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

382 radius. This circle is called Kadamba-Bhrama-Vritta or the circle in which the pole of the Ecliptic revolves round P (due to diurnal revolution of the Earth). In that circle Hsine of an angle will be H sin δ............Or again draw the great circle with the planet's position as the pole, called the horizon of the planet. The arc intercepted on this circle between the Ecliptic and the celestial Equator will be Āyana Valana and that intercepted between the celestial Equator and the horizon is the Ākṣa Valana; and the arc intercepted between the Ecliptic and the horizon is the Sphuṭa Valana. Or again draw a circle with K as centre and radius ω = 24°. This circle is called the Jina-Vritta where the word Jina means 24. Let a secondary to the Ecliptic passing through K and K' the poles of the Ecliptic revolve with KK' as fixed. When this revolving circle passes through Cancer (Sāyana) it will be passing through P. The angle turned through by this circle from Cancer, will be equal to the angle turned through from P. The Hsine of that angle in the Jina Vritta will be H sin δ of a longitude equal to that angle. This is the Āyana Valana and it arises at the end of Dyujyā, since the north-polar distance of the planet is (90-δ) whose Hsine is Dyujya ie. H cos δ. The corresponding Āyana Valana in a circle of radius R is got by multiplying by R and divided by H cos δ. Let us clarify Bhāskara's mind. (Ref. fig. 81) Let PBD be the Jina Vritta drawn on the sphere with K, the pole of the Ecliptic as centre and ω=24° as radius. Let a revolving secondary to the Ecliptic coincide initially with KC where C is Cancer. Let it occupy subsequently the position KM where M is the centre of the Moon's disc taken to be on the Ecliptic as is the case very approxi- mately at the moment of an eclipse. Now the Āyana Valana is the angle KMP. Let MA be the declination of M. Produce MP to L such that MK=ML=90°. Hence

383 Fig. 81 PL = δ since PA = 90° and LM = 90°. The Āyana Valana K̂MP is measured by the arc ML where ML is an arc of the Grahakṣitija or the horizon of the planet M (ie. the circle with M as centre and 90° as radius drawn on the sphere or what is the same the great circle whose pole is M). PB is an arc of the small circle parallel to KL which is an arc of a great circle. Then in the Jina Vritta sin PB = sin PK × sin P̂KB = sin ω sin (90 - λ) = sin ω cos λ ∴ sin KL = sin ω cos λ/cos PL = (sin ω cos λ) / (cos δ) = sin PMK.

384 Here sin ω cos λ is called Sa-thribha-graha-ja-kranti or the declination of a point whose longitude is 90+λ where λ is the longitude of M. As we have the formula sin δ = sin ω sin λ, sine of the declination of such a point is equal to sin ω sin (90+λ)=sin ω cos λ. When δ is very small sin PMK may be taken to be sin ω cos λ or what is the same Sa-thribha-graha-ja-krānti as is formulated by Sūrya- siddhānta. It may be doubted how sin PB=sin PK sin 90-λ. (Ref. fig. 82). Let K′ be the centre of the circle PBD, Fig. 82 K′ being in the plane of PBD. K′P and K′B are radii of this circle. Since the arc PB stands for 90-λ PK′B= 90-λ. Draw the H sine of arc PB, which is PB′. Now K′P=H sin ω, as PK′ is ⊥ar drawn on OK ∴ PB′=PK′ sin PK′B =H sin ω ✕ sin PK′B = (H sin ω ✕ H sin PK′B) / R = (H sin ω H cos λ) / R ∴ H sin KL = [(H sin ω H cos λ) / R] ✕ H cos δ = (H sin ω H cos λ) / (H cos δ) as given.

385 [चित्र: Fig. 83 — Equator, Ecliptic, Prime vertical] Fig. 83 Note 1. Bhāskara says PB' (fig. 82) is Krānti-Sinjanī; so it is because PB' = PK' sin (90 - λ) = (H sin ω H cos λ) / R = H sine of the declination of a point whose longitude is 90 + λ = Satribha-grahaja-krānti as is mentioned by Bhāskara and Sūrya-siddhānta. Note 2. If M be the centre of the Moon's disc and ABC its horizon defined above, the arc AB intercepted between the Ecliptic and the Equator is Āyana Valana, the arc BC intercepted between the Equator and the prime- vertical is Ākṣa Valana and the arc AC intercepted between the Ecliptic and the prime-vertical is Sphuta- Valana. Note 3. The analysis of Ākṣa Valana proceeds on similar lines, only we have P and N in the place of K and P. Note 4. The mistake of Lallāchārya alluded to by Bhāskara is as follows. The Āyana Valana, we have seen is zero at the Ayanas ie. the solstices and maximum at ♈ and ♎ ie. the equincotial points removed by 90° from the Ayanas. Now Hversine = R — H cosine so that when 90 — λ = 0 ie. λ = 90°, Hversine (90 — λ) = R — H cos (90 — λ) = R — H sin 90° = R — R = 0 and when 90 — λ = 90° ie. 49

386 λ = 0 Hversin (90 — λ) = R — H cos (90 — λ) = R — H sin λ = R — H sin 0° = R — 0 = R. Hence Lallācharya took by mistake that the Āyana Valana varies as Hvers (90 — λ) instead of H sin (90 — λ) since both Hversine and H sine of 90 — λ̄ are zero at the Ayanas and maximum at r and ♎. The same mistake was committed by Lallācharya in the context of the Moon’s phase also as criticised by Bhāskara as we shall see later. In fact, this latter criticism is not so justified as the former, as will be shown in that context. Note 5. If instead of taking the Āyana Valana to vary as H sin (90 — λ) we happen to take according to Lallācharya that it varies as Hversine, then in places (Ref. verses 38, 39 Valana Vāsanā, Golādhyāya) removed by 90° from the points of intersection of the Ecliptic and the prime-vertical, where there should be no Sphuta- Valana, we do get that there is some Sphuta Valana there, since the value of Hversine differs from Hsine, though these two functions happen to be zero simultaneously and maximum simultaneously. Bhāskara continues in verses 66-68 (Ibid) “ I shall now depict Ākṣa Valana by means of the hour-angle. Take the sum or difference of S'anku-Agrā and S'anku-tala according as they are of the same direction or not ; compute √(R² — B²) where B is the result ; then H sin ϕ H sin h ─────────────── is equal to H sin ξ where ξ is the Ākṣa Valana ”. √(R² — B²) Comm. We saw before in the Tripraśnādhyāya that A = S + B where A = S'anku-Agrā, S = S'anku-tala, and B = S'anku-bhuja = H sin µ where µ is represented in fig. 79. Hence √(R² — B²) = H cos µ so that the above formula gives H sin ϕ × H sin h H sin ξ = ───────────────────── which is the same as got by H cos µ the modern formula in Equation II.

887 Note. A small circle parallel to the prime-vertical is called Upa-Vṛtta. Also secondaries drawn to the Ecliptic, Equator and the prime-vertical are called Kadamba-Sūtra, Dhruva-Sūtra and Sama-Sūtra. They are also called occasionally as Kadamba-prota-Vṛtta, Dhruva- prota-Vṛtta, and Sama-prota-Vṛtta. Bhāskara proceeds to find the Āyana Valana in a very ingenious way in verses 69-74. We shall first give it a modern treatment so that we may better appreciate his genius. Let (S) be the Sun's disc. (It does not matter whether we take the Sun or the Moon). EQ is its diameter along the diurnal circle, and CL along the Ecliptic. LM is the difference of the declination of L and S. Let SL = Δ λ and LM = Δ δ. We have sin δ = sin λ sin ω ; differentiating cos δ Δ δ = sin ω cos λ Δ λ ∴ Δ δ = (sin ω cos λ / cos δ) × Δ λ ; put Δ λ = b, the angular radius of the disc ∴ Δ δ = (b × sin ω cos λ / cos δ) = (b H sin ω × H cos λ / R × H cos δ) . This gives the Valana in the disc of radius b. If that be so, what will it be on the sphere of radius R? The result is (b H sin ω × H cos λ / R × H cos δ) × R / b = (H sin ω × H cos λ / H cos δ) as got before. Let us hear Bhāskara, " put the disc of the Sun at the point of intersection of the Ecliptic and the diurnal circle. The Valana (LM of fig. 84) at the periphery of the disc is the difference of the declinations of L and S. To get the value of this let us first get the value. SL in terms of λ, the longitude of S. It is (b × B / 225) ; so that LM

388 Fig. 84 will be equal to (b × B) / 225 × (H sin ω) / R where B is the Bhogya- khanda of λ. To obtain the value of the above for a circle of radius R from a circle of radius b, we have to multiply by R/b. So, the result is (b × B) / 225 × (H sin ω) / R × R / b = (B H sin ω) / 225 . But the value of B is got as follows. 'If for H cos λ equal to R we have the first Bhogyakhanda equal to 225, what shall we have for H cos λ?' The result is (225 × H cos λ) / R . Substituting for B, we have (H sin ω) / 225 × (225 × H cos λ) / R = (H sin ω × H cos λ) / R Now, on account of declination, the Sun's disc is inclined like an umbrella. So LM of fig. 84 will take a position like L'M as shown in fig. 85 where the triangle MLL' is similar to SMO, S being the centre of the Sun's disc, O the centre of the sphere. Hence (L'M) / LM = R / (H cos δ) ∴ L'M = R / (H cos δ) × (H sin ω H cos λ) / R = (H sin ω × H cos λ) / (H cos δ) as got before '.

389 Comm. Bhāskara terms SL as the Dorjyāntara ' or the variation in H sin λ, which he knows to (H cos λ Δλ) / R. [Fig. 85: Triangle L'LM] [Fig. 86: Triangle SMO, Hypotenuse = R, SM = H Cos δ, MO = H Sin δ] Fig. 85 Fig. 86 But proceeding from first principles, as he always does, he asks us to consider the Bhogyakhanda at λ namely B. If this be for an interval of 225', what will be it be for ‘b’? The result is (b × B) / 225 . Then to rectify B, the proportion used is as used above. Bhāskara says many a time that the variation in Hsine is proportional to Hcosine. This concept he might have derived by looking at the Hsine table of 90 Hsines. Hence the argument advanced by him to rectify B is ‘If for Hcosine equal to R (at zero-value of the argument) the initial Bhogyakhanda is 225, what will it be for an arbitary H cos λ? The result is (H cos λ) / R × 225. Substituting this for B in the above, we have (b H cos λ) / 225 × 225 = (b H cos λ) / R . This expression we perceive as no other than (H cos λ Δλ) / R as equal to Δ (H sin λ), for b is to be taken as Δλ. This (H cos λ × b) / R is called by Bhāskara as Dorjyāntara meaning thereby Δ (H sin λ).

390 Then the next argument is 'If for H sin λ equal to R we have the declination equal to H sin ω, what shall we have for the above Dorjyāntara. The result is (H sin ω / R) × (H cos λ × b / R). Since this is in modern terms sin ω cos λ × b and b = Δλ, we perceive that this expres- sion is Δ (H sin δ) where δ stands for LM of fig. 84, ie. the difference of the declinations of the points S and L of the disc of fig. 84. The next argument advanced by Bhāskara, namely that on account of declination, the disc is slanted and LM gets thereby enlarged into L'M of fig. 84 and adduces proportionality from fig. 86. But this argument seems to be faulty. In fact, the magnitude of LM is got for the diurnal circle of radius H cos δ. To get its value for radius R, the result would be [(H sin ω H cos λ × b) / R²] × [R / (H cos δ)] = (b. H sin ω H cos λ) / (R. H cos δ) . Then the argument is 'If in the disc of radius b, we have Δδ equal to the above what will it be for radius R? The result would be (H sin ω H cos λ) / (H cos δ) . Note. Our argument is based on the idea that lines of the small circle namely the diurnal circle get enlarged for a circle of radius R in the porportion R: H cos δ. Bhāskara's concept of enlargement on account of slanting does not seem to be plausible because, on account of decli- nation, the disc may occupy an overhead position when the Equator is itself inclined. Thus slanting does not arise out of declination. The question might be asked as to how Bhāskara got the right answer by such an argument. He got the answer up his sleeves through the other methods he gave and he adduced this argument to get at that answer.

391 Second half of verse 21 and first half of verse 22. H cos λ × H sin ω ───────────────── = H sin θ where θ is the Āyana Valana H cos δ The direction of this Valana is that of the hemisphere north or south in which the Moon lies. Comm. This is the formula we have already derived. Regarding the direction of the Āyana Valana, the conven- tion is that it is to be considered north, if the Moon be in the northern hemisphere, otherwise south. The reason is that at the time of a lunar eclipse, the Moon being in opposi- tion, if he be north, the Sun will be south of the Equator, and the line BA of fig. 80 representing the Ecliptic which is roughly the join of the Sun and the Moon will be north of the line JI which is parallel to the equator. Thus the direction of the angle IMA gives the direction of the Āyana Valana. Latter half of verse 22 and verse 23. Sphuta Valana. The Hsine of the sum or difference of the two Valanas according as they are of the same or opposite directions, multiplied by the sum of the angular radii of the Moon and Rāhu, and divided by the radius gives the Hsine of the Sphuta Valana. Those who said that the Valana is proportional to the Hversine, do not know spherical geometry properly. Comm. The direction of the Akṣa Valana, was defined in verse 20 that it is north if the hour angle is east, other- wise south. The meaning of this convention is that the diameter of the Moon's disc parallel to the Equator when the hour angle is East, is north of the diameter which is parallel to the prime-vertical. Thus combining the two conventions regarding the directions of the Valanas, it is clear that if both the Valanas are north, the line MI is north of MG, and MA is north of MI (fig. 80) so that the

392 Sphuta Valana is equal to the sum of the angles G M̂ I and I M̂ A. Suppose MA is south of MI either falling within the angle GMI or south of MG, then clearly the Sphutr Valana G M̂ A = G M̂ I - A M̂ I or A M̂ I - G M̂ I as the case may be, which is obtained as the difference of the two Valanas. Having obtained G M̂ A as the Sphuta Valana, H sin GMI multiplied by (P+r) and divided by R gives Hsine of the Sphuta Valana to be represented in a circle of radius equal to P+r. This latter convention of repres- enting the Sphuta Valana in a circle of radius P+r is only a convention. The expression H sin (Sphuta Valana) × P+r --------------------------- gives us RN of fig. 87, R where GMA is Sphuta Valana. In other words, we are to draw fig. 87 to show the point of first contact namely A in relation to MG the line parallel to the prime-vertical. Fig. 87

393 Verse 24. Conversion of liptas into what are called Angulas. H cos z of the eclipsed body at the moment of eclipse being divided by the radius and the result being added to 2½ gives the number of liptas per angula. The time elapsed after the rise of the body being divided by the rising hour angle (both being expressed in the same units of time) and the result being added to 2½ also gives the same. Comm. While a parilekha or a geometrical drawing of the eclipse is attempted at, the problem arises as to how many liptas or minutes of arc giving the measure of the disc are to be taken to be equivalent to one angula. For example, suppose the diameter of the disc is 30'. With what radius shall we draw the disc on a board or paper? In this behalf, a convention based on observations is being mentioned. The discs of the Sun and the Moon are observed to be big at rise and small when they are on the meridian. So, taking the measure of the disc to be 30' for example, if the eclipse takes place at the rise of the disc, it is laid down to draw the disc with a radius of 15/2½ ie. 6 angulas at the rate of 2½ liptas per angula. {The word angula here mentioned might not be what we take it to be today in our daily parlance as one inch. The gnomon or S'anku was taken in those days to be of a length equal to 12 angulas. Bhāskara mentions in the beginning of Lilavati that 8 Yavas are together equal to one Angula, twentyfour angulas to one hasta, four hastas to one danda and 2000 dandas to one Kros'a, and 4 Kros'as to one Yojana. Also a vamsa is equal to ten hastas. This system discloses that, a Yojana equals 5 modern miles according to Bhāskara's estimate of the diameter of the Earth as compared with its modern estimate. (The method given by Bhāskara as to how the diameter of the Earth could be measured is found to be quite scientific as men- tioned by us before). 50

394 33 modern inches or angulas are equal to 80 angulas of Bhāskara as per the above. At this rate the gnomon's modern length would be just 5″.} Reverting to our subject, we are asked to represent the disc of 30 liptas when the eclipse takes place at noon by 30/3½ angulas counting at the rate of 3½ liptas per angula. Then the question arises as to what should be the correspondence between the liptas and angulas when the eclipse takes place in between the rise of the Moon (or Sun) and its noon. The directive is that one angula = 2½′ + (H cos z) / R = 2½ + cos z I This means, supposing Z=zenith-distance of the body to be 60°, one angula is to be taken to be equal to 2½ + cos 60 = 2½ + ½ = 3′ or 3 liptas. The reason given by Bhāskara reiterating what Sri- pati said in that behalf, as to why the Moon's or Sun's disc appears to be big at the moment of rising and small on the meridian, is that the disc is immersed in its own rays at noon and rendered small in appearance, whereas, most of the rays are swallowed by the earth or its atmos- phere at the moment of rising, making the disc appear large and easily visible. Note. Bhāskara gives the proof of the above formula I as follows. Since at the time of rise, we are taking 2½ liptas of the measure of the disc to be equal to one angula and while the disc is on the meridian, 3½ liptas are to be taken as one angula, there is an increase of one lipta for an increase of H cos z from zero at the horizon to a value equal to the Radius. So, the argument adduced is 'If for an increase of H cos z equal to the radius, there is an increase

395 of one lipta in addition to 2½, what should the increase be for an arbitrary H cos z '? The result is (H cos z / R) × 1 = cos z. Note 2. As finding cos z at the time of an eclipse implies additional arithmetical calculation, and as we have already with us the data of (1) the time elapsed after rise of the celestial body (Moon or Sun) till the moment of the eclipse and (2) half the duration of the day of the body ie. the rising hour-angle converted into time at the rate of 6° per nadi, so a rough formula is given using h in the place of z. The rule of three now used, is 'If for an unnata equal to the dinārdha or rising hour-angle converted into nadis, we have an increase of 1 lipta per angula (over and above 2½ liptas) what shall we have for an arbitrary un- nata ?' The result added to 2½ liptas, gives the formula one angula = 2½ + Unnata / Dinārdha . Dinārdha corresponds to H, the rising hour-angle and Unnata corresponds to (H-h) where h is the hour-angle at the time. Bhāskara uses the word 'Angula-liptas' meaning thereby the liptas that are to be taken to be equal to one angula while drawing the parilekha of the disc at the time of its eclipse. Verse 25. Converting Valana etc. into Angulas. The measures of the Valana (defined above) or the Sara ie. the celestial latitude of the Moon or the Rāhu Bimba or the Bhuja (defined) are to be converted into Angulas at the rate given by the above formula. While drawing a figure of the solar eclipse, the celestial latitude of the Moon is to be drawn in its own direction whereas in a lunar eclipse, it has to be drawn in the opposite direction. Comm. The first part is clear. Regarding the second statement, in as much as the centre of the Rāhu Bimba lies

at the foot of the Moon's celestial latitude, the latter has to be drawn in the opposite direction, since in the pari- lekha, the centre of the Moon's disc is to be at the centre. Verses 26 to 29. How to depict the eclipse in drawing. Draw a circle with radius equal to that of the radius of the disc of the eclipsed body and also a circle of radius equal to r+p, the sum of the radii of the eclipsed and eclipsing bodies; let directions (east etc.) be marked in the figure. In the outer circle, draw the Valanajya or the Hsine of the Sphutavala with respect to the East point, Valanajyā pertaining to the moment of first contact. In the case of the Moon, the Valanajyā pertaining to the moment of first contact should be marked from the East point and that pertaining to the moment of last contact should be marked from the West point. In the case of the Sun the reverse is to be done. If the Valana is south, it should be marked in the clockwise direction, otherwise anticlockwise. Having marked the Valanajyā in the form of a Hsine, draw the line joining the centre to the top of the Valanajyā, ie. to the point of intersection of the Hsine with the outer circle. The celestial latitude of the Moon is to be drawn from this top of the Valanajyā in the form of Hsine again. If the latitude pertains to the moment of first contact, it should be drawn from the top of the Valanajyā pertaining to that moment, and if it pertains to the moment of last contact, it should be laid off from the top of the Valanajyā pertaining to the moment of last contact. The celestial latitude pertaining to the middle of the eclipse should be drawn from the centre along the line of Valanasūtra or the line joining the centre to the top of the Valanajyā. Taking the extremities of these latitudes, circles are to be drawn with the radius of the eclipsing body to depict the eclipse at the respective moments.

397 Fig. 88 Comm. Let M₁, M₂, M₃ be the positions of the centre of the Moon's disc at the moment of first contact, at the middle of the eclipse and the moment of last contact respectively. Draw a circle with radius M₁ R₁ = r+p which is called the Manaikyārdha Vritta. Let M₁ E₁ represent the Eastern direction known as the Sama- mandalaprāchī. Draw E₁ V₁ equal to Hsine of the Valana, so that M₁ V₁ is the Krānti-Vritta-prāchī ie. the point of intersection of the Ecliptic with the Eastern horizon. Draw N₁ R₁ perpendicular to M₁ V₁ in the form of Hsine, which is the latitude (Vikṣepa) of the Moon at the moment of first contact. With R₁ as centre draw the Grāhaka- Vritta with p as centre ; this circle represents the Rāhu- Bimba. Similarly let M₃ be the centre of the Moon's disc at the moment of last contact. Draw a circle with M₃ as centre and r+p as the radius which is the Manaikyārdha- Vritta. Let M₃ W₃ be the direction to the West, the Samamandalapratichī. From W₃ draw W₃ V₃ the Hsine of the Valana, so that M₃ V₃ is now the Krāntimandala- pratīchī ie. the point of intersection of ecliptic with the western horizon. Let, the Vikṣepa N₃ R₃ be drawn as a Hsine of the Mānaikyārdha Vritta. With R₃ as centre and radius p, draw the Rāhu-Bimba. Let M₂ be the centre of the Moon's disc at the middle of the eclipse. Let M₂ S₂ represent the South with respect to the prime-vertical. From S₂, draw the Hsine of the Valana S₂ V₂ so that M₂ V₂ is the Krānti-Vritta-Dakshiṇā ie. the South with respect to the Ecliptic. Now the

398 Vikṣepa or the celestial latitude of the Moon M₂ R₂ is to be drawn along this Valanasūtra M₂ V. With R₂ as centre and radius p drawn the Rāhu-Bimba. Fig. 89 Depiction of Fig. 88, keeping the Moon fixed The slight flaw in this figure is that ML₁ L₃ implied as the path of the Moon is taken to be parallel to the ecliptic R₁ R₂ R₃ the path of the eclipsing body the Grāhakamārga, in as much as latitudes are drawn perpendi- cular to ML₁ L₃. This figure depicts a total eclipse of the Moon. If M coincides with R₂ at the middle moment of the eclipse, then the eclipse is called central. The duration of a central eclipse will be on the average the time that the Moon's disc takes to cross the diameter of the Rāhu-Bimba with its relative velocity. Hence the mean duration of a central eclipse is Average diameter of Rāhu + Average diameter of the Moon ────────────────────────────────────────────────────────── Relative velocity of the Moon with respect to the shadow = [(81+64) × 24] / [790′-35″ — 59′-8″] hrs = 145/732 × 24 = 290/61 = 4 hrs–45 minutes approximately.

399 Verse 30 and first half of 31. Geometrical depiction of the eclipse at the beginning and end of totality and also of the magnitude of the eclipse. The Bhuja is to be laid from the centre of the Moon along its Valanasūtra or the line indicating the direction of the ecliptic; the latitude is to be drawn from the end of the Bhuja and perpendicular to the Bhuja. The hypo- tenuse is to be drawn from the centre of the Moon. Taking the point of intersection of the latitude (Kōti) and the hypotenuse, as centre and radius p equal to that of the eclipsing body, if circles be drawn, from these circles could be known the points where totality begins and ends as well as the magnitude of the eclipse at any given moment. Or these could be found in another way as follows. Comm. The method given above for depicting the phases of an eclipse geometrically, could be applied for any moment during the course of the eclipse and depends upon before-hand computed Bhuja and Kōti. Refer to fig. 90. Let M be the centre of the Moon's disc. Mark Eω the East-west line drawn through M. Compute the Valana for the required moment, either for the moment when totality begins or for that when totality ends or for Fig. 90

400 any arbitrary moment whatsoever. With this Valana primarily laid in the Manaikyārdha Vritta, decide the Krānti Vrittaprachī or the East-West direction of the eclipse. Thus in the figure V₁ V₂ is this direction. Then lay off the computed Bhuja along this V₁ V₂ from M, say MA for the Sammīlana moment or the moment when totality begins or MC for the Unmīlana moment or the moment when totality ends or MD for an arbitrary moment. Draw AR₁ or CR₂ or DE equal to the latitude at the particular moment, perpendicular to the Valana- sūtra. In the figure drawn the Valanasūtra is shown to be the same. This does not mean it will be the same throughout. It will be changing because the position of the Ecliptic changes from moment to moment. So Bhās- kara uses the word ie. 'the respective Valanasūtra'. Also the latitudes will be differing from moment to moment as well as the Bhujas both of which are to be computed for any moment along with the Valana. (The method of computing the Bhuja was given in verse 15). Computing the respective Karṇas or the hypotenuses from the formula K = √(Bhujā² + Kōti²), (Kōti is here the lati- tude) with centre M and radius equal to the Karṇas, if arcs be drawn to cut the latitudes, the points of intersection would be no other than R₁, R₂ or E. Join MR₁, MR₂ and ME. With centres R₁ and R₂ and radii equal to p - r, (where p is the radius of the Rāhu-Bimba, and r the radius of the Moon's disc) if circles be drawn, they just touch the Moon's disc at F and G which are the points where totality begins and ends respectively. With centre E and radius P, if a circle be drawn, that will show what amount of the disc is shadowed as well as the measure of the magnitude of the eclipse (defined in verse 11). Note. In the above commentary and figure we have depicted MD as the Iṣṭa-Bhuja or the Bhuja at a given moment, taking a moment prior to the Unmī- lanakāla, for showing the magnitude of the eclipse.

401 If a moment in between the Sammīlana and Unmīlana were taken, the then Bhuja and Koṭi could be no doubt computed, but the question of magnitude of the eclipse does not arise as the entire disc has been plunged in the shadow. Second half of verse 31 verse 32 and first half of verse 33. Alternative method of depicting the eclipse geo- metrically. Joining the upper end of the latitude of the middle moment of the eclipse to those of the first and last contacts, we have what are called the Pragrahamārga and Mokṣamārga ie. the path of the centre of the eclipsing body from the first contact to the middle moment of the eclipse and that from the middle moment to the last contact. The lengths of these paths could be computed and they could be drawn before hand. Then with the centre of the Moon as centre and radius equal to p—r, if a circle be drawn, it cuts the paths described above each in one point. With those points as centre and radii equal to p, if circles be drawn, they will touch the Moon’s disc each in one point which are respectively the points of Sammīlana and Unmīlana. Comm. In as much as the latitude of the Moon differs from moment to moment, the Pragrahamārga and the Mokṣamārga are separated to achieve a little more accuracy than could be got by joining the upper extre- mities of the initial and final latitudes. The remaining statement is evident, for, at the moments of Sammīlana and Unmīlana, the distance between the centres of the eclipsing body and the eclipsed will be p—r, so that the points of intersection of the Pragrahamārga and Mokṣa- mārga with the circle whose centre is the centre of the eclipsed body and radius p—r will give the centre of the eclipsing body. 51