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

TABLE 2

101 [L

SPAṢṬĀDHIKĀRA — RECTIFICATION OF PLANETS Introduction. In the Bhagaṇādhyāya section of the previous chapter Bhāskara gave under the Caption Bhaga- ṇopapatti his proofs as to how the ancient is might have obtained the number of sidereal revolutions of the planets and the planetary points called apogees or aphelia and Nodes. But in trying to give those proofs, he was aware and he confessed also in so many words that some of his proofs at least were obsessed by what is called Itarēta- rāśraya-Doṣa i.e. “answer begging the question” It is worth hearing his words in his commentary under verses 1–6 of the section cited above—“That the planets, and the planetary points perform so many revolutions in a Kalpa, is essentially conveyed by the Āgama i.e. the Śāstra (which is to be taken on faith). That Āgama, got diversified i.e. there are many versions of that Science, due to the defects of scribes, the teachers and the students and due to a long lapse of time from the originators of the Āgama. That being so, the question arises as to which of the versions is to be trusted as the right authority. If it be said so, in mathematics only an āgama which could be proved also should be taken as authority. Such a number of revolu- tions as are obtained by proof, is to be accepted. Even that could not be (a proof); for, a great scholar could just understand the proof and by that proof alone, it is not possible to know the exact number of revolutions (in a kalpa), for, a man's longevity is not much. In the proof that could possibly be given, the planet's position is to be observed and noted every day, during the entire course of its revolution. Thus Saturn Completes its sidereal re- volution in about 30 years. The apogee of the Sun and the aphelia of the planets have their revolutions running into hundreds of years. Hence the observation of one complete revolution (of such a planetary point) is beyond the capa-

103 city of a mortal. Hence great astronomers accept such an āgama as would give results which accord with observations during their times, and such a one as was formerly accepted by a very intelligent astronomer. Then they produce their own works exhibiting their own Skill in the Science and refuting wrong notions of others. Their idea is 'Let the Āgama we take as an authority be whatever it would be Let us show our own skill in the course of our work', just as in this work, the āgama accepted by Brahmagupta is taken on faith as the authority. Then it might be argued "Better not attempt at trying to prove how the numbers of sidereal revolutions were arrived at. Even if a proof be attempted, that proof would be obsessed by the ' Itarētara- śrayadōṣa' (cited above). Nevertheless we shall give a brief proof, That 'itarētarāśrayadōṣa' is apparently a dōṣa i.e. an apparent defect; for, different proofs could not be adduced simultaneously. The proof will now be given". These words indicate that even such a highly rational and supremely intelligent astronomer like Bhāskara could not set aside his faith in our āgama and attempt at a pure and rigorous proof, which would not invoke the āgama-Let us see where in his proof he does commit the so-called itarētarāśrayadōṣa and where he invokes the āgama. Also we shall try to construct a proof, of course to a good extent on the lines on which Bhāskara tries to give his proof, but at the same we shall not invoke the āgama. where he does, but try to proceed purely on a rational basis. We shall take up the proof under verse 18, in its appropriate context. We shall now proceed with the text upto that point, which gives a brief sketch of the Hindn trigono- metry.

104 Verse 1. In as much as true positions of the planets alone are required to decide auspicious moment; for journeys, marriages, celebrations pertaining to temples, astrology and the like, we shall now give the methods of rectifying the mean positions of the planets so as to accord with their observed positions. Comm. Clear. Verse 2-9. Obtaining the sines of the angles and tabulation of the sines. The planet deflected to the true position from the mean lies at the end of a half-chord (which is the Hindu sine of an angle) so that many processes pertaining to a planet are carried through Sines of angles; hence the word half-chord alone is connoted in this work wherever the word Jyā meaning a chord is used. The lengths of these half-chords (or the Hindu Sines) for angles increasing from 0° to 90° at intervals of 3¾° are as follows—225', 449', 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2977, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438. The ut-kramajyāis or the Hindu versed-sines are respectively 7, 29, 66, 117, 182, 261, 354, 461, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438. The word Tribhajyā or Trijyā is half-diameter. The word jyā khandas used by pandits connote the differences between successive sines. Comm. There is a difference between modern trigono- metrical sines and the Hindu sines as detailed below (Ref.

105 Fig. 6 fig. 6 overleaf). Let (O) be a circle i.e. a circle with centre ‘O’. Let AB be an arc called ‘Chāpa’; let BC be drawn perpendicular on OA; then BC is half of the full chord BD (known as jyā). The half-chord Ardha-jyā is itself spoken of as jyā for convenience and is the Hindu-sine of the arc or chāpa AB. In Hindu trigonometry ‘angle’ is connoted by the arc corresponding to it and as such spoken of as chāpa. OC is spoken of as the Hindu-cosine or Koti- jyā and CA is called the ut-kramajya or the Versed-sine. The radius O B is called trijyā and let us connote it by R. To differentiate between the modern terms and the Hindu terms, we use the words H. Sine, H. Cosine, H. vers-sine for the Hindu sine, the Hindu cosine and the Hindu vers- sine respectively. Also the radius R is generally taken to be 3438′ which, we know to be the approximately the minutes in a radian. To talk of a length in minutes ap- pears rather odd but no confusion need be there, for, an arc of length R subtends 3438′ at the centre. It is called Trijyā for the reason that it is the H. sine of 3 Rasis or 90°. A Rasi is equal to 30° because the ecliptic circle of 360° is divided into 12 Rasis Mesha, Vrishabha etc. meaning 14

106 Aries, Taurus etc. The names of the Rasis in Sanskrit and the modern English words we use for them have the same meaning, which raised a suspicion in the minds of many orientalists that the Hindu Astronomy drew upon the Greek. Many scholars of India assert that the Greeks derived this knowledge from the ancient Hindus; but we shall not enter into the controversy here. It may be noted also that the Sanskrit names of week-days have the same meaning as Sunday, Monday etc. On the basis of taking trijyā equal to 3438', the other H. sines or half-chords are also expressed in minutes. Generally twenty-four H. Sines are given in a quadrant and to obtain the H. sine of an angle intermediate, a for- mula for interpolation also is given. Also the method of calculating the H. sines for every degree is given, as we shall see shortly. In the table of 24 H. sines, the first is H. sine 3°-45' or H. sine 225' and this is approximately taken as 225' because in fig. 6 if AÔB=3°-45', the H. sine BC will be almost equal to the arc AB. The H. Vers-sines are also given to get the corresponding H. Cosines easily, for, H. Vers sine 3°-45 = R — H. Cos 3°-45 = 7' means H. Cosine 3°-45'=3431=H. sine (90°-3¾°) = H₂₃ where we use the notation H r to mean the rth H. Sine. Now we propose to give here some essential formulae used by the Hindu astronomers, as given in the golādhyāya by Bhāskara under the caption jyotpatti-krama. Inciden- tally it may be noted that H. sine θ = R sine θ where sine θ is the modern sine of the angle θ°. Similarly H. Cos θ=R Cos θ and H vers θ = R vers θ. Thus when we have an equ- ation of the type. Sin δ = Sin ϕ Cos z + Cos ϕ sin z sin a in modern astronomy arising out of the famous spherical triangle PZS where P is the Celestial Pole, Z the Zenith and S the position of the Sun or a Star, the Correspond- ing Hindu formula would be R³ H Sin δ = RH Sin ϕ H cos Z + H cos ϕ H sin Z H sin a. Occasionally the

107 radius is taken to be 120, and the corresponding H sines are called Laghu-jyās or simpler H sines used where great accuracy is not required. Śrīpati took the radius to be 3270 units in addition to 120 as did Brahmagupta. Muñjāla took 488′ and some others some other values also. Out of these 3438′ alone has a right significance (Vaṭēśwara took 3272) Bhāskara says under verses 1 to 5 under Jyotpatti- Vāsanā in the Golādhyāya that the Hindu astronomers got the values of the main H sines of 30°, 45°, 60°, 18° and 36° by inscribing regular polygons in a circle. They are called the pancha-jyakās or the fundamental H sines. From these the others were calculated according to the methods given by Bhaskara as follows. To start with, we have the fundamental formula H sin²θ + H cos²θ = R² (from fig-6) I In addition to this formula, Bhāskara gives another formula (verse 10, 11 Ibid) H Sin θ/2 = √(H sin² θ + H vers² θ) = √(½ R. H vers θ) II In the commentary under the above verses, he has given the method by which II was obtained (Ref. fig. 7) BM = H sin θ where AÔB = θ ; also AM = H vers θ and AB² = AM² + MB². Let N be the mid-point of AB. ½AB = AN = H sin θ/2 ∴ H sin θ/2 = ½AB = ½ √(AM² + MB²) == ½ √(H sin² θ + H vers² θ) which proves the first part of II. Again from the right—angled triangle ABC, AB² = AM · AC = H vers θ × 2R ∴ H sin θ/2 = ½AB = ½ √(2R H vers θ) = √((R H vers θ) / 2) which proves the second part.

108 Fig. 7 In the Commentary under verses 1—25 ibid, Bhāskara tells us how formulae I and II are used to construct the table of 24 H sines. To start with, the four H sines of 30°, 45°, 60° and 90° which may be denoted by the symbol Hr where r=8, 12, 16 and 24, are known. Now using the formula II, H₄ is obtained from H₈, H₂ from H₄ and H₁ from H₂. Similarly from H₁₂, H₆ and H₃ are successively obtained. Now using formula I, H₂₀, H₂₂, H₂₃, H₁₈, H₂₁ are obtained respectively from H₄, H₂, H₁, H₆ and H₃. Now again from H₂₀, H₂₂, H₁₈, we obtain using formula II H₁₀, and H₅, H₁₁, H₉ respectively. Formula I gives again H₁₄, H₁₉, H₁₅, H₁₃ from the above. H₁₄ gives H₇ and H₇ gives H₁₇ using formula II and I respectively. Thus the table is Completed. Then Bhāskara poses the problem as to how a table of the H sines could be computed when a quadrant is divided into 30 equal parts. He says that formula I and II do not suffice in this behalf and shows how they do not, as follows

109 in the same commentary cited above. To start with, the H sines of 18°, 30°, 36°, 45°, 54°, 60° are known. They are respectively H₆, H₁₀, H₁₂, H₁₅, H₁₈, H₂₀. Also H₃₀ i.e. H sin 90° = R is also known. Formulae I and II will help us to derive, from H₆, H₃ and from H₃, H₂₇ ; also from H₆, we derive H₂₄. From H₁₀, H₅ and from H₅, H₂₅ and again from H₁₈, H₉ and from H₉, H₂₁ are derived. The remaining H sines sixteen in number cannot be got from either of the formulae. To meet the situation Bhāskara gives other formulae of his own discovery as he says “ प्रवक्ष्येऽथ विशिष्टमस्मात् ” i.e. “ I shall tell something more than this. These formulae he gives in the verses 12 to 15. They are H Sin ((90 ± x) / 2) = √((R² ± R H Sin x) / 2) III H Sin ((x − y) / 2) = √((H sin x + H sin y)² + (H cos x − H cos y)²) IV √(((H Cos x − H Sin x)²) / 2) = H Sin (45 − x) V R − (2 H Sin² x) / R = H Sin (90 − 2x) VI These formulae correspond to the modern formulae Sin (45 ± x/2)° = √((1 ± Sin x) / 2) Sin ((x − y) / 2) = √(((Sin x + Sin y)² + (Cos x − Cos y)²) / 2) √(((Cos x − Sin x)²) / 2) = Sin (45 − x) 1 − 2 Sin² x = Cos 2x respectively

110 These formulae imply a knowledge of the expansion of Sin (x ± y) which is given in verses 21, 22 in the form H Sin (x ± y) = (H Sin x H Cos y ± H Cos x H Sin y) / R VII The formula H Cos (x ± y) is got from VII by putting 90 - (x ± y) for x ± y. To construct the remaining sixteen H Sines Bhaskara directs us to use his formula IV wherein taking x = 27°, and y = 15°, we have H₁₂ which gives H₇₈. From H₇₈ we have H₁₄, H₇, and H₁ from H₄. Then H₁₆, H₂₃ and H₂₉ are have H₈, and H₄ which in turn give give H₂₂ and H₂₆. H₂₆ gives H₁₃ which in turn gives H₁₇. H₂₃ similarly gives H₁₁ which in turn gives H₁₉. Thus the table is complete. Verses 16 to

111 δ (Sin x) = Cos x δx. Since H Sin (x+1)° = H Sin x + (H Cos x × H Sin 1°) / R approximately, H Sin (x + 1)° — H Sin x° = H Cos x × (H Sin 1° / R) = H Cos x × a constant. Hence Bhāskara could see that the variation in the function H Sin x is proportional to H Cos x. Let it be now required to find the increment in H Sin x for an increment δx in x where δx < 60'. Let H Sin (x+1)° — H Sin x = (60' × H Cos x) / R = y where y is called the Bhogya-Khanda. Then Bhāskara argues “If for an increment of 60', there is an increment of y, what shall we have for δx ?”. The answer is yδx / 60 = (60 × H Cos x / R) × (δx / 60) = (H Cos x × δx) / R. Hence H Sin (x + δx) — H Sin x = δ

112 षष्ठात्पञ्चदशादपि, सप्तमात् द्वादशात्सप्तदशाज्ञाधोत्तरं मतम् ” quoted from the Brahma Siddhānta by Ranganātha in his com- mentary of Sūryasiddhānta were alluded to in the articles cited but no satisfactory mathematical explanations were given by them. We shall give hereunder a satisfactory explanation of the matter discussed in the articles. In the first place it may be noted that in the table of those 24 H sines, the sixteenth as given by Bhāskara namely 2977 is more correct than that given in the Sūrya- siddhānta namely 2978 (Lakshmi Venkateswara press edi- tion 1955 Bombay). In the course of the Commentary under the verses 15, 16 of the Sūryasiddhānta, Ranganātha gives the hint which must have been at the back of the mind of the author of the Sūryasiddhānta when he gave the rule to construct the table cited. Just as δ (H Sin θ) = (H Cos θδθ) / R , Similarly the formula δ (H Cos θ) = - (H Sin θ δ θ) / R must have been known to the author. The negative sign means that the successive differences of the H sines namely 225, 224, 222, 219 etc. are decreasing and also that the successive differences of these differences are increasing according to the H sine. Just as Bhāskara could see that the H sines were increasing and the successive differences of the H sines were in Kotijyānupāta i.e. in direct ratio to the H Cosine at their respective place, similarly, the author of the Sūryasiddhānta could see that the second differences cited above were in Kramajyānupāta as hinted by Ranga- nātha. From the formula δ ((H cos θ δ θ) / R) = - (H sin θ δ θ²) / R² putting θ = 90°, we have the second difference numerically

113 equal to δθ³ / R = (225 × 225) / 3438 = 14' — 43" — 30". Here Ranganātha made a mistake in taking this to be 3438 / 222 = 15' — 16" — 48" — Even δθ³ / R is approximate and a more correct value of the second difference would be 14' — 47" approximately. Ranganātha then argues that taking this second difference to be 15 for the H sine 3438 ‘what will it be for the H sine 225' ?’ The answer would be (15 × 225) / 3438 = (15 × 25) / 382 = 375 / 382 = 1' approximately. So, the second difference in the beginning of the table happens to be 1' ie 225 / 225. This led the author of the Surya siddhānta to use the words “तद्विभक्तलब्धोन”. This being an approximate formulation, naturally necessitated a second formulation where the approximation led to an error of 1' through the verse “एकविंशाच्च विंशाच्च etc.” This second formulation intended to make a correction, was done in the wake of a correct calculation through the formulae I & II which were known even prior to Bhaskara. Verses 10, 11. To find the H sine of an intermediate angle. Suppose it is required to find the H sine of an angle θ° ie θ × 60'. Divide this by 225; the quotient gives the previous H sine. Then (R × D) / 225 where R is the remainder, and D the difference between the previous and next H sines, added to the previous H sine gives the H sine required. Comm. The formula is evidently based on an applica- tion of rule of three. Verse 11. To find the angle when the H sine is given Suppose the H sine of an angle is given to be x'. Subtract the greatest H sine that could be subtracted from this. 15

114 Suppose the H sine of θ° could be subtracted. Let the remainder be r. Then (r × 225) / D where D is the difference between the previous and next H sines, added to θ gives the angle corresponding to x'. Comm. Evidently this is the converse of the previous process and this also is based on Rule of three.' Verses 12—15. The H sine of the obliquity of the ecliptic taken to be 24° is 1397. Now, the successive diffe- rences of the H sines will be given (on the basis of taking R = 120) which are known as Laghu-Jyās intended for ease in Computations, namely 21, 20, 19, 17, 15, 12, 9, 5, 2. These are given for intervals of 10°, so that if it be required to find the H sine of x°, let q be the quotient and r the remainder when x is divided by 10. q gvies the number of the previous H sine. Then (r × D) / 10 where D is the next difference or jyākhanda as it is called, added to the previous H sine gives the required H sine. In this table the H sine of 24° is 48' — 45". Also the H versines in this table are got by the reverse differences. To get the angle θ° for a given H sine say x' subtract the sum of as many differences (Jyā—Khandas) as could be from x. Let the remainder be r. Then (r × 10) / D where D is the next jyā—Khanda added to the previous angle upto which the jyākhandas have been subtracted, gives the required angle. The H sine will be more accurate if the Bhōgya-Khanda or the next H sine—difference is rectified (as per the rule of interpola- tion next given). Comm. H vers θ = R—H Cos θ = R—H sin (90—θ) so that H verse 3¾° = 3438—H sin (86¼°) = 3438—3431=7 as given in the previons table. Similarly in the above table of Laghu-Jyās, H vers 10° = R — H cos 10° = 120 — H sin 80° = 120 — (21 + 20 + 19 + ... + 5) = 2 so that the

115 above differences in the reverse order give the H versines. The rest of the contents of the verses is simple, the proces- ses being based on the ‘Rule of three’. Verse 16. Rectification of the next H sine difference known as Bhōgya—Khanda. The difference of the previous and the following H sine —differences being multiplied by the remaining degrees and divided by 20, the result is subtracted from the arith- metic mean of the previous and following H sine-differences to give the rectified H sine —difference, in question. Comm. This is a formula for interpolation which agrees with the interpolation formula given by Ball in his spherical astronomy on page 18 in the form y = yo + x/h (y₁ — yo) + x (x — h) / 2h² (y₂ — 2y₁ + yo). This formula is a re-statement of the formula enunciated by Brahmagupta in his work Brahma Sphuta Siddhānta as well as Uttara- Khandakhādya in the form “गतभोग्यखण्डकान्तरदलविकलवधात् शतैर्नवभिराप्तैः, तद्युतिदलं युतोनं भोग्यादूनाधिकं भोग्यम्” wherein in the place of ten-degree-interval, a fifteen-degree-interval ie 900'-interval was taken. Rule of three is a linear for- mula of interpolation, whereas the above is a quadratic formula reflecting much credit on the mathematical genius of Brahmagupta. We shall now see how the formula is applied and what mathematical significance it has. Suppose it is required to find the H sine of 24° from the previous table of H sine — differences given for intervals of 10° — from the table H sin 10° = 21, H sine 20 = 41, H sine 30 = 60 where R is taken to be 120. Now to find H sin 24°, we are asked to rectify the next H sine —difference namely 19', where the table is 21, 20, 19 etc. As a first approximation, applying rule of three H sin 24° = 41 + ⁴/₁₀ × 19=48·6=48'—36"

116 This is a crude approximation, the actual value being 48' — 48" — 14"'. Application of rule of three is justified if the H sine—differences are uniform, but they are not so, being in a decreasing order. So, the following H sine— difference namely 19' is to be rectified so as to be applicable at 24°. In other words we have to take such a H sine— difference which will hold good at 24°, not at 20° or 30° — from 10° to 20°, the H. S. d. (H Sine—difference) is 20', and from 20° to 30° it is 19'. If that be so what will be exactly at 24°? It should be less than 20' and greater than 19'. Now the argument advanced by Bhāskara is that the H Sine—difference at the mid-point of the 2nd and 3rd differences namely 20 and 19 should be (20 + 19)/2 = 19·5. The H sine—difference at the end of the third interval is 19. Then by the rule of three ‘If there is a decrease of 19·5 — 19 = ·5, during the course of the 10° of the third interval, what should the difference be for 4°? (where we have to find the H sine at 24°) The result is 4/10 × ·5 = (4 × 1)/20 which is given by the words “यातैष्ययोः खण्डकयोर्विशेषः शेषांश निघ्नो नखहृत् ”. This decrease makes the H sine— difference at 24°, 19·5 — 4/20 = 19·3 at 24°—Now the argu- ment to find the H sine at 24° is ‘If for an interval of 10°, the H sine—difference is 19·3, what should it be for 4°?’ The answer is 4/10 × 19·3 = 7·72. Hence the H sine of 24° is 21 + 20 + 7·72 = 48·72 = 48' – 43" – 12"' which is nearer the truth 48' – 48" – 14"' than what was obtained by the crude rule of three namely 48' – 36". Here we have to explain Bhaskara’s words more elabo- rately, because, Kamalākara happened to criticise Bhas- kara’s words in this context. Bhaskara Says “The H sine difference at the end of an interval is the arithmetic mean of the preceding and succeeding differences, whereas the succeeding one is that which holds good at end of the suc- ceeding. In between, we have to apply the rule of three to

117 obtain the rectified difference." What Bhāskara means is this. At the end of an interval, to obtain the H sine, it is enough to add the H sine difference belonging to that inter- val to the preceding differences. But when it is required to find the H sine in the interior of an interval, we have to construe that the difference at the end of the previous inter- val is the arithmetic mean of the previous and succeeding differences. There is apparently a self-contradiction in Bhāskara's words; for, at the end of the interval, according to his own words, the difference is that belonging to the previous interval and not the arithmetic mean as postula- ted. The contradiction will not be there when we read Bhaskara's mind that he means "When we require to find the H sine in the interior of an interval only, the difference at the beginning of that interval is to be taken as the arith- metic mean of the previous and the current differences, and that at the end of the interval the current difference holds good." The truth of Bhāskara's statement could be seen analytically as follows. The arithmetic mean of the prece- ding and succeeding H sine differences is (The context is to rectify the third H sine difference namely 19, for, we were finding the H sine of 24°) (AB + BC) / 2 (Ref fig. 8) where OA, AB, BC etc are the successive differences. (AB + BC) / 2 = (H sin 20 - H sin 10 + H sin 30 - H S 20) / 2 = (H sin 30 - H S 10) / 2 = R ((Sin 30 - S 10) / 2) = R × Cos 20 Sin 10 = (H Cos 20 H Sin 10) / R The numerical difference of the preceding and succeed- ing H sine differences is (i.e. यातैष्यखण्डविशेष). AB - BC = (H sin 20 - H S 10) - (H sin 30 - H sin 20) =

118 (2 H Cos 15 H sin 5) / R — (2 H Cos 25 H sin 5) / R = (2 H sin 5) / R (H Cos 15 — H Cos 25) = (2 H sin 5) / R × (2 H sin 20 H sin 5) / R Let now X be the point where we are to find the H sine (Here let us take it as x° after the previous interval for generalisation). Then Bhaskara's formula would give (H Cos 20 H Sin 10) / R — (2 H sin 5) / R² × (2 H sin 20 H sin 5) / 20 × x = (H Cos 20 H sin 10) / R — x / (10 R²) × 2 H sin 20 H sin² 5 = (2 H Cos 20 H sin 5 H Cos 5) / R² — 1 / R² × x / 10 × 2 H sin 20 H sin² 5 = (2 H sin 5) / R { (H Cos 20 H Cos 5 — x / 10 × H sin 20 H sin 5) / R } put now successively x = 0° and 10° to get the rectified differences at B and C respectively; then those rectified differences would be respectively H Cos 20 H sin 10 and (2 H sin 5) / R × H Cos 25. But (AB + BC) / 2 = (H Cos 20 H sin 10) / R (found above) and BC = H sin 30 — H sin 20 = (2 H Cos 25 H sin 5) / R In other words the rectified differences at B and C are respectively what exactly has been stated by Bhāskara. Hence Kamalākara's condemnation of Bhāskara is quite unjustified. Verse 17. To rectify the arcual difference to obtain the arc for a given H sine. Subtract as many H sine-differences as could be sub- tracted from the given H-sine. Half of the remainder multiplied by the difference of the preceding and succeed- ing H sine-differences and divided by the succeeding and

119 the result being subtracted from or added to as the case may be (added in the case of Hversines) the arithmetic mean of the preceding and succeeding H sine-differences gives the rectified H sine-difference while finding the arc for a given H sine. Comm. Let the given H sine be that of 24° found before i.e. 48·72. We could subtract 21 and 20 from this and the remainder is 7·72; half of this is 3·86 which multiplied by (20-19) is 3·86. This divided by the succeed- ing difference namely 19 is ·2 approximately. The arith- metic mean of the preceding and succeeding si (20 + 19) / 2 = 19·5. If the above result is subtracted from this, we have 19·5 − ·2 = 19·3. If for 19·3 we have 10° increment what shall we have for 7·72. The answer is (10 × 7·72) / 19·3 = 10 × ·4 = 4°. Hence the required arc is 20° + 4° = 24°. The proof is analogous to the previous proof. Having subtracted 21 and 20, the remainder is 7·72, (Ref. fig. 8) Fig. 8 so that BX = 7·72. Now during the course of the succeed- ing interval of 19, there has been a decrease of 19·5 − 19 = ·5 or to put it in general terms, during the course of y the succeeding interval BC there has been a decrease

120 (x + y)/2 - y = (x - y)/2 where x is the previous interval AB and (x + y)/2 is the H sine difference at B. Hence the argu- ment is "If for y, there has been a decrease of (x - y)/2, what will it be for δ (Here δ = 7·72) "? The answer is δ ((x - y)/2) × 1/y = δ/2 (x - y) × 1/y = अवशेषकार्धम् × गतैष्यान्तरम् ÷ एष्यम्. This result is to be subtracted from (x + y)/2 i.e. ·2 is to be subtracted from 19·5 to give the rectified H sine-difference. Verse 18. Definition of Kēndra and assignment of sign thereto. The excess of the longitude of the mean place over that of the apogee or aphelion as the case may be is called the mean anomaly. The excess of the longitude of the point called Sīghrōccha over that of the planet rectified by the first equation known as Manda-phala or equation of centre is known as the Sīghra-anomaly. The equation of centre is positive or negative according as 180 < m < 360 or 0 < m < 180 where m is the mean anomaly. The case will be reverse in the case of the sign of Sīghraphala, the second equation. Comm. “चन्द्रसूर्यौ स्फुटौ स्यातां मान्देनैकेन कर्मणा” i.e. The Moon and the Sun could be rectified by the equation of centre alone. This means that the Moon revolving round the Earth directly and that the Sun revolving rela- tively round the Earth are subject to only one correction namely the equation of centre for rectification. In the case of the Sun, though the fact is that he is revolving round the Earth relatively, assuming as Hindu astronomy does that he is revolving directly round the Earth and subjecting him to the correction of the equation of centre