā is in a circle of radius R, and if it be re- duced to a circle whose radius is K, is will be (K / R) × (R sin δ / cos ϕ) = (K sin δ / cos ϕ) which is called Karṇāgrā. Hence we have Karṇāgrā = s + b which we shall write as a = b + s (6). This is an important formula which is going to be formulated later in verses 72, 73. In the above formula, s being constant, by differentiating δa = δb which means that the variation in the bhuja is on account of the variation in the Karṇāgrā. If in Fig. 35, Cω', CE″ be morning and evening shadows when they are equal as per the verse under comment, Mω' = b the morning bhuja, NE″ = b', the evening bhuja dE' is the variation in the bhuja ie. b — b' = δb which is formulated and equal to δa. But δa = δ(K sin δ / cos ϕ) = K δ (sin δ) / cos ϕ, ϕ being con- stant and K also being constant because the shadow S is constant and K = √(S² + 12²) = constant on both the occasions. ∴ δb = δK = (K / cos ϕ) (sin δ₁ — sin δ₂) as stated by Bhāskara.