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

349 unable to adduce a proof he declares (2) 'उपलब्धिरेव वासना' meaning thereby (1) 'In this branch of science, we reckon only such an authority which has a proof behind it ie. which could be substantiated' and (2) 'There is no proof in this but accordance with observations alone has to be taken as a proof'. Though in the case of formulating the equation of centre Karṇānupāta was not stipulated to simplify matters, as there was not much difference, the Equation of centre being generally small. It is to answer such contexts as this that Bhāskara said in the Golādhyāya 'स्वल्पान्तरत्वात्, अबहूपयोगात्, प्रसिद्धभावाच्च बहुप्रयासात्, ग्रन्थस्य तद्वैर्गुरुताभयेन यत्त्यज्यतेऽर्थो न स दूषणाय" ie. 'If we in some particular context do not mention certain things it should not be condemned because in such contexts, (1) there is not much difference or (2) no useful purpose is served to a good extent or (3) it is too clear as does not require to be mentioned (4) the procedure implies a lot of cumbrous calculations and the result is after all negli- gible and (5) Mention will make the work on hand too unwieldy and brevity which is the soul of wit is to be sacrificed. In the present context this procedure of 'rectification of the Karṇa' is sought to improve matters. Bhāskara's words 'यदा ग्रहस्य कर्ण उत्पन्नः तदा कर्णो व्यासार्धं ग्रहकक्षायाः' seem really to imply that in the formula r/R H sin m for the equation of centre in the place of R, we are called upon to substitute really K. Though we had been in default for not doing so in the context of the Equation of centre, there is no reason why we should not make up the deficiency in this context. So, therefore, this rectification of Karṇa is stipulated here.

350 The procedure originally called for a rectification is that taking K to be R, we have to compute r and again taking the resulting K to be R, we have to compute r and so on repeating the process till an invariable value for K is obtained. This means that we should go on substituting for r, r K / R . Instead of following this laborious process of ' Asakṛt-Karma ' ie. method of successive approxi- mations, Bhāskara gives an alternative in the verse, whieh is as follows. Let K be the value of the Karṇa, for a value r of r. Since we are directed to make this K as R, ie. we have to add R—K to K thus making it R, we add also R—K to R, to keep the relative position of R and K to be almost the same. In other words considering the fraction K / R , adding R—K to both the numerator and denominator we have R / (2 R—K) whieh means that for a radius 2 R—K, the Karṇa will be R; that being so for a Radius R what will be the Karṇa ? The result is R² / (2 R—K) as given. It will be noted that the above interpolative procedure is adopted as a short cut technique to the otherwise laborious process. The mathematical correctness of this procedure will be seen from the following analysis. The problem is to change the Mandaparidhi to a radius K of the deferent by the formula (as indicated by Bhāskara in the course of the commentary) r' = r K / R so that δr = r¹ — r = r K / R — r = r (K—R) / R (1) Now we have K² = R² + r³ + 2 R r cos m construing R and m as constants we have to find δ K for δ r

351 Differentiating 2 K δ K = 2 r δ r + 2 R cos m δ r ie. δ K = [δ r (r + R cos m)] / K . But from fig. 65 (which is a portion of the epicyclic figure) M̂₁ = 180—m Ô₁ = θ = Mandaphala M̂₂ = m—θ Fig. 65 r = K cos m̅—̅θ̅ — R cos m so that r + R cos m = K cos (m — θ). Substituting in the above, δ K = [δ r × K cos m̅—̅θ̅] / K = δ r cos m̅—̅θ̅ But δ r from (1) is [r (K—R)] / R ∴ δ K = [r (K—R)] / R cos m̅—̅θ̅. But from the tri- angle of fig. 65. K = R cos θ + r cos m̅—̅θ̅ so that r cos m̅—̅θ̅ = K—R cos θ. But θ being small cos θ may be taken to be unity so that r cos m̅—̅θ̅ = K—R. Again substituting in the above δ K = [(K—R)²] / R (2) Now as per the formulation of Bhāskara K¹ = R² / (2 R—K) = R² / (R + R — K) = R / [1 + (R—K)/R] = R / [1 — (K—R)/R] = R (1 — (K—R)/R)⁻¹ Since |(K—R)/R| < 1, expanding binomially,

352 K¹ = R (1 + (K - R) / R + (K - R)² / R²) = R + K - R + (K - R)² / R = K + (K - R)² / R ∴ K¹ - K = δ K = (K - R)² / R as found above. Note. Bhāskara, having formulated this, appeals to ' Dhulikarma' ie. arithmetical computation, for convin- cing those who may not be able to follow his logio. Here one may note also the wrong directive given by the Samśodhaka in the text. The proof furnished by us above gives a mathematical veracity to Bhāskara's formulation. Verse 5. To rectify the Yōjanakarṇa or the spatial radius Vector. The above Kalākarṇa multiplied by the Karṇa given in Yōjanas and divided by the Radius gives the rectified Yōjanakarṇa. Comm. In the formula given above δ K = (K - R)² / R which is in units of spatial minutes (on the scale of R=3438). (K' × y) / R where K' is the rectified Kalā-karṇa, and y the Yōjanakarṇa given in verse 3, gives the rectified Yōjana- ·karṇa. Second half of verse 5. The spherical radii of the Sun and the Moon. The spherical diameters of the Sun and the Moon are respectively 6522 and 480 Yōjanas. Comm. The word- ‘ Bimba ’ is used to connote the spherical diameter. The diameter of Moon as given will be equal to 480×5=2400 miles in modern terms which is not far from truth. Once we accept that the method indicated by us in the Kakshādhyāya of chapter I was that

353 followed by the ancient Hindu Astronomers to estimate the distance of the Moon, the spherical radius the hori- zontal parallax, the orbital radius pertaining to the Moon could all be deduced and the magnitudes so deduced accord with their average values in modern astronomy. The magnitudes pertaining to the Sun however, should be deemed as parameters. Verse 6. e - [(S - E) Kₘ / Ks] = 2 α where e = Earth's diameter, s = The Sun's diameter; Km = Moon's dis- tance from the Earth's centre; Ks = The Sun's distance from the Earth's centre, and α = radius of the Earth's shadow cone at the lunar orbit. Comm. In fig. 66, let CD be the radius of the Earth's shadow cone at the lunar orbit. Required to find the magnitude of CD. Triangles DEF and ESG are similar Fig. 66 Fig. 67 45

354 Fig. 68 Fig. 69 Fig. 70 where DF and EG are drawn parallel to VBA, the common tangent. ∴ EF/DE = SG/ES ie. (½e - a) / Km = (½s - ½e) / Ks

355 where ɑ = FB = CD required ∵ ɑ = ½ { e - Km/Ks (s - e) } I as formulated 2ɑ being what is called Rāhu-Bimba or diameter of the Earth's shadow cone at the lunar orbit which is called Ku-bhā Vistṛti in the verse (Ku = Earth ; Bha = Shadow ; Vistṛti = diameter). Note (1) We shall prove that this formula accords with the modern formula given for the radius of the shadow cone. Divide I throughout by 2 Km, so that ɑ/Km = e/2Km - (s - e)/2Ks II But from fig. 67. ɑ/Km = sin Ê = Ê = angular radius of the shadows cone expressed in radius. From fig. 68, e/2Km = Horizontal parallax of the Moon ; s/2Ks = sin Ê = Ê = angular radius of the Sun expressed in radians from fig. 69 ; and e/2Ks = (from fig. 70) Horizontal parallax of the Sun. Thus Equation II means ρ = P - σ + P¹ III where ρ = angular radius of the shadow cone, P = Horizontal parallax of the Moon ; P¹ = that of the Sun and σ = angular radius of the Sun. Note (2) If we don't divide I by 2 Km, we have the radius of the shadow-cone in Yojanas, substituting the values of e and s, Km and Ks. Note (3) It is worth hearing Bhāskara in his commentary under this verse. Observe the Sun's disc while rising on the day when his true motion is equal to his mean, with a compass composed of two rods hinged at one end and carrying a protractor at the other. We get the mean diameter of the Sun equal to 32' - 31" - 33"'. Similarly, observe the Moon's disc on a full-moon-day

386 when his true motion equals the mean. It will be 32'- 0'' - 9'''. Note (4) Substituting the values of P, P' and σ in III ρ = 52'-42'' + 3'-57'' - 16'-16'' = 40'-23'' = Angular radius of the shadow-cone at the lunar orbit which almost accords with the modern value 41'-49''. Note (5) One may wonder as to how, taking wrong values for s and Ks, such a correct value could be obtained for P. In equation II, α, Km, e are all near the truth so that the terms effected are s/K and e/Ks; but both s and Ks being parameters s/Ks comes off alright, which is the angular radius of the Sun's disc which could be measured. The only vitiating term is e/Ks which is the horizontal parallax of the Sun which was overestimated unwittingly by a wrong supposition as indicated in the Kakshādhyāya. However e/Ks comes to be 3'-57'' and this overestimate is mitigated to some extent that the Earth has an atmos- phere which boosts the angular radius of the shadow cone by about 1' and the remainder of the overestimate makes amends for the smaller value of P taken. Verse 7. To convert spatial measures into angular measure. The diameters of the Sun, the Moon, and Rāhu in Yōjanas multiplied by R = 3438', and divided respectively by Ks, Km and Km give their angular measures. Comm. From fig. 69, (s/2) / Ks = sin Ê = Ê expressed in radians = E' / 3438 ∴ E' = (3438 × s) / (2 Ks) which means that the angular radius of the Sun is got by multiplying s/2 ie. the spherical radius of the Sun by R = 3438 and divid- ing by Ks as mentioned. Similar is the case with respect to the other two. Note. - The word Kalākaraṇa is used to signify To convert into angular measure.

367 Verse 8. An alternative method of obtaining the angular radii. The daily motion of the Sun increased by one-tenth of its value and halved, gives the angular diameter of the Sun. The Moon's daily motion multiplied by 3 and divided by 71, gives the angular diameter of the Moon. Or the daily motion of the Moon being decreased by 715 and divided by 25 and the result being added to 29 gives the angular diameter of the Moon. Comm. This method gives in an easy way the true angular radii. The formulae given are s' = ½s₁ (1+¹⁄₁₀) ; and m' = (3 m₁) / 74 = (m₁ - 715) / 25 + 29. This may be eluci- dated as follows. The argument used is "If the spherical diameter of 6522 Yojanas corresponds to a spatial daily motion of 11858¾ Yojanas, what angular diameter corres- ponds to the angular daily motion s₁?". The proportion- ality is clear and the result is (s₁ × 6522) / 11858¾ = 26088 / 47435 . Converting 26088 / 47435 , into a continued fraction, it is 1/(1+) 1/(1+) 1/(4+) 1/(1+) 1/(1+) 1/94 . The penultimate convergent is 11/20 = ½ (1+¹⁄₁₀). The formula follows. Similar calculation gives m'. Note. The advantage of these formulae is that they are not only easy but also adopting the true daily motion we have the true angular radii. This procedure was adopted by Bhāskara from Brahmagupta. The latter, how- ever, prescribes a nearer convergent namely 10/247 but actually 17/420 is the nearest convergent. The next formula namely m' = (m₁ - 715) / 29 + 29 is approximate. This may be elucidated as follows. Let the

366 daily motion be 715 ; then as per the previous formula the angular diameter should be 3/74 × 715 = 2145/74 = 28 73/74 = 29' very approximately. The mean daily motion is 790 which corresponds to 32' of angular diameter. Taking advant- age of this arithmetical correlation namely that the excess of 3' over 29' corresponds to 75' of daily motion. Bhāskara gives the formula m' = (m₁ - 715)/25 + 29. This formula correctly holds good when m₁ = 740, for, equating 3x/74 = (x - 715)/25 + 29 = (x + 10)/25 , x will be equal to 740. For other values between 715 and above it holds very approximately. Thus, when m₁ = 715, m' = 28 73/74 ie. 29 when m₁ = 740, m' = 30, when m₁ = 765, m' = 31 1/75 (error 1/75) when m₁ = 790, m' = 32 1/37 (error 1/37) and so on. Verse 9. An alternative method of finding the angular diameter of the shadow cone. 2 ρ = 2/15 m₁ - 5/12 s, where m₁ and s₁ are the daily motions of the Moon and the Sun respectively. Comm. In the previous verse, we had formulae to compute the diameters of the discs of the Sun and Moon, knowing their daily motions. Since in practice we have these daily motions computed for every day, so the compu- tation based upon those daily motions conduces to ease in the matter of calculation. Now in this verse, the radius of the shadow cone is also calculated in terms of the daily motions of the Sun and the Moon, which is more an ingenious device adopted in practice. The elucidation of the formula depends on the following technique as conceived by the Hindu astronomers. In as much as the Sun's sphere is far bigger than that of the Earth, the shadow of the Earth assumes the form of a cone. From a knowledge of the decrease in the diameter, as we proceed from the

309 Sun to the Earth and a knowledge of the Sun's distance, we can compute similar decrease as we proceed from the Earth to the lunar orbit knowing the distance of the Moon. Such a decrease measured in Yojanas is termed by Bhāskara as 'अपचययोजनानि' ie. Yojanas of decrease in diameter. It was this concept that led to the formulation of 2a in Yojanas in the form 2a = e — ((s — e) Km) / Ks of verse 6 and is indeed based upon the similarity of triangles as proved by us in that context. Dividing the above equation by 2 Km, we have a/Km = e / (2 Km) — 1/2 ((s — e) / Ks) I Dividing a, e and (s—e) thus by the distances Km and Ks is termed 'Kalā-Karaṇa' ie. converting spatial distance into angular measure. Thus dividing a by Km is con- verting the radius of the shadow cone into angular measure at the lunar orbit ; dividing 1/2e by Km is estimating the angular measure of the earth's radius as seen from the Moon's distance or what is the same the horizontal parallax of the Moon, whereas dividing 1/2s by Ks is getting the angular radius of the Sun's disc as seen from the Earth and dividing 1/2e by Ks is getting the angular radius of the . Earth's disc as seen from the Sun or what is the same the horizontal parallax of the Sun. Equation I which gives a/Km the angular radius of the shadow cone, may also be interpreted as follows. e / (2Km) = Horizontal parallax of the Moon as mentioned above which is equal to the angle (fig. 66) EĈB, for, 1/2e = EB ; Km = EC so that e/2Km = EB/EC = sin EĈB = EĈB expressed in radian measure. Also 1/2 (s—e) / Ks = (SA—GA) / SE = SG / SE = sin SÊG = SÊG = EṼB

360 (alternate angle) = Semivertical angle of the shadow cone (say θ). Now EĈB − ÊVB = CÊD = angular measure of CD ie. the angular radius of the shadow cone (expressed in radian measure). Converting ½e/Km into angular measure, the propor- tion used by Bhāskara is "If by the daily spatial motion of 11859¾ Yojanas of the Moon, we have its daily motion in arc, what shall we have for e = 1581 Yojanas?" The result is 1581/11859¾ m₁. Converting the coefficient into a continued fraction we have 1/(7+) 1/(1+) 1/(1+) 1/175 ......... The penultimate convergent is ²/₁₅. Hence e/Km = ²/₁₅ m₁. Converting ½ ((s − e)/Ks) into arc, the proportion used is "If by the daily spatial motion of 11859¾ Yojanas, we have the daily motion of s₁, what shall we have for s − e = 6522 − 1581 = 4941 Yojanas?" The result is 4941/11859¾ s₁. Converting the coefficient into a continued fraction, we have 1/(2+) 1/(2+) 1/(2+) 1/146 . The penultimate convergent is ⁵/₁₂. Hence, the result is ⁵/₁₂ s₁. Note (1) It might be asked whether the Hindu astronomers used the theory of continued fractions. The answer is, they did though they did not write the con- tinued form in the form we do now. They arranged the successive quotients in a vertical line and called the column as a 'Valli' or 'creeper'. One may refer to the chapter in Bhāskara's Bijaganita on 'Kuttaka' in this context. Note (2) The formula derived above to obtain the Rahu-Bimba or diameter of the shadow-cone at the lunar orbit, is one which could be conveniently used in practice,

361 for, as mentioned before, the daily motions of the Moon and the Sun are ready computed for every day. Also the advantage in using this formula is that besides the fact that we need to deal only with small quantities instead of the big numbers of Yojanas, the true value for the day of eclipse is got by using the true daily motions. Using the mean daily motions, however, we have for the mean diameter, ²/₁₅ m₁ — ⁵/₁₂ s₁ = ²/₁₅ × 790′–35″ — ⁵/₁₂ × 59′–8″ = 105 ⅔ — 24 ⅔ = 81 very approximately. Note (3) In obtaining a convergent for (s − e)/Ks , since the values of s and Ks are parameters s/Ks is got alright, where e/Ks is more exaggerated as the value of the hori- zontal solar parallax. By this term the result is increased to an extent of 3′ out of which 1′ is mitigated by the fact that we have to take the earth's atmosphere also into

362 the Moon with respect to the ecliptic where λ is the longi- tude of the Moon with respect to the nearer node and 270' or 4½° is taken to be the inclination of the lunar orbit to the ecliptic or what is the same, the maximum latitude of the Moon. Comm. The formula is evident and similar to that for calculating the declination of the Sun. Thus H sin β = (H sin λ × H sin 4½°) / R . But since β is small and also 4½°, we can take β = (H sin λ × 270') / R . The argument used by Bhāskara, however, is ‘ If by a H sin λ equal to R, we have the maximum latitude of 270', what shall we have for H sin λ?’ The result is as given. Note. The modern value for i the inclination of the lunar orbit to the ecliptic is given to vary between 4°-58' and 5°-18'. Verse 11. The definition of the magnitude of a lunar eclipse. Sthagita or the magnitude of an eclipse is defined as P+r+β where P and r are respectively the radii of the eclipsing and eclipsed bodies and β is the latitude of the Moon. If the Sthagita is greater than 2r, then the eclipse is total. [Diagram: Two intersecting circles representing the Moon (center M, radius r) and Earth's shadow (center C, radius P), along lines labeled "LUNAR ORBIT" and "ECLIPTIC"] Fig. 71

363 Comm. In figure 71, the eclipsing body is just con- tacting the eclipsed body. Taking the case of a lunar eclipse, the latitude then of the Moon is evidently P+r ie. β=P+r holds good at the moment of first con- tact. (C) is the cross-section of the shadow-cone at the lunar orbit and (M) is the Moon. (fig. 72), CB+AM-CM=CB+AB+ BM-CM=AB+(CB+BM)-CM= AB+CM-CM=AB ∴ P+r-β=AB=Sthagita. Thus Sthagita gives the portion of the diameter of the eclipsed body which is shadowed. The eclipsed body is termed the Chādya, the eclipsing body as the Chādaka and P+r as the Manaikya-ardha ie. half the sum ofthe diameters of the eclipsing and eclipsed bodies. Fig. 72 When the Sthagita exceeds the diameter of the eclipsed body the eclipse is evidently total ie., when P+r-β>2r ie. P-r>β. Verse 12. Duration of the eclipse and duration of its totality. Sthiti-Khanda = √(P+r)²-β² × 60 = ½ Duration of —————————————— the eclipse m₁-s₁ Marda-Khanda = √(P-r)²-β² × 60 = ½ Duration of —————————————— totality m₁-s₁ where P is the radius of the shadow-cone, r the radius of the Moon's disc, β its latitude taken to be constant during the eclipse, m₁ and s₁ the daily motions of the Moon and Sun respectively. Comm. (1) The time between the moment of first contact and the middle of the eclipse or the moment of

opposition or conjunction as the case may be is called Sparśa-Sthiti-Khaṇḍa. (2) The time between the middle of the eclipse and the moment of last contact is called the Mokṣa-Sthiti- Khaṇḍa. (3) The time between the commencement of total eclipse and the middle of the eclipse is called the Sammi- lana-Marda-Khaṇḍa. (4) The time between the middle of the eclipse and the end of total eclipse is called Unmilana-Marda-Khaṇḍa. The suffix Khaṇḍa meaning ‘half’ is generally omit- ted while refering to these phases. In the above verse we are given formulae for Sthiti-Khaṇḍa and Marda- Khaṇḍa only without specifying whether they pertain to Sparśa or Mokṣa. Though the same formulae serve for both the Sparśa phase as well as the Mokṣa phase under the supposition that β does not vary, it will be noted that the Sparśa-Sthiti-Khaṇḍa will not be equal to Mokṣa-Sthiti-Khaṇḍa and that the Sparśa-Manda- Khaṇḍa will not be equal to the Mokṣa-Marda Khaṇḍa in as much as β changes from moment to moment. In fig. 73, let C₁ be the position of the eclipsing body at the moment of first contact and C₂ its position at the moment of last contact. In the figure is shown only one position of the Moon's disc signifying that we may con- sider the motion of the eclipsing body keeping the eclipsed body fixed (or what is the same relative to the position of the eclipsed body). It is evident from the fig. that C₁M = C₂M = P + r so that C₁M C₂ is an Isosceles triangle. Let MN be the ⊥ᵃʳ dropped from M on C₁ C₂. C₁ C₂ is the ecliptic because the centre of the shadow will be moving along the ecliptic, for, in fig. 66, SE the ecliptic passes through D the centre of the cross-section of the shadow- cone, as well as through the vertex V of the shadow-cone.

868 Fig. 73 MN is therefore the latitude of the Moon. Since the lati- tude is not the same at the moment of first contact and that of the last contact, the figure drawn does not represent the true figure but only a figure drawn on the supposition that β remains the same and C₁ Moves relative to M. From the figure C₁N² = (P+r)² − β² = C₂N² I The Sthiti-Khanda defined in this verse is the time taken by C₁ to reach the position N ie. the position at the moment of opposition, and again from the position N to the position C₂. The velocity of C₁ relative to M is no other than the excess of the velocity of, the Moon over that of the Sun. (The velocity of the Earth is the relative velocity of the Sun with respect to the Earth and this is equal to the velocity of the shadow moving along the ecliptic). So, the time taken by C₁ to reach the position of N relative to the Moon is equal to [ 60 × √(P + r² − β²) ] / [ m₁ − s₁ ] Similarly the time taken by the centre of the shadow from N to C₂ ie. from the point of opposition to the moment

386 of last contact has also the same formula where in each case β is the latitude at the moment of opposition. The path taken by the centre of the shadow is called 'ग्राहकमार्ग' ie. the path of the eclipsing body. The actual case when both C and M are both moving and when β is considered as a non-changing quantity is shown in fig. 74. In this Fig. 74 case, three positions are shown, (1) that at the first con- tact (2) that at opposition and (3) that at last contact, where C₁, M₁, C₂, M₂ and C₃, M₃ give the positions of the centre of the shadow and that of the Moon's disc respe- ctively, both the centres being shown as moving. Since the Moon moves faster than C and as such overtakes C, the path of M from M₁ M₃ which synchronizes with the path of C from C₁ to C₃, is shown to be longer. But, one may wonder, how C₁ N₁ and C₃ N₃ represent the Sparśa- Sthiti-Khanda and Mokṣa-Sthiti-Khanda respectively. The distance overtaken by M with respect to C from the point of first contact to the point of opposition is M₁ M₂ — C₁ C₂ = C₁ N₁. Hence we compute C₁ N₁ by the formula C₁ N₁² = (P+r)² — β². Similarly from the point of opposition to the point of last contact M overtakes C by the distance M₂ M₃ — C₂ C₃ = C₃ N₃ = √(P+r)² — β². Fig. 75 shows the situation when β changes as is the actuality. When the opposition takes place after the Moon crosses the node, then β₃ > β₂ > β₁, whereas if

367 [चित्र: Fig. 75] Fig. 75 the opposition precedes the Moon’s position at the node β₃ < β₂ < β₁. Also, when β changes, M₁ M₂ does not exceed C₁ C₂ exactly by C₁ N₁. So, on both the counts, the formulae, given in verse 12 are approximate. What is done in practice is that β is computed for the moment of opposition and estimating the Sparsa-Sthiti-Khanda by the formula given above, and subtracting it from the time of opposition the moment of first contact is got. Then β is computed for that time and again the formula is applied to get the Sparsa-Sthiti-Khanda. Repeating the process, we rectify the Sparsa-Sthiti-Khanda. Even then, we do not have the actual value of the Sparsa-Sthiti-Khanda, because M₁ M₂ does not exceed C₁ C₂ exactly by C₁ N₁. A more correct procedure would be to compute the time between the moment of first contact and the moment of opposition and by that time, to compute the length of M₁ M₂ and take [m' (β₁ ~ β₂) / (M₁ M₂)] in the place of m' and use the formula of verse 12. This nicety, however, need not be attended to with respect to the duration of totality, for, it does not make much difference. Another way of obtaining a better value for T, the Sparsa-Sthiti-Khanda is to take average values for β₁ and β₂, m₁ and m₂, s₁ and s₂ where m₁ and m₂ are the values of the Moon’s daily motion and s₁ and s₂ are those of the

368 Sun's at the point of first contact and the moment of con- junction respectively. We may also use calculus to obtain δT, the variation in time for a variation of δβ in β and a variation of δm₁ in m₁ ignoring the small variation in s₁, as follows. T² = ((P + r)² - β²) / (m₁ - s₁) ∴ 2T δT = [(m₁ - s₁) × -2β δβ - ((P + r)² - β²) δm₁] / (m₁ - s₁)² ∴ δT = - β δβ / [T (m₁ - s₁)] - δm₁ ((P + r)² - β²) / [T (m₁ - s₁)²] The first term on the Right hand side gives the variation for δβ and the second for δm₁. Fig. 76 shows the case of totality. [Diagram: Fig. 76] Fig. 76 M₁ M₂ = N₁ C₁ + C₁ C₂ ∴ The Moon has to over- take C from the moment of the beginning of totality to the moment of opposition by the distance C₁ N₁ with a relative velocity of m₁ - s₁. Hence the time of Sammilana- Marda-Khanda is equal to [√(C₁M₁² - β²) × 60] / (m₁ - s₁) = [√((R - r)² - β²) × 60] / (m₁ - s₁) as given, taking β to be constant. Similarly the Un- milana-Marda-Khanda from the position (M₂ C₂) to the position (M₃ C₃) will also be the same, taking β to be constant.