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

131

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-

135 real period equal to the heliocentric sidereal period which again means that the geocentric longitude of the Uccha is the heliocentric longitude of the Planet. Thus the radius vector to the planet from the Sun is parallel to the geocen- tric radius vector of the Uccha. This accounts as to how the heliocentric sidereal periods of Mercury and Venus could be found under a geocentric concept and also as to how the heliocentric planets are themselves spoken of as their respective Ucchas, while their mean planet is the same as the Sun. On this count it was mentioned by the Hindu Astronomers ज्ञशुक्रयोः ग्रहः सूर्यो भवेत् तौ शीघ्रनाकौ ie. The mean Planet of Mercury and Venus is the Sun himself where as they are themselves spoken of as their Ucchas. The phrase 'they are themselves' in the above statement is significant as it connotes that the word 'they' stands for the heliocentric planets, though it was uot stated in so many words. Shortly we shall see also that the centre of the eccentric circle coincides with the centre of the Sun also and applying Bhāskara's statement 'यस्मिन्वृत्ते भ्रमति खचरो नाऽस्य मध्यं कुमध्ये' i.e. the centre of the circle in which the planet moves does not coincide with that of the Earth', the Sun was, though unwittingly taken as the centre of the Planetary motion. Thus we see how even the geocentric postulation also could help computation of the Planetary positions, the mathematics behind revealing heliocentric motion. What Copernicus achieved was that he identified that the point about which the planets revol- ved which was construed by the Hindu astronomers as an imaginary point not coinciding with the earth's centre, was no other than the Sun himself. In the case of Mercury and Venus the so-called Sīghra-phala came to be discovered first and we shall pre- sently see why their elongation was called Sīghra-phala and how the Uccha postulated as above came to be termed Sīghroccha. Since initially the equation was to be zero, when the Planet and the Uccha coincided with the Sun

136 and then the elongation has to increase as the Uccha gained over the Sun, the initial conjunction was the modern Superior conjunction. The other position of the Uccha when again the elongation i'e. the equation is Zero must be therefore the Inferior conjunction. Also at the motion of Superior conjunction, the planet must be having the maximum daily motion, as it is clear from a heliocen- tric figure that at that point the relative motion of the planet with respect to a geocentric observer is the sum of the velocities of the planet and the earth. Hence this Uccha is spoken of as the Śīghroccha also because the Uccha being a geocentrically moving point having helio- centric angular motion, its velocity is always greater than that of its planet namely the Sun. The word Uccha is applied because at the Superior conjunction the planet is farthest or highest from the earth. The excess of the longitude of this Uccha over the longitude of the planet ie. the Sun is known as the Śīghrakendra or anomaly as it is said in the verse under commentary ‘चलोच्चं ग्रहोनं भवेत् शीघ्रकेन्द्रम्’. Thus in the case of Mercury and Venus, the Śīghraphala came to be discovered first. This being dis- covered, formulated as will be shortly shown, and applied to the mean Sun as the planet, still it was found that there was a difference between the computed position and the observed position. Such differences were tabulated. By analogy from the case of the Sun and the Moon, it was thought that there should be also a Mandoccha here also, so that the point indicated by the position of the mean planet after being corrected by the equation where the above tabulated difference was zero, was identified as the Mandoccha. From the H Sine of the maximum difference taken as the radius of the Manda epicycle, its circumference was then computed. In the case of the superior planets, we have already said that the geocentric sidereal periods accord with their heliocentric ones. Calculating the mean position of the planet and finding its difference from the observed true

137 position, such differences were tabulated. It was discovered that these difference almost vanished when the planet was in conjunction with the Sun and attained a maximum when the elongation was nearly a right angle from an analogy from the Manda-Karma i.e. process of obtaining the Manda-phala. Since the differences attained their maxi- mum value when the elongations were nearly a right angle it could be seen that the Sun played the part of the Uccha in this case. As the Sun has a quicker motion than the planet and also as at conjunction the planet has the quick- est motion relative to the Earth while it is farthest from the Earth the Uccha ie the Sun here, is termed Śīghroccha. The excess of the longitude of the Sun over that of the planet is termed accordingly the Śīghra-kendra and the Śīghra-phala the equation was formulated as will be shown. Applying this Śīghra-phala to the mean position, the diffe- rences still found between the position so obtained and the observed true position were tabulated. The point indicated by the above position where the difference was found to be zero, was identified as the Mandoccha, and through the maximum difference, the Mandaparidhi was formulated. In the above discourse, we have tried to give an account of how the originators of Hindu Astronomy could give us a workable system. We never assumed that an Āgama gave us the numbers of sidereal revolutions of the planets or the measures of the epicycles Manda or Śīghra. But in the explanation given by Bhāskara under Bhagaṇōpapatti, one will notice that when Bhāskara gave the proof of the Moon's sidereal revolutions, he said that having got the true positions of the Moon on two conse- cutive days, the mean positions were computed from the true by an inverse process of applying the equation of centre, and getting the mean daily motion of the Moon from those mean positions, the number of sidereal revolu- tions in a Kalpa were obtained. Here the Upapatti or the proof adduced by Bhāskara was not a proof but only a verification in as much as (1) he assumed the formulation 18

138 of the Mandaphala from the Āgama without pointing out how it was formulated and (2) he assumed the period of a Kalpa and that at the beginning of the Kalpa the planets were all in conjunction at the first point of Aries. Similarly in the Upapatti adduced by Bhāskara with respect to the Mandocchas of the planets, he assumed the formulation of Sīghra-phala on the basis of Āgama without proving how the concept of Sīghra-phala was arrived at by the founders of Astronomy and how the difference between the observed apparent positions and the computed mean positions, was resolved into two equations the Mandaphala and the Sīghra- phala. In the proof adduced with respect to the Sīghroccha of Mercury and Venus also Bhāskara did not mention anything as to how their heliocentric sidereal periods could be obtained but simply assumed the formulation of the Mandaphala and Sīghra-phala as already being there on the basis of Āgama. We shall now proceed to describe the method of formu- lation of the Sīghra-phala with respect to the five Tāra- grahas, star planets namely Mercury, Venus and Mars, Jupiter, Saturn and show how so different a set of geomet- ries of the ancients and the moderns the one geocentric and the other heliocentric could give identical formulation with respect to Sīghra-phala. Let us consider the case of Mercury and Venus in the first instance. Having taken the mean Sun to play the part of the ‘Graha’ in the case of Mercury and Venus and having formed a concept of Sīghroccha as mentioned before, whose geocentric period of revolution was determined, without suspecting it to be the heliocentric sidereal period of the planet the ancient Hindu astronomers assumed by analogy from the case of Mandaphala with respect to the Sun and Moon, that the centre of the circle in which these planets revolve does not coincide with the centre of the Earth. In other words, they continued eccentric circle theory here also so that without suspecting heliocentric revolution

139 their mathematics led to them to make the centre of the eccentric circle coincide with the Sun himself. On this basis alone it was given to Copernicus to formulate helio- centric theory, sponsored by a thought that the Heavenly Sun could not be deemed as a satellite of the ‘ Mundane ’ Earth. (Ref. fig. 9). The same figure 9 will also serve the pur- pose to obtain the Sīghraphala, only M₁ M₂ will be now on the right hand side of the Sīghrocchas A₁ A₂, for, the latter will be taken to be in advance of the mean planets Kakshā- Vrittīya Madhyagraha M₁ (i.e. mean planet of the deferent) and prati-Vrittīya Madhyagraha M₂ (i.e. mean planet of the eccentric). The points A₁ add A₂ are themselves called the Kaksha-Vrittīya Sīghrōccha and prati-Vrittīya Sīgh- rōccha respectively. As was shown in the case of the Mandaphala from the eccentric figure 9, M₂ N₂ = r/R H sin (Kendra) so that PN₁ = r/R H sin (Kendra) × R/K = r/K H sin (Kendra) = (Antyaphalajya × Sīghrakendrajyā) / Sīghrakarna . We shall take this for elucidation in the appropriate context. Verse 19. Three Rasis each of 30° constitute a quad- rant, and there are four quadrants in a circle which are res- pectively odd, even, odd and even. In the odd quadrants the Kendra covered is itself called Bhuja whereas in the even ones, the complement thereof is called Bhuja. Also, the complement of the Bhuja is called the Koti. Verse 20. R − H sine = Co. H versine and R − H cosine = H versine and R − Co. H versine = H sine, R − H versine = H cosine. Verse 21. Also √(R² − H sine²) = H cosine, √(R² − H cosine²) = H sine. Similarly √(R² − Krantijya²) = Dyujya, √(R² − Dyujya²) = Krāntijya; √(R² − Drig-jyā²)

140 = Śaṅku, √(R² — Śaṅku²) = Drig-jyā. In all the cases cited above, the radins R happens to be the hypotenuse. Comm. The convention in verse 19 corresponds to saying in modern trigonometry that sin 90 + θ = cos θ sin (180 — θ) sin θ, sin 180 + θ = — sin θ, sin 270 — θ = — cos θ sin 270 + θ = — cos θ, sin 360 — θ = — sin θ. In Hindu trigonometry the sign is understood and not explicitly mentioned. Krāntijyā, Dyujyā, Drig-jyā and Śaṅku, are respecti- vely H sin δ, H cos δ, H sin Z, H cos Z where δ is decli- nation and Z the Zenith-distance of a celestial body. Taking the radius of the celertial equator to be R, the radius of the diurnal circle of a celestial would be equal to R × cos δ = H cos δ which is called Dyujyā because it is the radius of the diurnal circle. Verse 22. The lengths of the circumferences of the Manda—epicycles are respectively 13° — 40', 31° — 36', 70°, 38°, 33°, 50,* for the Sun, Moon, Mars, Mercury, Jupiter, Venus and Saturn. Comm. Bhāskara has given these measures reportedly on the basis of Āgama or ancient authority. The peculiarity of measuring the circumferences in degrees less than 360°, is due to the idea that these circumstances are measured in relation to that of the deferent or Kakshā Vritta taken to be 360°. In other words, circumference of the epicycle of a planet as given above : circumference of the mean orbit :: x : 360 = radius of the epicycle : radius of the mean orbit where x is the measure of the circumference of the planet- ary epicycle. It may be mentioned once again that the radius of a planetary epicycle is the measure of the greatest

• The printed book of Brahma Sphuta Siddhānta gives in the case of Saturn 30° only which might have been the mistake of the scribe (Vide verse 36, Spashtadhikāra B. S.). In the place of शून्यरामाः it ought to have been शून्यबाणाः.

141 equation of centre pertaining to the planet which may be taken to be equal to 2e as a first approximation where e is the eccentricity of the elliptic orbit of the planet. It may be further mentioned here that in Sūrya- siddhānta, as well as elsewhere in this work, the circum- ferences are given to vary continuously. This variability curiously achieves ellipticity in the orbit as may be seen as follows. In the case of the Sun, the epicycle has a periphery of 14° when m = 0 or 180° and of 13 ⅔° when m = 90° or 270 according to Sūryasiddhānta, where m is the Manda- kendra or mean anomaly. At any arbitrary point, where Fig. 10

142 the mean anomaly is m the periphery is given to be 14° — (20' H sin m) / R. The corresponding radius will there- fore be r — (20' / 2π) · (H sin m / R) = r — λ H sin m (say) where r = 14° / 2π. [λ = 20' / 2πR] (Ref. fig. 10) Let E be the earth's centre, A the posi- tion of the apogee EE₁ = the radius of the epicycle meas- ured along EA, B any arbitrary position of the mean planet and b the position of the planet in the epicycle. Here the radius Bb is not equal to the max. radius equal to EE₁ i.e. r but equal to r — λ H sin m. Take E₁ as the origin and E₁A as the y-axis and a perpendicular to E₁A through E₁ namely E₁x as the positive direction of the X-axis. If the mean anomaly BEA be m, then the coordinates of the true planet are given by x = BL = H sin m (1), y = E₁L + Bb = EL — EE₁ + r — λ H sin m = H cos m — r + (r — λ H sin m = H cos m — λx. ∴ y + λx = H cos m (2) Squaring and adding (1) and (2) x² + (y + λx)² = H sin²m + H cos²m = R² ∴ x² (1 + λ²) + 2λxy + y² = R² which is an ellipse with centre E₁ Verses 23, 24, 25. The peripheries of Śīghra epicycles. The peripheries of the epicycles of the star-planets Mars, Mercury, Jupiter, Venus and Saturn are respectively 243°-40', 132°, 68°, 258° and 40°. The H sine of the Manda mean anomaly of Venus being multiplied by 2 and divided by 343, and the result being subtracted from the periphery gives the rectified Manda periphery. The H sine of its Śīghra mean anomaly being multiplied by 5 and divided by R, and the result being added to the Śīghra periphery gives the rectified periphery. The smaller of the H sine or H cosine of the Śīghra anomaly of Mars being multiplied by 6⅔ and divided by H sin 45° and the result in degrees being subtracted from or added to as the case may be,

143 the aphelion gives the rectified aphelion. The S'ighra peri- phery being reduced by the above degrees gives the rectified S'ighra periphery in case the S'ighra anomaly is 90° < m < 180 or 270° < m < 360°. Comm. In the commentary Bhāskara adds that in the case of Venus the Mandaperiphery of 11° as given is at the end of even quadrants whereas at the end of odd quadrants it is 9°, wherefore the enunciated rectification. Similarly in the case of his S'ighraphala, the periphery of 245° men- tioned is at the end of even quadrants whereas at the end of odd quadrants it is 263°, and so the suggested rectifica- tion. Again in the case of Mars, the aphelion as computed is the same at the end of all quadrants whereas in the middle of the quadrants it is to be increased or decreased by 6⅔° when the anomaly is as stated. Also in the case of this Mars, the S'ighra periphery mentioned is at the ends of quadrants. In the middle of the quadrants the periphery is to be reduced as suggested. In all these interpolations Bhāskara accepts the Āgama as enunciated by Brahma- gupta. We shall deal with the geometrical nature of these S'ighra peripheries shortly in the appropriate place. Verse 26. To obtain what are called Bhujaphala and Kotiphala both in the case of Mandaphala as well as S'ighra- phala. The H sine and H cosine of the Manda or S'ighra anomalies multiplied by the respective peripheries and divided by 360°, or multiplied by r and divided by R gives the Bhujaphala or Kotiphala where r and R are respectively the radius of the Manda or S'ighra peripheries and R the radius of the deferent taken to be 3438′. If the radius 3438′ be respectively multiplied by the Manda or S'ighra peripheries and divided by 360°, the result will be the H sine of the maximum Mandaphala or S'ighraphala, known as Antyaphalajyā in either case.

144 Comm. As per the formulation. Bhujaphala = (H sin m × c) / 360 = (H sin m × r) / R Kotiphala = (H cos m × c) / 360 = (H cos m × r) / R in the case of Mandaphala or Sīghraphala where c = peri- phery of the Manda or Sīghra periphery, r=Antyaphalajyā defined above R = 3438' and m stands for the Manda or Sīghra anomaly. These Bhujaphala and Kotiphala will be used in their respective contexts. Verses 27, 28, 29. Calculation of what is known as Sīghrakarṇa. (H Cos m ± r)² + H Sin² m = K² (1) (R ± Kotiphala)² + Bhujaphala² = K² (2) R² + r² ± 2 R × Kotiphala = K² (3) R² + r² ± 2 r × H Cos m = K² (4) The arc of the H Sine of the equation of centre is called the Mandaphala. Comm. Ref. fig. 9. From triangle E₁M₁M₂ (E₁M₁ + M₁N₂)² + M₂N₂² = E₁M₂² = K². But M₁N₂ = Kotiphala and M₂N₂ = Bhujaphala defined previously so that we have the second formula for K enunciated above. From the similarity of the triangles E₁M₁Q₁ and M₁M₂N₂ we have M₂N₂ / M₁Q₁ = M₁M₂ / E₁M₁ so that M₂N₂ = r / R × M₁Q₁ ; Since M₁Q₁ is called the Bhuja, the corresponding M₂N₂ in the Antya- phalajyā triangle is called the Bhujaphala. Similarly M₁N₂ is called the Kotiphala. Again (E₁Q₁ + Q₁Q₂)² + M₂Q₂² = E₁M₂² where E₁Q₁ = H Cos m, Q₁Q₂ = M₁N₂ = r and M₂Q₂ = M₁Q₁ = H Sin m. From this we have the first formula enunciated.

145 Again expanding (E₁M₁ + M₁N₂)², (E₁M₁ + M₁N₂)²

• M₂N₂² = E₁M₂² we have the third formula; similarly expanding (E₁Q₁ + Q₁Q₂)², (E₁Q₁ + Q₁Q₂)² + M₂Q₂² = E₁M₂² we have the fourth formula. Since the S'ighraphala has been defined to be equal to r/K H Sin m, we have had the necessity of knowing the value of K. The convention of signs mentioned in the formulation in the words 'योगो मृगादावथ कर्कटादौ केन्द्रेऽन्तरम्' is due to the fact that cosine is positive in the fourth and first Quadrants and that the Kotiphala becomes negative in the 2nd and 3rd Quadrants as could be seen by draw- ing the figure in those Quadrants. Now we shall prove what is most important, namely that postulating an entirely different geocentric motion how the Hindu Astronomers could formulate the S'ighra- phala which accords exactly with the heliocentric theory, assuming of course coplanar circular orbits. Let figures 11 and 12 pertain to the modern heliocentric geometry, Fig. 11 Fig. 12 the former with respect to the Inferior planets Mercury and Venus signified by V, and the latter to the superior planets Mars, Jupiter and Saturn signified by J. Let fig. 13 pertain to the Hindu geocentric geometry dealing with both the Inferior and Superior planets as well. In the heliocentric figures let S = Sun, E = Earth, SA = directon to Aswini, the Zero-point of the Zodiac from the Sun, EA' = 19

146 [चित्र: FIG. 13] FIG. 13 geocentric direction towards Aswini. Let SV, EV¹, be the heliocentric and geocentric directions of the S'īghroccha where V is the actual planet and V¹ an imaginary point. Draw EJ' ∥ to SJ. Let the radius of the inner and outer heliocentric circles be respectively r and R. Let K be the radius vector to the planet in both the figures. Let in fig. 13, E₁=Earth's centre, E₂=the centre of the eccentric circle which we shall presently show to be coinciding with the Sun's position. Let M₁, M₂ represent the mean planets in the deferent and the eccentric known as Kaksha-Vrittīya Madhyagraba and prati-Vrittīya Madhya- graha. Let A₁, A₂ be the S'īghrocchas in the deferent and the eccentric. Let E₁E₂ = r known as Antyaphalajyā and R the radius of both the deferent and the eccentric. Let K be the radius vector to the planet known as S'īghra-Karṇa.

147 We shall prove that m the Sīghra anomaly of the Hindu figure will be the same as 'm' as marked in the heliocentric figures. Sīghra anomaly is defined as longitude of Sīgh- roccha - longitude of the planet = a Ê₁ A₁ - a E₁ M₁ = a¹ Ê₂ A₂ - a¹ Ê₂ M₂ = m(fig. 13.) In the heliocentric fig. 11, since A¹EV¹ is the longitude of the Sīghroccha, and A¹ Ê S the longitude of the Sun treated as the Madhyagraha of the Inferior planet A¹ÊV¹ - A¹ÊS = V¹ÊS = V Ŝ S¹ = m = the Sīghra ano- maly. In fig 12, m = S¹ŜJ = SÊJ = A¹ÊS - A¹EJ¹ = A¹ES - A Ŝ J = Longitude of the Sun treated as the Sīghroccha of the Superior planet minus longitude of the heliocentric planet known as Mandasphutagraha or the planet rectified for the Mandaphala or equation of centre = Sīghra anomaly In the case of the Inferior planet the heliocentric direction of the planet is equal to the Sīghroccha. Now consider the triangles ESV, JSE, E₁M₁M₂ of the three figures. Evidently E Ŝ V = J Ŝ E = E₁ M̂₁ M₂ = 180 - m = Supplement of Sīghra anomaly. If, further it is shown and (it will be shown subsequently) SV / SE = SE / SJ = M₁M₂ / E₁M₁ the similarity of the triangle E₁M₁M geocentric figure separately with ESV and JSV will have been established. Taking this similarity to have been established, M₁ Ê M₂ known as Sīghraphala will be equal to SÊV in fig. 11 and SĴE in fig. 12. In fig. 13, M₂ the prativritta Madhagraha is also known as the pāramārthikagraha or the actual planet where as p its geocentric position on the Kakshāmandala is taken to be the true planet or apparent position of the planet. In figures 11 and 12, EV and EJ are the directions to the true planets V and J so that the angles between the Sphuta- graha and the Madhyagraha (ie the Manda Sphutagraha =

148 A¹ÊV - A¹ES = SÊV (in fig 11) and = A¹ÊJ - AŜJ = S Ĵ V̂ᴱ = Sighraphala. Once the similarity of the triangles fig. 12 is established, the equality of the Sighraphala will be establi- shed. Also due to the similarity mentioned above the for- mulae for K as given in Hindu Astronomy should also accord with that in the heliocentric figures. In fact in the heliocentric figures K² = R² + r² + 2Rr cosm = R² + r² + 2RH cosm which is indentical with the four formulae given before as per verses 27, 28, 29. It will be seen that the epicycle (M₁) with radius M₁M₂ will be identical with the inner circles in the heliocentric circles, whereas the Kakshamandal (E₁) with radius R will be identical with the outer circles of the heliocentric figures. Before we proceed further, we shall annex the table wherein the ratio r/R as given in Hindu Astronomy will be seen to accord with that in modern astronomy.

149 Formula for Sīghraphala from the heliocentric figures In fig. 11, r / K = sin SÊV / sin EŜV so that sin SÊV = r / K × sin m In fig. 12 r / K = sin SĴE / sin EŜJ so that sin SĴE = r / K sin m Both these accord with the Hindu formula. It will be interesting to point out here that in fig 11, keeping the earth constant and supposing the Sun S to go in a circle with centre E and radius ES, the orbit of the Inferior planet V will play the part of the epicycle of Hindu Astronomy. Thus in the case of the Inferior planets, the epicyclic theory is only a different version of the helio- centric theory. In the case of the Superior planets, how- ever, (fig. 12) cut off EJ¹=SJ along EJ" parallel to SJ; then J¹J will be parallel to ES just as M₁M₂ is parallel to E₁E₂ in fig. 13. Then the circle with E as centre and EJ¹ as radius corresponds to the deferent of fig 13, whereas the circle (J¹) with centre J¹ and radius J¹J corresponds to the epicycle. The circle with S as centre and radius SJ corresponds to the eccentric. Verse 30. The equation of centre pertaining to the Sun and the Moon using a simpler table of H sines where the radius = 120 units. The H sines of the mean anomaly as found from the simpler H sine table where radius = 120, multiplied by 20, and divided by 1103 and 477 respectively gives the equation of centre of the Sun and the Moon in degrees. Comm. The maximum equation of centre with respect to the Sun is 2°-10'-31". Then the argument is "If by the H sine of the anomaly equal to the radius 120, we have the above max. equation what shall we have for H sin m?".

160 The answer is (H sin m × 2°-10'-31") / 120 = (2 21/120°) / 120 × H sin m very approximately = 261 / 14400 H sin m = 20 / 1103 H sin m Similarly in the case of the Moon, the maximum equation of centre is 5°-2'-8". By the same argument as above we have (H sin m × 1133) / (225 × 120) = (H sin m × 20) / (54000 / 1133) = (H sin m × 20) / 477 Verse 31. Rectification of the mean daily motion of the Sun and the Moon. The H cosine of the mean anomaly divided by 54 in the case of the Sun and in the case of the Moon multiplied by 4 and divided by 7 gives the increment or decrement in the respective mean motions according as 90 < m < 270 or 270 < m < 360 + 90. Comm. We have Equation of centre = r / R H sin m = E (say) so that differentiating δE = r / R H cosm δm / R . But r / R H cosm is called kotiphala and δm is called Kendra gati so that δE = Kotiphala × Kendragati / R . Since Koti- phala is negative when 90 < m < 270 δE is negative but in Hindu Astronomy we measure the Kendra not from perigee as in modern astronomy but from aphelion so that the equ- ation of centre is strictly - r / R H sin m if sign is also taken into consideration. Hence δE must be + ve. Since M + E = S where M is the mean planet, E the equation of centre and S the true planet δS=δm+δE so that the true motion is equal to the mean motion plus δE. As δE is +ve when 90 < m < 270 as mentioned above we have to add this to the mean motion to get the true motion. This δE