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

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.

235 Fig. 35 Verse 9. Having determined the East-West line as mentioned, by drawing with a compass with the same radius on either sides of Eω (Eω is the East-West line through the centre of the circle drawn parallel to the rectified East- West line namely E′ ω′ where E″ ω′ is the approximate East-West line obtained by the join of the extremities of the shadows) two arcs which intersecting each other form a fish-like enclosure as shown in the figure and as such called Matsya meaning a fish, and joining the ends of the fish namely f, g and producing fg, we have the south and the north points s and n. Or again, the north-south line could be had from Fig. 36, where EC is a rod held in the direction of the north-pole as seen by the eye at e and EA and CB the lines indicated by the plumb-line called अवलम्बकसूत्र A and B being on the plane and AB joined passing through N, the north- point. Or again the directions could be determined as follows from a single shadow S namely Cω′ in Fig. 35,

236 calculating the bhuja ω'M and Koti CM; Bhuja and Koti being known, and the shadow Cω' being drawn, holding two rods whose lengths are equal to the bhuja and Koti perpendi- cular to each other, one extremity of the bhuja-rod being held at ω' and one extremity of the Koti-rod being held at C, and the rods making a right angle at M, then the bhuja-rod determines the north-south direction and the Koti-rod the East-West dire- ction. Fig. 36 Verse 10. The Bhuja is defined as the distance of the extremity of the shadow from the east-west line where the S'anku or gnomon is placed at the intersection of Eω and NS. Koti = √(S² − b²) ∴ Koti will be in the East-West direction. So Chayā-koti = K sin z cos a (7). Comm. Easy. Verse 11. The Chayākarṇa, K is equal to √(S² + 12²), so that √(K² − 12²) = S or √[(K + 12) (K − 12)] = S. Comm. Easy. Verse 12. The S'anku is also called Nara or Nā. The zenith-distance of the Sun at Noon when the Sun is in ♈ is the latitude of the place, called pala or Aksha; the altitude then is called lamba or colatitude. Comm. The word S'anku we have previously used for the gnomon. It is also used for the H cosine of the zenith-distance and to differentiate it from the previous S'anku called Dwādasāṅgul'a-S'anku or twelve-unit-length S'anku, it is termed Mahā S'anku and occasionally Iṣṭa-S'anku. Mahā S'anku or Iṣṭa S'anku = H cos z (8). Thus in figure (37) ☉M = H cos z. In the fig. where gn = gnomon, go = S

237 the shadow, ☉ = the position of the Sun whose zenith- distance is ☉ z. If z' be taken as the zenith ☉ z' meas- ures the zenith-distance whereas if z be taken as the zenith ☉ z is the zenith-distance. The apparent inconsistency that both ☉ z' and ☉ z are taken as the zenith-distance is not there if we consider the ^ ^ zenith-distance as the angle ☉ nz' = ☉ oz. O ☉ = R, ☉ z = z, so that ☉ L = H sin z, and LO = ☉ M = H cos z = Śanku or Nara or Nā. it is called Nara or Nā which means man, because a man may consider himself as a gnomon, which is called Śanku. So the word is also applied to the parallel H cos z parallel to the gnomon and called Mahāśanku. H sin z is called Drigjyā (9) zenith-dis- t

238 Fig. 38 plane of the meridian - where E₁ is the centre of the sphere. (4) The fourth latitudinal triangle is E₁ S₁ F₁ the projection of ESF (Fig. 21) on the meridian plane where E₁ F₁ is the Sama-Sanku or the Sanku of the celestial body when it is on the prime vertical. E₁ S₁ is the Agrajyā as mentioned, S₁ F₁ is what is called Taddṛti. (5) The fifth latitudinal triangle is E₁ B₁ F₁ where E₁ B₁ is Krāntijyā or H sin δ, E₁ F₁ is the Sama- Sanku defined above and B₁ F₁ is what is called the higher segment of the Taddṛti which is equal to Taddṛti minus Kujyā or Kshitijyā. (6) The sixth latitudinal triangle is E₁ D₁ B₁ where E₁ D₁ is called the first segment of Agrajyā, D₁ B₁ is what is called un-mandala Sanku or H cos z of the celestial body when it is on the unmandala or the Equatorial horizon and E₁ B₁ Krāntijyā.

239 (7) The seventh latitudinal triangle is D₁ S₁ B₁ where D₁ S₁ is the second segment of the Agrajyā, S₁ B₁ is the Kujyā, and D₁ B₁ is the unmandala Sanku. (8) The eighth latitudinal triangle is B₁ L₁ F₁ where B₁ L₁ is equal to the first segment of Agrajyā, L₁ F₁ is the higher segment of Agrajyā, L₁ F₁ is the higher segment of the Sama-Vritta-S'anku, and F₁B₁ is the upper segment of Taddṛti mentioned before. Comm. It was already mentioned that a latitudinal triangle is such a right-angled tri-angle constituted by the chords of the celestial sphere where the angles in the triangle are ϕ, 90 − ϕ, 90°. The side opposite to ϕ is called the Bhuja, that opposite to (90 − ϕ) is called Koti and the third Karna. Such triangles are not only eight as have been mentioned, but many more will be there as mentioned by Bhāskara. They are all formed as mentioned by him by the intersections of the diurnal paths and the celestial equator with the circles of the sphere namely horizon, prime vertical, meridian, Equatorial horizon and declination circles. These circles clearly intersect at ϕ or 90 − ϕ. The eight triangles mentioned are those whose elements will be entering computations. There is another important latitudinal triangle with which we have to deal later namely that formed by what is called Hriti, S'anku, and S'ankutala (Fig. 39). O₃K = Agrā ; O₃C = S'anku-tala ; CN = S'anku-bhuja = CK ∴ Agrā = Sankutala + Sanku-bhuja Agrā = R (H sin δ / H cos ϕ) ; S'ankutala = H cos Z tan ϕ S'anku-bhuja = (H sin z H sin a) / R (Ref. fig. 40') ∴ R (H sin δ / H cos ϕ) = H cos z tan ϕ + (H sin z H sin a) / R I

240 Fig. 39 or in modern terms sin δ = sin ϕ cos z + cos ϕ sin z sin a as derived from the triangle PZS. Formula I is with respect to the Mahā Śaṅku ⊙ L of fig. 40; if it be reduced to the Śaṅku of the gnomon the Śaṅku-bhuja ⊙ N becomes the Chāyābhuja pr which will be equal to (H sin z H sin a / R) × (K / R) = K sin z sin a whereas, Agrā becomes (R H sin δ / H cos ϕ) × (K / R) = (K sin δ / cos ϕ) which is called Karṇāgrā and Śaṅku-tala becomes (H cos z tan ϕ × K / R) = K cos z tan ϕ = 12 tan ϕ = s, which is a constant quantity. It will be noted that Karṇāgrā differs from point to point since K differs from time to time during a day. In fact Karṇāgrā is the perpendicular dropped from the extremity of the shadow

pages! The page itself is in English, with English text. We should transcribe the text as it is (English, transliterated Sanskrit, numbers, symbols).

Let's look at: ^ ^ ^ ^ vs the angles Â, B̂, Ĉ, D̂, the angles between the vertical Wait, what if we do: Wait, let's look at: If an OCR line detector is used, it detects: ^ ^ ^ ^ Wait, does Tesseract detect ^ ^ ^ ^? Usually Tesseract might detect it as ^ ^ ^ ^ or ^ ^ ^ ^. Wait, but if it's considered a line: Let's see what happens if someone DOES consider it a line: If we omit it, we have 2 fewer lines! Wait, let's count total lines: With carets as separate lines: 241 on the line parallel to the East-West line and north of it ... If we look at the image, those carets are very distinct, printed with substantial space: ^ ^ ^ ^ Wait, let's look at the first caret line: ^ ^ ^ ^ It is indented to match A, B, C, D. And the second caret line: ` ^ ^

242

Bb = Sama-S'anku, and Dd unmandala S'anku. O₂b is not actually the Agrajyā but parallel and equal to it, since Agrajyā is the H sine of SE of fig. 21, which is the per- pendicular from S on Eω. Similarly do₄ is not the second segment of Agrajyā but a parallel and equal segment. Thus Bb is the perpendicular distance between EE' and FF' ie. Eω and a parallel through F to Eω (fig. 21.) Dd is the perpendicular distance between DD' and BB'; bo₂ is the perpendicular distance between Eω and SS'; do₄ the perpendicular distance between DD' and SS' (fig. 21) ; Bo₂ the perpendicular distance between FF' and SS' ; Do₄ the perpendicular distance between BB', SS'. In this fig. 39, O₁A is called Hṛti, Co₃ = Ishta-Hṛti or any arbitrary hṛti. Taddhṛti Hṛti and Kujyā are special cases of Ishtahṛti. Hṛti is the maximum of Ishtahṛti. Calling Aa, Bb, Cc, Dd S'ankus in general ao₁, bo₂, co₃, do₄ are called S'ankutalas. Aa is called Dinārdha-S'anku or the S'anku of the mid-day ; whereas Dd is the unmandala- S'anku and Bb Sama-S'anku. Cc is called Ishta-S'anku. Perpendiculars from A, B, C, D on the plane of the prime vertical are called S'anku-bhujas. Since B is on the prime-vertical itself, the S'anku-bhuja is zero and at this point BO₂ is Agrajyā. In the arbitrary case at C, the perpendicular from C on the plane of the prime- vertical being S'anku-bhuja, which is equal to the per- pendicular from C on Eω, and Co₃ being the S'anku-tala, and since the perpendicular distance between ML and Eω is the Agrajyā, which is equal to the sum of O₃ C and the S'anku-bhuja S'anku-tala + S'anku-bhuja = Agrajyā. (10) which is a different expression of (5). We shall present this analytically. Putting S'anku = H cos z, and using the latitudinality of Cco₃ (H cos z) / 12 = Co₃ / s = Co₃ / K applying similarity with the first fundamental latitudinal triangle where s = equinoctial shadow, and K the Viṣuvat-Karṇa. Hence we have

243 Co₃ = Ishta-hṛti = K/12 H cos z = (H cos z)/(H cos φ) = (cos z)/(cos φ) (11) Co₃ = Śanku-tala = s/12 H cos z = H cos z tan φ = (H cos z H sin φ)/(H cos φ) = R cos z tan φ (12). From the triangle PZS, we have the formula sin δ = sin φ cos z + cos φ sin z sin a written under verse 8 ∴ (sin δ)/(cos φ) = cos z tan φ + sin z sin a or in the Hindu form (R sin δ)/(cos φ) = H cos z tan φ + (H sin z H sin a)/R . I. We saw under verse 8 that (R sin δ)/(H cos φ) = (RH sin δ)/(H cos φ) = Agrajyā (formula 5). Also we have from (12) above H cos z tan φ = Śanku-tala. From fig. 40, if SM be drawn secondary to the prime-vertical, the right-angled spherical triangle SMZ gives [L14

244 Hṛti = O₁A = OO₁ + OA = Kujyā + H cos δ or Dhujyā VI since OO₁ = DO₄ = Kujyā. The line in the Equatorial plane cor- responding to Kujyā is Charajyā which is equal to R tan φ tan δ (13) so that we can write Kujyā = R sin δ tan φ = H sin δ tan φ (13'). Also from VI [Fig. 40] Antyā = R + Charajyā = R + R tan φ tan δ (14) which gives us the duration of half the day where R gives 6 hrs and Charajyā the increment in day due to φ and δ. [Fig. 40] Verse 18. To obtain the magnitudes of the various chords or the elements of the latitudinal triangles. The elements or the sides of all these latitudinal triangles are mutually derivable from similarity. Comm. Easy. Second half of Verse 18. The radius multiplied by the Kotis or Bhujas and divided by the Karnas gives H cos φ and H sin φ. Comm. The seven triangles except the second, of the eight latitudinal triangles are compared here with the second. Remembering Bhujas are the sides of the triangles opposite to φ and Kotis opposite to 90 − φ,

245 Bhuja / Karṇa in any triangle = Bhuja / Karṇa in the second latitudinal triangle = (H sin φ) / R ∴ (R × Bhuja) / Karṇa = H sin φ Akshajyā or Palajyā (15) Similarly Koti / Karṇa in any triangle is equal to the Koti / Karṇa in the second latitudinal triangle = (H cos φ) / R so that (R × Koti) / Karṇa = H cos φ = lambajyā (16). Verse. The arcs of H sin φ and H cos φ are respec- tively the Akshamsas and lambamsas as they are called ie. latitude and colatitude. H sin φ and H cos φ are also obtainable thus √(R² - H sin² φ) = H cos φ and H sin φ = √(R² - H cos² φ) Or again (H sin φ × Koti) / Bhuja = cos φ and (H cos φ × Bhuja) / Koti = H sin φ where the Bhuja and Koti may belong to any latitudinal triangle. Verse 20. Agrajyā can be had by multiplying Krāntijyā by Karṇa of any lat. triangle and divided by its Koti. Also Sama-Sanku = Karṇa / Bhuja × Krāntijyā and (Sama-Sanku × Karṇa) / Koti = Taddhṛti. Comm. The first of these statements pertains to the similarity of the third triangles to the others. The second of the statements pertains to the similarity of the fifth latitudinal triangle to others whereas the third pertains to that between the fourth and the others.