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

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

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

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

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

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

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

121 does not alter the mathematics that goes into his rectifi- cation. According to modern astronomy the Sun and the Moon, one relatively and the other directly go round the Earth, in ellipses, the Earth being in one focus whereas in Hindu astronomy both the Sun and the Moon are taken to be going round the Earth in eccentric circles i.e. circles whose centres do not coincide with the centre of the Earth. In fact, Bhāskara says in so many words “ भुमेर्मध्यं खलु भवलयस्याऽपि मध्यं यतः स्यात् , यस्मिन् वृत्ते भ्रमति खचरो नाऽस्य मध्यं कुमध्ये, भूस्थो द्रष्टा न हि भवलये मध्यतुल्यं प्रपश्येत्, तस्मात् तद्ज्ञैः क्रियत इहतत् दोःफलमध्यखेटे i.e. The centre of the celestial sphere coincides with the centre of the Earth. The centre of the circle in which a planet goes does not coincide with the centre of the Earth. Hence an observer on the surface of the Earth finds the True planet’s position differing from that of the mean planet, so that what is called the correc- tion of Bhujaphala is to be made in the mean position of the planet to get the True position.” Here it has to be noted that the Bhujaphala mentioned stands both for the equation of centre and the second equation known as Sīghraphala as well. One may wonder how it could be so, but it may be noted that in the formulation of both the equations, the centre of the eccentric does not coincide with that of the Earth and also in both the cases the equa- tion contains the term H sine of the anomaly where the word anomaly whether it be of the first or second equation is known as ‘Bhuja’. . In the case of the other planets, the fact is that they go round the Sun in elliptic orbits, the Sun being in one focus. In Hindu Astronomy, we shall see that the centre of the eccentric circle in which these planets are taken to revolve coincides with the Sun. Hence, though the ancient Hindu Astronomy postulates geocentric motion, the mathe- matics that goes into the formulation of the second equa- tion, makes the Sun’s centre the centre of planetary revolu- tion. One may wonder again how the equation of centre 16

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formulated by Hindu Astronomy agrees with its formula- tion in Modern Astronomy which enunciates elliptic motion; but it will be seen that the eccentric-circle theory also gives very approximately the same formula for the Equation of centre. There is just one point of difference, which does not matter. Whereas in Modern Astronomy the mean anomaly is reckoned from the perigee or perihelion as the case may be, it is reckoned in Hindu Astronomy from the apogee or aphelion. The difference is made up by prefixing the appropriate sign to the equation. The word Mandōccha stands for the apogee in the case of the Sun and the Moon and for the aphelion in the case of the other planets and the word Manda kendra stands for the mean anomaly in both the cases. In Hindu Astronomy the word ‘graha’ stands for not only the five planets Mercury to Saturn but also for the Sun and the Moon. Why that word is applied to the Sun and the Moon as well is, that both the Sun and the Moon also while moving among the stars along with the five other planets Mercury to Saturn, wield an influence on the residents of the Earth. The etymology of the word ‘graha’ is गृह्णातीति वा गृह्यते अनेनेति वा ग्रहः ie. ‘that which seizes upon the fates of the resi- dents of the Earth’, with this etymological significance only, even the lunar orbital nodes known as Rāhu and Kētu are also taken to be grahas in Hindu Astronomy. Hence translating the word graha as a planet and criticis- ing Hindu Astronomy for taking the Sun, Moon and the lunar orbital nodes also as such is not right. In other words the translation should be pronounced wrong. Uranus, Nep- tune and pluto were not mentioned in Hindu Astronomy. We shall now elucidate the eccentric and epicyclic theories of Hindu Astronomy, which will be seen to give identical position to the planets. How they came to be postulated will be also elucidated. Incidentally we deal with the ‘Bhagaṇōpapatti’ or the proof of the numbers of sideral revolutions of the various grahas enunciated in the

123 beginning of the Madhyādhikāra, Bhagaṇādhyaya in verses 1 to 6. Even Bhāskara gave such a proof as appealed to Āgama ie. 'Authority', which proof therefore will not be acceptable to a student of Modern Astronomy, who is likely to question how the Āgama came into existence. How the Āgama came to formulate the number of sidereal revolutions of the grahas, we shall now see. The forefathers of Hindu Astronomy (have been re- ported to be eighteen in number in the famous verses " सूर्यः पितामहो व्यासो वसिष्ठोऽत्रिः पराशरः कश्यपो नारदो गर्गो मरीचिर्मनु- रङ्गिराः लोमशः पौलिशश्चैव च्यवनो यवनो भृगुः, शौनकोऽष्टादशैतेस्युः ज्योतिषशास्त्रप्रवर्तकाः " Of these Brahmagupta mentions Brahma-Siddhānta, which he reports to have resuscitated. Varāha Mihira gave a version of the old Sūryasiddhānta, mentioning that it accorded with observations. Āryabhata says that he revived his system from the then existing ocean of knowledge both good and spurious. The fact that none of these outstanding astronomers mentioned that they had derived their systems from a foreign source, and the reasonableness in presuming that all these three could not be impostors, make the author of this work feel strongly that there should have been some works in the name of Āgamas extant long before these Acharyās. The argument that the crude Vedānga-jyotiṣa alone should have existed before Āryabhata, simply because, no other work worth the name has been discovered, may not be correct. It is quite possible that crude works could exist side by side with advanced scientific works, just as even nowadays we have thinkers and works of a primitive type existing along with highly advanced thinkers as well as scientific works). Bhagaṇopapatti. In the first place, the forefathers of Hindu Astronomy must have noticed very easily that the Moon has a motion among stars, for, this could be detected

124 even by a lay man during the course of a single night. So, the period of a single sidereal revolution, could be roughly recognized by noticing the conjunction of the Moon with a luminous star. Having thus observed a good number of sidereal revolutions, which could be done even with the naked eye, the average period could be arrived at with sufficient accuracy within the course of a few years. Having thus obtained almost accurately the average of a sidereal revolution of the Moon, the sidereal revolution of the Sun could have been arrived at as follows. The moment of an eclipse solar or lunar could be observed with the naked eye.-Observing a good number of eclipses within the course of a few years, the average of a lunation could be easily arrived at very accurately, for, in between the eclipses of the same nature an integral number of lunations elapse. That the Sun also has a motion among stars must have been noticed clearly during the course of a few months, for, observing at Sunset the star that was rising, it should have been noticed gradually even during the course of a month, that the Sun must have been approach- ing the star or vice versa and as the stars were found to keep the distances amongst them constant, it was the Sun that was approaching the star and not the star it was that was approaching the Sun. Having decided thus that the Sun was moving among stars from west to East, the approximate period that the Sun took to complete a side- real revolution was arrived at. Then, as both the Sun and the Moon were having east ward motion, and as the Moon has a more rapid motion, the arc by which the Moon over- takes the Sun during a day was roughly noticed. Thus arriving at a rough estimate of a lunation within which a conjunction of the Moon with the Sun recurs, it was noticed that the excess of the sidereal revolutions of the Moon over those of the Sun gave the number of lunations. Since a correct estimate of both a sidereal revolution of the Moon as well as that of a lunation were previously arrived at, the number of sidereal revolutions of the Sun during

125 the course of a certain period were computed whereby a correct estimate of a sidereal solar year was arrived at. This period could also be checked simultaneously by obser- ving the interval between the heliacal risings or settings of a particular star of the Zodiac as well. Thus far, we have seen how the sidereal periods of the Sun and the Moon were determined very accurately. It may be noted that these periods as determined by the Hindu astronomers were correct to a good number of decimal places. When once the Sun and the Moon were found to be having eastward motion among stars, and when it was discovered that there were other luminous bodies like the Jupiter and Venus etc. moving among stars, it was attempted to determine their sidereal periods. It must have been done as follows. In the first place, it was noticed that these other luminous bodies which were five in number, namely Mars, Mercury, Jupiter, Venus and Saturn, were found to be having retrograde motion also unlike the Sun and the Moon. As these five bodies were looking like stars they were named Tārā-grahas i.e. grahas looking like stars. Also a distrinction could be drawn very easily between Mercury and Venus on the one hand and the other three on the other, for, the former were always found oscillating about the Sun, never parting from him through long distances. Thus during the course of a suffi- ciently long interval, the geocentric sidereal periods of Mercury and Venus coincided with that of the Sun. In other words, it was taken that the geocentric sidereal periods of Mercury and Venus also were taken to be an year. This is clear from the statements made by all the Siddhāntas that in a Kalpa of 4320000000 years the Sun, the Mercury and Venus all the three make 4320000000 sidereal revolutions. Then with respect to the other three planets Mars, Jupiter and Saturn, it was noticed that they were having a pre-dominantly longer period of direct motion, though there was a retrograde motion for some time. This

126 gave the clue to arrive at an approximate estimate of their sidereal revolutions. But the correct estimates were arri- ved at not by observing their conjunctions with stars, for, that would take a very long period of observation in the case of Saturn, but, by observing a good number of their heliacal risings or settings. The interval between two consecutive heliacal risings or settings being a little grea- ter than an year, ten or fifteen observations could be done very easily by a single person. Then the aforesaid argu- ment given in the case of finding the sidereal revolution of the Sun, was also advanced in the case of these three planets Mars, Jupiter and Saturn. Let x° be the arc that the planet covers during the course of a day. Let a° be the arc covered by the Sun during the same period which was previously known and that correctly. So, during the course of a day the Sun overtakes the planet by (a - x)°. Hence to overtake 360°, the period S was computed. This period was observed as the interval between two consecu- tive heliacal risings or settings and known as the synodic period. So, from the equation 360 / (a - x) = S, the value of x could be arrived at, wherefrom p the sidereal period was determined. This sidereal period could be also determined in another way. Noting the distance covered among stars by a planet during the course of a synodic period, using rule of three, the sidereal period could also be arrived at with a good accuracy, for, the retrograde motion affects equally each synodic period. An average of such determi- nations made in two ways could give the sidereal periods of Mars, Jupiter and Saturn with a good amount of accu- racy. It will be noted here, that the average of a good number of geocentric sidereal periods in the case of these planets (called Superior) is also the heliocentric sidereal period (for a proof of this statement reference may be made to page 80 of the author's 'A critical study of the ancient Hindu astronomy, published by the Karnatak University Dharwar). It is why the sidereal periods of these planets

127 as given by Hindu astronomy tally with the heliocentric sidereal periods given by modern astronomy. This is also one of the reasons why heliocentric motion of the planets could not be detected by Hindu astronomy, and also why a statement was made that "In the case of Mercury and Venus the Sun was the planet. and they are termed as S'īghrōcchas, whereas in the case of Mars, Jupiter and Saturn, they are the planets while the Sun plays the part of S'īghrōccha "कुजजीवशनीनां तु रविः शीघ्रोच्चनामकः, ज्ञशुक्रयोः ग्रहः सूर्योभवेत्, तौ शीघ्रनामकौ" (An elucidation of this state- ment will be given shortly) In the case of the planets Mercury and Venus (Inferior planets) one may wonder how under the geocentric theory, their heliocentric periods could be arrived at, though they were not recognized as such but were pronounced as the "geocentric periods (not considered as heliocentric)" of two points known as their S'īghrōcchas. Here we come across the peculiar concept of a S'īghrōccha which arose out of the fact of postulating a geocentric system in the place of the heliocentric. This concept is to be elaborated, in as much as confusion is there in the minds of many interpreters of Hindu astronomy in this behalf. In the first place let us consider as to how the rectifi- cation of the Sun and the Moon, known as sphutikaraṇa was achieved. Having got their sidereal periods, their mean daily motions were calculated. Also a period was conceived, during which this Sun and the Moon would per- form an integral number of revolutions. This period was termed as a Mahāyuga (or simply a yuga as we hereafter name it) whose duration was estimated as 4320000 solar years. That the yuga is an integral L.C.M. so to say of the sidereal periods of the Sun and the Moon (also of the other planets as we shall see shortly) could be seen from the statement of the Surya Siddhānta युगे सूर्यज्ञशुक्राणां

128 खचतुष्करदार्णवाः, कुजार्किगुरुशीघ्राणां भगणाः पूर्वयायिनाम्, इन्‍दो रसाग्नित्रित्रीषुसप्त भूधरमार्गणाः '' i,e. ‘In a yuga, the Sun the Mercury and Venus perform 432000 sidereal revolutions as well as the Śīghrōcchas of Mars, Jupiter and Saturn, whereas the Moon performs 57753336 revolutions' (It may be recalled here that the Mercury and Venus are oscillating about the mean position of the Sun; also it will be noticed that the Sun playing the part of the Śīghrōcchas in the case of Mars, Jupiter and Saturn, their Śīghrōcchas are also deemed as making the same number of revolutions as the Sun. In as much as the Śīghrōcchas in the case of Mercury and Venus are looked upon as different from the planets, so in the case of Mars, Jupiter and Saturn also the Sighrocchas are taken as different Divine entities though coinciding with the Sun in position). However smaller periods could be conceived as integral L.C.M's of the side- real revolutions of the Sun and the Moon, but a presump- tion sponsored by a sense of orderliness in the Cosmos, that the planets should all have been started from the Zero point of the Zodiac, made the integral L.C.M. to be of such a dimension as 4320000 solar years in which period the other planets also would have made an integral num- ber of sidereal revolutions. Here in this point the tradi- tional Hindu astronomers place their faith in the Āgama, which said that the planets were all started at the Zero- point of the Zodiac in the beginning of the yuga and were ordained to return to the same point at the close of the yuga. Even a rational astronomer like Bhāskara, appa- rently placing faith in the Āgama, while adducing a proof in the name of Bhagaṇōpapatti, states that after obtaining the mean daily motions of the planets, calculates them for the period of a Kalpa taking it on trust that the planets were started at the initial point of the Zodiac in the begin-

129 ning of the Kalpa, A modern astronomer, however, ques- tions the assumption that the planets were all started at the first point of the Zodiac, and even though they might all have been in conjunction at that point in some remote past, whether it was the initial point of the reported Kalpa. Proceeding on the basis of the reported initial conjunction of all the planets at the first point of the Zodiac, and calcu- lating the number of days that have elapsed from the begin- ning of the yuga, the mean positions of the Sun and the Moon were computed. Noticing that these mean positions did not exactly accord with the true observed positions, the ancient astronomers tabulated the differences between those mean and true positions. These differences were found to be zero at two diametrically opposite points, and maximum roughly at two points differing by a quadrant from them. To account for these differences, the thought that occured to their minds was that probably the Sun and Moon did not move in a circle whose centre coincided with that of the Earth but were moving in an eccentric circle i.e. a circle whose centre is at some other point than the Earth's centre. This surmise could be made because unequal motion was accountable only on varying distance from the Earth's centre and a celestial body appearing to move fastest must be nearest whereas the same appearing to move slowest must be farthest. Thus in the first place seeing no reason for non-circular motions and also expec- ting the celestial bodies to move only in circles, for, a circular motion appealed to them as the most ideal motion, the ancient astronomers later postulated an eccentric circular motion with respect to the Sun and the Moon. This postulation appeared to give good results as seen below. 17

130 Fig. 9 Eccentric circle theory.—Let E₁ be the earth's centre; let M₁PA₁ be the circular orbit in which the planet (here the Sun or the Moon) moves with a uniform motion. This planet is termed the Madhyagraha or the mean planet. Let M₂A₂P₂ be the actual orbit of the planet whose centre E₂ is removed a little away from E₁. Since the centre E₂ is moved in a vertical direction away from E₁, every point of the eccentric circle (E₂) will be vertically over the corresponding point of the mean circle. Thus M₂ will be the position of the actual planet where M₂ is vertically above M₁ the mean planet. Join E₁M₂ to cut the mean circle in P. Since A₂ is the position of the actual planet farthest from E₁ the Earth's centre, the planet should have

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the slowest motion there. So this point A₂ is termed Mandoccha, Manda because it is point where the planet is slowest and Uccha because it is the highest or the farthest point from the Earth’s centre. Corresponding to this Mandoccha in the eccentric circle A₁ is termed the Uccha in the mean circle. Also p the point where E₁M₂ the line joining the Earth’s centre to the actual planet and called the Mandakarna, cuts the mean orbit is taken to be the position where the apparent planet is situ- ated. Thus ‘p’ is seen to be deflected from the mean planet M₁ towards the Mandoccha on which account the Mandoccha is considered to be attracting the planet “ उच्चोद्याःकर्षको भवति ” as Bhāskara puts it. The angle A₂Ê₂M₂ is spoken of as the Manda-Kendra or the mean anomaly and it is equal to Z₂Ê₂M₂ – Z₂Ê₂A₂ = longi- tude of the planet minus the longitude of the Mandoccha, where E₁Z₁ and the parallel E₂Z₂ are directions towards the Zero-Point of the Zodiac. This accounts for the statement ‘मन्दूच्चेन हीनो ग्रहो मन्दकेन्द्रम्’ (of the verse under elucidation) i.e. the excess of the longitude of the planet over that of the Mandoccha is termed Mandakendra. While M₁ is termed the Madhya-graha in the mean orbit, M₂ is termed the prativritta-Madhyagraha, and not spaṣṭagraha as might be deemed, while p is spoken of as the spaṣṭagraha or the True planet or apparent position of the planet. The word Prativritta stands for the eccentric circle. Now M₁P the difference between the mean and True positions is spoken of as the Mandaphala which corresponds to

132 generally a small quantity M₂N₂ is taken to be equal to PN₁. If, however, this approximation is not made, PN₁= (M₁N₂ × E₁P) / (E₁ M₂) (by the similarity of the triangles E₁ PN₁ and E₁ M₂ N₂) so that the actual equation of centre is (r / R) H sin E₂ × (R / K) = (r / R) H Sin E₂ where K = Mandakarna E₁ M₂. As M₂ moves from A₂ to P₂, the equation of centre as given by (1) gradually increases from Zero to a maximum r when E₂ = 90° and decreases from this maximum to Zero when E₂ = 180°. Thus from what was noticed from the tabulated differences between the computed mean positions and observed true positions, the fact that those differences vanish at A₂ as well as P₂ the diametrically opposite point of the Mandoccha (not called Sīghroccha, for this word Sīghroccha will be seen to have altogether a different connotation) was verified. The maximum

133 contradistinction to the word Sīghra which we shall shortly deal with. In modern astronomy the equation of centre is given approximately to be equal to 2e sin m where ‘m’ stands for the mandakendra so that r = 2e. It will be shortly seen from a subsequent table that this formulation of the equa- tion of centre gives results which closely accord with their modern values. The true or apparent positions of the Sun and the Moon could be obtained fairly well from the above formulation, so that it is stated that चन्द्रसूर्यौ स्फुटौ स्यातो मान्देनैकेनकर्मणा i.e., the Moon and the Sun could be re- ctified by the equation of centre alone.” This is quite in order for, the Sun and the Moon may be taken to be going round the Earth in ellipses, with the earth in one focus, the former relatively and the latter directly. After having formulated the method of rectification in the case of the Sun and the Moon, the next question was with respect to the Tārā grahas i.e. Mercury, Venus and Mars, Jupiter and Saturn. As these are going round the Sun and the Sun going round the earth relatively, the process of rectification got complicated. In the first ins- tance, the ancient astronomers must have tabulated the differences between the mean computed positions and the observed true positions. In the case of Mercury and Venus, the case appealed different from what it was in the case of the other three planets, for the simple reason that the mean positions of the former were taken to coincide with the mean Sun. This meant that for rectification, the elongation had to be computed and added to or subtracted from the mean position of the Sun to get the apparent geocentric positions of Mercury and Venus. The analogy of the method of the formulation of the Mandaphala is taken here also by imagining (1) eccentric circular motion and (2) postulating an Uccha. In the case of the Manda- phala, the equation was zero when the mean planet coinci- ded with the Mandoccha. Here the equation is zero when

134 elongation is zero, i.e. when the apparent geocentric posi- tion of the planet coincides with the Sun, who is taken to be the mean planet. Naturally therefore the Uccha is taken to coincide with the Sun the mean planet, when the planet is in conjunction with the Sun. The maximum equation was had in the case of the Mandaphala when the arc between the Uccha and the mean planet was a right angle. So, here also, the maximum equation i.e. the maxi- mum elongation should be had when the Uccha is a at right angle from the Sun. Thus an Uccha was postulated with the following criteria namely (1) It should be a point moving in a geocentric circle (2) It should coincide in direction with the Sun when the planet is in conjunction with the Sun (3) It should be removed by a right angle from the Sun when the elongation is maximum (3) It should be removed from the Sun by 180°, again when the planet coincides in direction with the Sun (4) It should have a longitude exceeding that of the Sun by 270 when again the elongation is a maximum on the other side and finally (5) It should complete a circle with respect to the Sun when again the planet coincides in direction with the Sun. When such a point was conceived it is clear that this Uccha is not the same as the planet, as some have mis construed, because while the planet oscillates about the Sun by a particular angle (29° in the case of Mercury and 45° in that of Venus) in Uccha completes a circle with respect to the Sun and further as Hindu Astronomy postu- lated geocentric motion, the Uccha is a point construed as going in a geocentric circle. By the above postulation the synodic period of the Uccha is equal to the period of oscil- lation of the planet about the Sun. But the latter period is no other than the synodic period of the planet so that the synodic periods of the Uccha and the planet coinciding their sidereal periods should be equal. In other words the Uccha so conceived is a point other than the planet going round in a geocentric circle and having a geocentric side-