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

of 29-30 and as such falls short of a lunation and (4) that the minimum length of a Solar month is 29-20-40 only. Then he gives a general and broad explanation as to how a Kshayamāsa occurs on the following lines. Let us call NS as the interval between a New Moon and the subsequent Saṁkrānti. In other words it is the Suddhi. Suppose Bhādrapada goes without a Saṁkrānti thus becoming an Adhikamāsa. Then in Āswayuja NS will have a small value, because the Saṁkrānti S which should have occured at the general rate of one per a lunation, has not occured before the lapse of Bhādrapada and being belated a little must have occured close on the heels of the New Moon of Bhādrapada. Let this interval NS have a particular value say x. Thereafter the Solar months grow gradually smaller i.e. the Saṁkrāntis occur earlier ; so in Kārtica the value of NS decreases. It might decrease to such an extent that the next Saṁkrānti might occur before the next New Moon. Suppose this does not happen in Kārtica ; then in Marga- sira the value of this NS is still smaller and there is a greater likelihood of another Saṁkrānti occuring during this Mārgasira. Then the value of NS being the smallest in the month of Pausha, another Saṁkrānti is bound to take place during this month at the latest. In other words S now occurs before the next N i.e. the Saṁkrānti occurs just a little before the next New Moon. Let now this SN be a small quantity say y; or what is the same the next NS or the Suddhi will be nearly a lunation. There after the Solar months begin to gain in length i.e. S occurs later and later. This means NS gains in length. Being nearly of a length equal to a lunation, and now gradually increasing it is bound to exceed a lunation shortly thereafter. This again means that S occurs later than the next New Moon, which is to say that one more lunation goes without a Saṁkrānti. Thus it is that another Adhikamāsa occurs shortly after the occurence of a Kshayamāsa. Hence in the course of one year there occur two Adhikamāsās and one Kshayamāsa which means again that the balance of an

79 Adhikamāsa is there in that year. In other words, the occurence of a Kshayamāsa during the course of an year as per a stipulated convention does not preclude the average occurence of an Adhikamāsa which is normally due after an average lapse of 32½ Solar months. That the occurence of a Kshayamāsa entails the occurrence of two Adhikamāsās on either side may be also seen in another way. Twelve lunations put together have a length of 354⅓ days approxi- mately during the course of which only eleven Saṁkrāntis could occur in case the initial Śuddhi is 21 days, for, sub- tracting this 21 from 354⅓, there remain 333⅓ days only which could contain only ten Solar months and not eleven. Ten Solar months are contained if there be only eleven Saṁkrāntis. Out of these eleven, two are consumed in a single lunation which happens to be a Kshayamāsa. Hence the remaining eleven lunations have to contain only nine Saṁkrāntis which means that two lunations have to go without Saṁkrāntis. In other words there are to be two Adhikamāsās during that year. Further both these Adhi- kamāsās could not occur on one side of the Kshayamāsa for the following reason. Since a Kshayamāsa could occur only during the course of Kārtica or Mārgasirsha or Pansha which alone are longer than the corresponding solar months, in the event of both the Adhikamāsās taking their place on one side of these months, they are to take place in nine lunations which are on one side of the three months beginning with Kārtica. This means that only seven Saṁkrāntis are to occur during the course of 9 × 29½ days = 265½ days. Even seven Solar months fall short of this period so that it is impossible that only seven Saṁkrāntis should occur in this period. Hence, the two Adhikamāsās which are to occur during the year containing a Kshaya- māsa have to take their place on either side of the Kshaya- māsa. Bhāskara mentions that when the daily motion of the Sun equals 61', the length of the Solar month will be 1800/61 = 29–30 approximately. He also states that the minimum

80 length of a Solar month will be 29-20-40. This will be so when the average daily motion during the course of a month is 61-20-30. Verse 7. A mention of the years past and future that had and will have Kshayamāsās. A Kshayamāsa occured in the Saka year 974 and will occur in the years 1115, 1256, 1378. Thus generally it occurs once in 141 years or even 19 years. Comm. We have seen before that according to the convergents given above a Kshayamāsa generally occurs in 19 years or 122 years or 141 years. In the course of the commentary Bhāskara says "19 years before or after" so that 141-19=122 years was also meant by him. He gives how in the course of 141 years or 19 years the Suddhi at the be- ginning of the year happens to be approximately 21 days. In this behalf, he invokes his formulation of the Adhika- māsās in the verse 6 under Pratyabdasuddhi. As per that formula the number of Adhikamāsās in 19 years will be 7-13-37-30 taken by Bhāskara as 7-13-40. In other words the Suddhi increases by 13 ghatis and 38 palas from what it was 19 years ago; thus practically the value of the Suddhi recurs in periods of 19 years which means that if a Kshayamāsa occurred this year, there is a likelihood of its recurrence after nineteen years. Of course the recurrence would not be certain because there is an excess of 13 ghatis Suddhi which might preclude its occurrence. Similarly in 122 years and 141 years the numbers of Adhikamāsās would be respectively 45 minus 7-15 ghatis and 52 plus 6 ghatis. 22 palas and 30 Vipalas, the latter being taken by Bhaskara to be 52 Adhikamāsās plus 6 ghatis and 20 palas. Thus Kshayamāsas are more and still more likely to recur in intervals of 122 years and 141 years respectively, the Suddhis at the beginning of the year almost recurring and assuming the original value of 21 days for a Kshayamāsa to occur.

81 Bhāskara adds that a Kshayamāsa occured in 974 Saka i.e. 62 years before his birth which knowledge he must have derived from hearsay or even by calculation. He then predicted its recurrence in the Saka years 1115, 1256 and 1378 which were at intervals of 141 years, 141 years and 122 years respectively. In the year 974 Saka i.e. 4153 Kali year, applying the above formula Dwidhābdāḥ the number of Adhikamāsās elapsed from the beginning of the Kali were 1531-21-12-52-30 i.e. the S'uddhi at the begin- ning of that year was 21-12-52-30. This naturally entailed the occurence of a Kshayamāsa. Thereafter during the years 1115, 1256 and 1378, the S'uddhis in the beginning of the years should be as per the above analysis 21-12-52-30 plus 68-22-30 i.e. 21-19-15; 21-25-37-30 and 21-25-37-30 and 21-18-23 respectively so that Bhāskara could forecast the occurrence of the Kshayamāsās in those years also. In this context it is worth-noting that Gaṇeśa notified in a verse that Kshayamāsās would recur in the years 1462, 1603, 1744, 1885, 2167, 2232, 2373, 2392, 2524, 2533, 2655, 2674, 2796, 2815 according to Sūrya Siddhānta and accord- ing to Āryabhaṭīya during the years 1482, 1793, 1904, 2129, 2186, 2251. These years also may be verified by computing the S'uddhis as said before and also with respect to Ārya- bhaṭīya according to which the number of Adhimāsas during a yuga differs by a little and as such effects a differ- ence in the sequence of the Kshayamāsa years. Verse 8. Tell me, how, when and in the course of how many years do two Adhikamāsas occur as mentioned by the Rishis? Questioned accordingly by an expert in questioning, if a mathematician could know the answer, I would reckon him as no other than Bhāskara (either the Sun-god or he himself i.e. Bhāskarāchārya) who could make the lotus-buds of mathematicians blossom. Comm. Probably taking up this challenge alone Gaṇeśa answered the question and gave the years cited above which should bring in a recurrence of Kshayamāsās. 11

82 N.B. It does not suffice merely to compute the S'uddhis alone in the beginning of the years but a rigorous compu- tation necessitates the calculation of the moments of New Moons and Samkrāntis also during the particular years as well, in as much as, the motion of the Moon also comes in- to the picture and it differs from month to month on account of the rapid motion of his apogee. Here ends the section named Adhimāsādi-nirṇaya.

THE SECTION BHŪ-PARIDHI-MĀNĀDIKA- CIRCUMFERENCE OF THE EARTH Verse 1. The circumference of the earth's globe is 4967 yojanas ; its diameter 1581. A yojana is equal to (d × 360) / (C × δϕ) where δϕ is the difference in the latitudes of two places on the same terrestrial meridian in degrees, C the circumference of the earth's globe given above and d the distance between the two places. Comm. The second half of the verse gives the method of computing the circumference of the earth's globe, which is mathematically correct ; for, by the rule of three "If by a difference of 90° in latitude we have ¼ C, what shall we have for 1° difference in latitude ? " The answer is C / 360° Again "If by distance of d between two places on the same meridian, we have a difference of δϕ° in latitude, what shall we have for 1° difference in latitude ? ' The answer is d / δϕ. Both the answers must be the same ; so equating them, we have C / 360 = d / δϕ = number of yojanas = x say. Hence one yojana = x / x = (d / δϕ) ÷ (C / 360) = (d × 360) / (C × δϕ) N.B. Here it must be noted that a yojana's length is derived from the number of yojanas contained in the circumference as reported in the Āgama. In the course of the commentary under this verse, Bhāskara explains why he had recourse to this kind of definition, which is based upon āgama and as such does not contain a proof. He says that in as much as the defi-

84 nition of a yojana was given basing ultimately on the units of Angulas and yavas (grains of paddy) differently by differ- ent authorities, and in as much as the circumference of the earth’s globe was more or less unanimously accepted, he has chosen to define a yojana on the accepted measure of the circumference. In some other place Bhāskara says that a yojana is equal to 4 Krōsās, where the word Krōsa etymologically means that distance through which the topmost voice of a healthy person could be heard by another healthy person having good audition. In this context it may be recalled how Śrīpati defined a yojana under verses 69, 70 Madhyamādhyāya of his Sid- dhānta Śekhara. “The minute speck of dust that is seen flying in the light of the Sun’s rays entering a house through the windows, is called a paramāṇu (not an atom as is being translated now). Eight such paramāṇus equal one Reṇu. Eight Reṇus equal the breadth of the end of a hair known as Vālāgra or Kaccha-mukha or yūka. Eight Vālāgras equal one Likshā; Eight Likshās equal one Yava; eight Yavas equal one Angulā; twelve Angulās make what is called a Vitasti (i.e. the expanded length of a palm of the tallest person). Two Vitastis equal one Hāsta. Four has- tās make one Chāpa. Two thousand Chāpas make one Krōsa and four Krōsās are reported to be equal to a yojana by Astronomers” Verse 2. Rectification of the circumference .ie. find- ing the length of the circumference of the earth parallel to the terrestrial equator and passing through the locality in question. Also the definition of the primary meridian. The equatorial circumference of the earth multiplied by cos ϕ and divided by R or multiplied by 12 and divided by the hypotenuse of the right angled triangle formed by the gnomon and the equinoctial midday shadow thereof. (Hereafter called equinoctial hypotenuse) gives the circum- ference of the earth parallel to the equator and passing

85 through the locality (hereafter called the rectified circum- ference). Also the primary terrestrial meridian is that longitudinal line passing through the places (1) Lanka (2) Ujjain (3) Kurukshetra and (4) the north pole. Fig. 4 Comm. (Ref. fig. 4) Let ABC be the terrestrial Equa- tor, and QST be a circle parallel to this terrestrial equator and passing through the locality S. This QST is called the rectified circumference of the earth at the place S. The terrestrial equator is defined as follows. It is known that the earth rotates about herself about the axis P₁ P₂ where P₁ P₂ is called the polar axis or Dhruvayashti. In other words the entire heavens appear to revolve round the earth in such a way that any star will appear to be revolving in a circle called its diurnal circle which is parallel to the circle ABC. The points P₁ and P₂ are called the north and

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