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

95 resulted in a difference between the computed planetary positions and their observed positions. So, the originator of this Bīja-Saṁskāra, noting the differences in his own time devised a formula, which could account for those differences. But this formulation was bound to go wrong in later times as long as the daily motions are not corrected to the minutest extent possible and as long as the funda- mental basis of the conjunction of all the planets and pla- netary points is not proved. This seems to be the reason why so may texts were written incorporating small diffe- rences in different times as reported by Gaṇēśa (1507 A.D.) in his work Bṛihat-Tithi-Chintāmaṇi in the words "The calculations of planetary positions according to the methods indicated by Brahma, Vasishtha and Kasyapa Siddhantas held good in their own times, but grew obsolete later; Then Maya, the demon at the end of Krita obtained the science from the Sun God, which again grew obsolete in this Kaliyuga wherein parāśara began to hold the ground for a good length of time. Then Āryabhata rectified the methods; when even those methods grew obsolete, Durga- Simha, Varāha Mihira and others set them right. Again Brahmagupta came into the picture to rectify the methods by his own observations. Then Came Kēśava (Gaṇēsa's father) who rectified further. After a lapse of sixty years, his son Gaṇesa has now to correct the Science. If this also grows obsolete (as it is bound to) in course of time, let others again rectify it by observing conjunctions of the Moon and planets with the asterisms." Obsoleteness arises out of two contexts, one a justifi- able situation and the other based upon a wrong premise. The first is as follows. Suppose as a first approximation we take the length of an year as 365 days. We will have committed an error nearly ¼ of a day, so that the error accures to a day in 4 years. Thus the convention of the leap year arose so that during four years we give a day more to February. Here again we have overestimated the error by nearly 1/100th of a day. Hence in 400 years the

96 above correction leads to an error of a day. So, it is that we pronounced that out of the years 2000, 2100, 2200, 2300 A.D., the year 2000 A.D. alone is a leap year and not the remaining, the convention being that the number of the century, here 20, must be also a multiple of four. On this back-ground, suppose we prepare a manual called a Karaṇa grantha taking the length of the year to be 365·25 days. It works alright for some time but in the course of 400 years the error will have reached to as much as one day. Thus a manual like the above works only for a short time and the approximation made gradually brings in a divergence on account of which such a manual grows obsolete. That is why one Narasimha who happened to prepare a manual in 1333 Saka year (1411AD) opens his work with the words “तिथिचक्रं यत्प्रणीतं मल्लिका- र्जुनसूरिणा, कालेन महता तस्मिन् खिलीभूते तदादरात्, नौपुरीसिङ्गयार्यस्य नरसिंहेनसूनुना एतदेव स्फुटतरं क्रियते सौरसम्मतम्” i.e. “In as much as a manual named Tithi-Cakra prepared by one Mallikār- juna Sūri long ago, based upon the Sūryasiddhānta has now diverged far from the Sūryasiddhānta (on account of the approximations made transcending the limits of negligibi- lity) I, the son of one Singaya belonging to a place named Nau-puri (probably Vada-palle of the East Godavary Dt.) am rectifying it and bringing it to accord with the Sūrya Siddhānta again.” This kind of obsoleteness arising out of inevitable ap- proximations that have to be made in the preparation of manuals is permissible. But Suppose the premise of the manuals itself is incorrect, then the rectification of the manuals is no good so long as the data given in the pre- mise are not corrected. There are two fundamental defects in the ancient works according to a modern analysis namely (1) The Supposition that all the planets were in conjunction at the Zero-point of the Zodiac in the beginn- ing of a Mahāyuga (2) Small variations in the constants like the daily motion of the planets and the like. Accord- ing to the modern interpreters of Hindu Astronomy the

97 first premise was not correct. According to them, some astronomers having observed the daily motions of the pla- nets or what is the same the sidereal periods of the planets to a sufficiently good approximation calculated back or extrapolated a date on which these planets should have been in conjunction at the Zero-point of the Zodiac. The extra-polated date was naturally wrong to some extent be- cause the sidereal periods found could not but be correct only to a particular degree of approximation. Thus a little alteration in the number of sidereal revolutions alone or the number of days in a Mahayuga made to suit the obser- ved positions at a particular epoch would be only a tinker- ing of the problem and not a true solution. Thus Hindu Astronomy could be saved and its methods could still be followed provided instead of trying to presume a date at which all the planets were in conjunction (No doubt in the long bosom of time, such a presumption also could not be ruled out) correctly observed positions of the planets by the help of modern instruments were taken as the basis of an epoch and thereafter using more correct values of the constants such as the sidereal revolutions, maximum equa- tions of centre and maximum Sīghraphala, obliquity of the ecliptic etc. The second defect cited above thus being removed, and the original premise being changed, the methods of calculation still hold good and there would be no necessity to be going on with tinkerings of the problem. The Bīja-correction which we are commenting upon was rightly criticised by Kamalākara as irrational though he himself fanatically tried to uphold Surya Siddhānta. Even today there are a good number of the traditional Hindu Astronomers who do hold that the Sūrya Siddhānta was revealed to Maya at the end of Kṛtayuga in spite of the fact that scholars like M. M. Sudhākara Dvivedi pronoun- ced that it was an extra-polated work shortly after the time of Brahma-Guptāchārya. It is interesting to note that Bhāskara, a very rational astronomer, had before him the verse “त्रिंशत्कृत्वो युगे भानां चक्रं प्राक् परिलम्बते” of the 13

98 Sūrya Siddhānta (verse 9 ch. 3). He did not give it the interpretation that was later put upon it through the two subsequent lines "तद्दोः त्रिघ्ना" etc., which lines were not there evidently in Bhāskara's time. Without these two latter lines the rate of precession was too small to be accepted by Bhāskara and so he chose to follow Muñjāla rather than the Sūrya Siddhānta. Our Traditional astro- nomers today have no reservation to accept the greatness of Bhāskara and worship him though they do not question what necessity Bhāskara had to write another treatise and that too basing it upon the Āgama accepted by Brahma- gupta and not Sūrya Siddhānta, when there existed Sūrya Siddhānta before him and from which he had no objection to quote verses like "अदृश्वरूपाः कालस्य मूर्तयः" etc. (verse I ch. 2.) The Bīja-correction first incorporated by Brahma- gupta and later followed by a good number of astronomers because Śrīpati and Bhāskara accepted it, will not be accep- table to modern astronomers, though it might have worked well at the time of Brahmagupta and for some years later. The reason is that it is construed only as a tinkering of the defect as explained before. It is also to be noted that the originator of this Bīja- correction did not make it secular i.e. valid for all time increasing without a limit, for, then, the respective corrections transcend all limits and render the corrections meaningless. So, he said that the corrections would be increasing for 6000 years and thereafter begin to decrease to nothing. They were Zero at the beginning of the Kalī because all the yugas are multiples of 12000 years. Also the maximum correction is in the case of Mercury 6000/200 × 52 = 1560' = 26°. Let us see how far this is justifi- able. The daily mean motion of Mercury as given by Bhāskara is 4° - 5' - 32" - 18'" - 28"" whereas as per modern astronomy it is 4° - 5' - 37⅞" approxly. So there is a posi- tive error of 5" ²³⁄₄₀ which will accrue to 18' - 35" in the

99 course of 200 years. But as per the Bīja-correction it should be 52'. Hence it is a fact that there is a positive error but not so much as indicated. But it must be noted that Mercury's orbit has the highest eccentricity of as much as .2, and the observer who stipulated the correction must have observed when Mercury was near its perihelion, where the error could have been as much as indicated and even more. Similarly on close analysis it could be proved that the Bīja-correction should have been as indicated, say, roughly about 3600 Kali era, which might be roughly the date of its stipulation. Verses 9, 10. Concluding verses of the Madhyādhikāra. If the work is made more voluminous by describing various methods which are easy and interesting to un- intelligent people, learned men look down upon such a work as indulging in unnecessary verbosity. Hence the volume of a work does not add to its greatness; So I have made my work neither voluminous nor brief-worded. The reason is that both the intelligent as well as the unintelligent people are to be enlightened. For the sake of clarity of exposition, different ingeni- ous methods being used in such a way that the work does not exceed the normal limits of the previous works, and in corporating as far as possible unit numerators, fractions having numerators and denominators mutually prime, using methods of interpolation and reduction, making use of different kinds of denominators and numerators in many ways, this kind of treatment must be given to a work of this nature by an intelligent man. Comm. Easy. Before we proceed to the next chapter, we shall add here tables of astronomical constants as given by different authorities, which will help comparison and appreciation of the work.

100 TABLE 1 Number of sidereal revolutions of planets and planetary points in a Kalpa

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