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

86 south pole respectively. These two points evidently do not move, though the earth is herself rotating in the clockwise direction. If O be the centre of the earth, OP₁ produced meets the skies at a point called the north celes- tial pole and OP₂ produced meets the skies in the point called the south pole. It so happens that the north celes- tial pole is very near a star which is called the pole star. This star is known as the Dhruva-Tāra in as much as it does not appear to move at all while all the heavens (i.e. all the stars of the sky) appear to be revolving round the earth rising and setting as seen at any place. (The word Dhruva means fixed). Aryabhaṭāchārya mentioned in so many words that it is the earth that really rotates and so the stars which are themselves fixed appear to be going round the earth in circles parallel to the circle ABC. अनुलोमगतिः नौस्थः पश्यत्यचलं विलोमगं यद्वत्। अचलानि भानि तद्वत्समपश्चिमगानि लङ्कायाम्॥ (Explained under verse 7 Bhagaṇādhyāya). The celestial Equator is the great circle which is the circle of intersection of a plane perpendicular to the polar axis and passing through the earth's centre with the celestial sphere (celestial sphere is the sphere-like surface which shape the Sky takes and on which the stars and the planets appear to be studded.) Similarly the terrestrial equator ABC is the circle of intersection of the earth's globe with the same plane. Thus the terrestrial and the celestial equators are concentric coplanar circles with the earth's centre as the common centre. A great circle of a sphere is a circle whose plane passes through its centre. Thus ABC is a great circle on the earth's surface, because its plane passes through the earth's centre. Similarly the celestial equator is a great circle of the skies. The circle QST is called a small circle, just as the diurnal circle traced by any star in its diurnal rotation is also a small circle parallel to the celestial equa- tor. Thus small circles, an infinity of them can be drawn parallel to the terrestrial equator ABC and they will be in decreasing dimension as we proceed towards the pole. Hence the Sphutaparidhi or the rectified circumference

87 QST at the locality S is a small circle whose circumference is smaller than that of ABC. The problem is now to find the length of this circle QST. Evidently QST / ABC = 2 π r / 2 π R = r / R where r and R are respectively the radii of the two circles. But r / R = Cos ϕ from the triangle OSO', where Ô'₁SO = ŜOA = latitude of the place S. ∴ QST = ABC × Cos ϕ = (ABC × H Cos ϕ) / R I where H Cos ϕ is the Hindu cosine of ϕ known as lambajyā and stands for OS₁ = O'₁S = r. This lambajyā is also called Dyujya if the small circle is the diurnal circle of a Star. The diurnal circle of a Star is called Dyujya-Vritta and its radius is called Dyujyā-Equation I proves the first statement of the Fig. 5 verse. In fig. 5 Gn N is called the fundamental gnomonic triangle where Gn is the vertical gnomon pointing to the Zenith Z of the celestial sphere and is considered to be of 12 units (Angulas as they are called), GN the midday- shadow of the gnomon cast on an equinoctial day when the

88 Sun is at the point Q where Q is the point of intersection of the celestial equator with the meridian of the place. If the equinoctial shadow be denoted by ' s ' and the hypo- tenuse of the triangle nGN namely nN be denoted by K (called the Vishuvatkarṇa or the equinoctial hypotenuse) then s / K = Sin ϕ and 12 / K = Cosϕ. If H Sin ϕ be the Hindu sine of ϕ called the Akshajyā H Sin ϕ = R Sin ϕ so that s / K = H Sin ϕ / R and 12 / K = HCosϕ / R = Lambajyā / Trijyā . Substituting for H Cos ϕ / R the value 12 / K in equation I above, we have QST = Sphuṭa - paridhi = (Bhū - paridhi × 12) / K II which proves the second statement given in the verse. Regarding the third statement, which defines the Bhu- Madhya-Rekhā (In modern text books of geography the terrestrial equator is spoken of as the Bhu-Madhya-Rekha. The terrestrial equator is called Niraksha-Rekha in Hindu Astronomy which means the circle of Zero-latitude), it is the primary meridian taken by Hindu Astronomers. In modern astronomy the primary meriodian is taken as the Greenwich meridian. Bhāskara has given four places locat- ed on the Hindu primary meridian, but, Śrīpati gives many more places located on this primary meridian under verse 96 Madhyamādhyāya namely (1) Laṅkā (2) Kanyākumārī (3) Kāñchī (4) Pannāta (5) the six-faced white mountain (6) Sri-Valma-gulmam (7) Māhishmatī (8) Ujjain (9) An Āsrama (10) Pattasiva, a town (11) Sri Gargarāna (12) Sthānviswara known also as purohita (13) Sītagiri and (14) Sumeru. Some of these places cannot be properly identi- fied but the following remarks may be made (a) Pannāta is one of the fifty-six small countries into which India was divided in ancient times according to the purāṇic literature

89 (b) It is not clear what places are indicated by (5) and (6) cited above (c) In some works Māhishmatī and Ujjain are used synonymously. (8) is not clear. Regarding (9) there is one pattasīva near Rajahmundry but S'rī pati does not seem to have meant it. Again (10) is not clear, Regard- ing (11) it is to be noted that the place is now pronounced (probably mis-pronounced) as Sthānes'wara. If (12) means the Himalaya mountain, there is not much meaning to say that it lies on the primary meridian; only a cross-section of it could lie there upon. The entire mountain extends from west to East over more than a thousand miles. Verse 3. To find the correction known as Desāntara. The distance between two places on the same latitude multiplied by the daily motion of a planet and divided by the rectified circumference is a correction subtractive in the east and additive in the west of the primary meridian in the planetary position obtained. Comm. In Hindu Astronomy the mean planetary positions are first calculated for the Sun-rise at the primary meridian. Now suppose a place lies to the east of this meri- dian. Then the Sun-rise at the place happens to occur earlier than on the primary meridian. Hence the correc- tion in the mean computed position of the planet is nega- tive if the position were to be calculated for the local Sun- rise. If the place happens to be on the western side of the primary meridian the reverse holds good i.e. the correction is to be additive. The amount of the correction is the amount of the motion of the planet in between the two Sun-rises. Let the planet move an arc equal to δm per day i.e. it moves δm when the earth rotates once about her axis. The time between the two Sun-rises above is the time by which the local meridian is carried through the distance between the locality and the primary meridian's point of inter-section with the latitudinal line or what is the same through the arc of the rectified circumfer- 12

90 ence of the earth pertaining to the locality. If d be this distance then the rule of three to be used is “ If the length of the rectified circumference viz. C rotates by the time the planet moves a distance δm, what is the arc traversed through by the planet if an arc ‘d’ of C rotates through ? ’’ The answer is (d × δm) / C which is the cor- rection required. Verses 4, 5, 6. The Correction Desāntara expressed in time. The eclipse of the Moon occurs at a place situated on the east of the primary meridian later than on the primary meridian and vice versa. The time in between the two moments is the Deśāntara expressed in time. The distance of Desāntara ie. the distance of the locality from the pri- mary meridian measured along a parallel to the terrestrial equator or Niraksha Rekhā is obtained by multiplying the rectified circumference by the Desāntara measured as above in ghatis and dividing by 60. Also the above time in ghatis multiplied by the planets’ daily motion and divided by 60, gives the correction in arc in the computed mean planetary motion. Further the week-day begins after or before the local Sun-rise by that Desāntara expressed in time according as the locality is on the east or west of the primary meridian. Also the week-day begins after or before the local Sunrise by the ghatis of the correction known as chara according as the Sun is in the northern or Southern hemisphere. Comm. An eclipse is first computed for the primary meridian. If an observer wants to know whether he lies east or west of the primary meridian and to know the Desāntara correction in time, the following procedure is to be adopted. Let a lunar eclipse begin x ghatis after the Sun-rise of the primary meridian. Let the observer note the time y ghatis which have elapsed after Sun-rise at his own place when the eclipse begins. Since a lunar eclipse

91 begins simultaneously for any place of the earth, if y > x, then he should know that he lies on the east of the primary meridian because his Sun-rise happens to be earlier than the Sun-rise on the primary meridian. Also the difference y—x gives the Desāntara correction in time for his place. The converse is the case if he happens to lie on the western side of the primary meridian. The time at which the eclipse takes place on the primary meridian after the Sun- rise there which is obtained by computation is called Drik- grahaṇa - Kāla ; whereas the local time after Sun-rise observed by the observer is called pragrahaṇa-Kāla. Their difference is therefore the Desāntara correction in time. If the Desāntara is to be got in yojanas, (T × C) / 60 is the answer, where T=y—x and C is the rectified circumference of the earth, for, if a difference of 60 ghaṭis be there for C yojanas, what should be the distance in yojanas in order that the difference is T”? The answer is as given above. Hence to obtain the positions of the Sun and the Moon at the beginning of the eclipse at the locality we have to add or substract as the case may be (T × δm) / 60 where δm is the daily motion of the Sun or the Moon, and T is y—x cited above, for, “ If in 60 ghaṭis the motion be δm, what would it be in T ?” is the rule of three for which the answer is as stated above. Now the question is when the week-day begins for the locality. It must be noted clearly, that in Hindu Astro- nomy the moment of Sun-rise at the primary meridian alone is to be reckoned as the beginning of the week-day universally. This convention is adopted for convenience. Thus the astronomical week day for any locality does not begin from the Sun-rise of the locality, but may begin earlier or later. This difference is given by y — x cited above.

92 There is yet another subtlety in the commencement of the week-day, arising out of the latitude of the place. The former analysis pertains to the longitudinal difference. The difference arising out of latitude between the local Sun-rise and the Lanka-Sun-rise is given by what is called Chara-Kāla. Since the week-day begins at Lanka Sun-rise and the local Sun-rise differs from the Lanka Sun-rise not merely by a longitudinal difference but also by a latitudi- nal difference, to compute the actual beginning of the week-day before or after the local Sun-rise, we have to take into account both the differences cited above. In other words, computing the local Sun-rise and also the Lanka Sun-rise, we have to decide the beginning of the week day before or after the local Sun-rise. Verses 7, 8. The correction called Bijakarma for the planetary positions. The number of years from the beginning of the Kalpa divided by 12000, the remainder, or the difference of the divisor and the remainder whichever is less is to be divided by 200. The quotient in minutes of arc, multiplied by 3, 5, 5, 15, 2 respectively is a negative correction in the positions of the Sun, Moon, Jupiter, Venus, and the lunar apogee and multiplied by 1, 52, 2 and 4 gives the positive ocrrection in the positions of Mars, Mercury, the lunar Node and the Saturn respectively. Comm. By the phrase 'The remainder or the diffe- rence of the remainder and the divisor', it is plain that the corrections positive or negative increase for 6000 years and decrease for the next 6000 years. Bhāskara gives no reason for these corrections, but, we have to construe these correc- tions on the following rational grounds. Bhāskara, however, says that the corrections were accepted by him on the basis of Āgama. This Āgama-stipulation was there in Brahma-Sphuta-Siddhānta and was later incorporated by Sripati also in his Siddhānta-Sekhara and as such was

98 accepted by Bhāskara also. However, in Brahma Sphuṭa Siddhānta both as first published as an edition of M. M. Sudhākara Dwivedi and later by the late Rāmaswarūpa Śarmā in 1966, the verses 59, 60 of Madhyamādhikāra suggest that the corrections are negative in the case of all the planets; whereas both Śrīpati and Bhāskara make them positive in the case of the latter four viz. Mars, Mercury, the lunar Node and Saturn. By this we have to construe that Śrīpati and Bhāskara must have had before them a text which should have read ‘स्व’ in the place of ‘च’ in the last pāda of verse 61. M. M. Sudhākara-Dwivedi did not notice this anomaly of the positiveness of the correction with respect to the latter four, but he remarked, however, that there was a prosodial lapse in the last pāda of verse 61, for which he offered a suggestion that instead of वेदैर्, we had better read वेदैः:—This suggestion, no doubt, rec- tifies the prosody of the verse, but not the the anomaly cited above which was not noticed by M. M. Sudhākara Dwivedi. So, we have offered our own suggestion namely that in the place of च as mentioned above if we read स्वं, we not only rectify the prosodial error but also the anomaly referred to. Rāmaswarūpa Śarma noted the anomaly but did neither refer to the prosodial error nor offer a correc- tion. It seems that Rāmaswarūpa Śarma did not verify the corrections stipulated from the verses 91, 92, 93 of Madhyamādhyāya of Siddhānta Śekhara. In this latter work, there is another anomaly namely that in the case of Mercury, the number 62 is the multiplier and not 52. Makkibhaṭṭa, the ancient commentator had before him a text which read 62 in the place of 52, in all probability, a mistake of the scribe. M. M. Sudhākara Dwivedi is repor- ted to have later pronounced that 52 must be the correct figure when this was brought to his notice as this number 52 was found both in Brahmagupta and Bhāskara. As re- ported by the editor of Siddhānta Śekhara Pandit Babuaji Mishra, who mentions this latter pronouncement of Sudhā- kara Dwivedi his teacher, also says that Sudhākara Dwi-

94 vedi suggested the reading द्विशर in the place of द्विरस of verse 93 of Siddhānta Śekhara. The fact that Makkibhaṭṭa commented द्विरससङ्गुणं as द्विषष्टिसङगुणं shows that he did not consult Brahmasphuta Siddhānta in this place; also, he must have had a manuscript before him which scribed द्विरस in the place of द्विशर. Using श ष, स, indiscretely is not uncommon in many books of North India, from a long time and the scribe of the manuscript probably having used स in the place of श and then by an oversight a latter scribe having inverted सर as रस, Makkibhatta must have commented like that. Incidentally a remark may be made here about Makki- bhatta. He was evidently a keralite because he used letters to signify numbers as was a common practice among the Kerala Astronomers, and as he also commented upon Brihad- Bhāskarīya. Further, it is interesting to note that he wrote in his commentary under verse 39 of the Sādhanā- dhyāya of Siddhānta Śekhara viz. “भभ्रमोऽर्कमण्डलान्तरं सावनानि कुदिनानि तानिवा”, “भूमेः प्राङ्मुखी भ्रमति” etc”. This idea shows that he accepted Āryabhaṭa's verse “अनुलोम गतिः etc” implying that the earth is rotating. Bhāskara says that the Bīja correction mentioned was purely based on Āgama and Upalabdhi (meaning authority and observation'). M. M. Sudhākara Dwivedi seems to have reiterated the same as reported by Babuaji Mishra, in a foot-note. Kamalākara, condemned this Bījakarma as it was unwarranted and had no proof. A rational explanation as to why this Bīja-Karma was prescribed either by Brahmagupta himself or some autho- rity which he seems to have accepted may be given as follows. The small differences in the numbers of sidereal revolutions or what is the same the minute differences in the accepted daily motions of the planets and the assump- tion of a conjunction of all the planets and planetary points at the beginning of Kalpa, which is beyond proof,

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