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

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

151 is called gatiphala or what is to be added to the mean motion to give the true motion. Incidentally we have commented on the contents of verse 37. Verse 32. The Śīghra phala with respect to the Star- planets. (H sin m × r) / R being multiplied by the radius R or the product of H sin m and r being divided by K the arc of the result gives the Śīghraphala. Comm. In the verse it is mentioned that the product of the Bhujaphala and the radius is divided by K so that the formula for the Bhujaphala being (H sin m × r) / R the Śīghra- phalajyā will be (H sin m × r) / K which is stated in the alter- native. We have already derived this formula before where we got H sin E₂ = (H sin m × r) / K. The arc of this will be E₂ i.e. the Śīghra-phala, (E₂ because we take E₁ as the Mandaphala). Verses 33 and 34. An alternative formulation of Śīghra-phala. H sin m being multiplied by R and divided by K and the difference between the arc of the result and H sin m will be the Śīghra-phala. Here H sin m belongs to the eccentric. The arc of the maximum Śīghra-phalajyā added to or subtracted from 90° will give respectively the Quadrants and the H sine will have to be taken of the elapsed Kendra or its Koti according as the Quadrant is odd or even. Comm. Ref. fig. 13. From the similarity of the tri- angles E₁ M₂ N and E₁ PM, PM / M₂N = E₁P / E₁M₂ = R / K ∴ PM = (R / K) × M₂N

152 But PM is the H sine of m¹ where m¹ is called Sphuta- kendra (m is called the madhya-Kendra). Hence H sin m¹ = R/K × M₂N = R/K × Bhujajyā (in the eccentric) ∴ m¹ = H Sin⁻¹ (R/K × Bhujajyā) = PA₁ ∴ M₁P = Sīghraphala = M₁ A₁ — PA₁ = Kakshya- mandala Bahu minus the chāpa m¹ — Here a clear under- standing of the word Bāhu or what is the same Bhuja should be had. The arc pertaining to the angle m in the eccentric is known as the Bāhu in the eccentric and that to the angle m in the deferent as the Bāhu in the deferent. When 0<m<90, m is itself spoken of as Bāhu; when 90<m<180, 180—m is spoken of as the Bāhu; when 180<m<270, m — 180 is the Bāhu and when 270<m<360, 360—m is the Bāhu. Thus the Bāhu is that angle whose H sine will be H sin m numerically. When it is said in the verse ‘त्रिज्याहता कर्णहृता भुजज्या’ the word Bhuja is the arc M₂A₂ as is mentioned in the same verse ‘ज्ञेयोऽत्र बाहुः प्रतिमण्डलस्य’. The second part of the verse divides the eccentric circle into such quadrants that in them S'īghraphala increases from Zero to a maximum, decreases again from a maximum to zero, again increases from zero to a maximum and again decreases from a maximum to zero. Thus at A₂ of the eccentric the S'īghraphala is zero; at a₁, it is a maximum namely the arc b₁c₁ where H sin b₁c₁ = a₁c₁ = r; Thus in the course of A₂a₁ the arc of the eccentric the S'īghraphala gradually increases from zero to a maximum and in the course of a₁a₂ the S'īghraphala decreases from a max to zero. Again from a₂ to a₃ it increases from zero to a max and from a₃ to A₂ it decreases from the maximum to zero. Thus the quadrants in the case of S'īghrapbala are not of 90° but arcs A₂a₁, a₁a₂, a₂a₃ and a₃A₂ which are respectively of mag- nitude 90° + H Sin⁻¹r, 90 — H Sin⁻¹r, 90 — H Sin⁻¹r and 90+H Sin⁻¹r. In the case of Mandaphala also, the quadrants should have been of the same magnitude if the so-called Karnānupāta has been postulated i.e. reducing the Manda-

153 phala from the extremity of the Karna to the extremity of the radius in the deferent; but as this Karnānupāta is not adopted, the difference being negligible the quadrants are all of equal magnitude i.e. each of 90°. In the course of the commentary of this verse Bhāskara mentions that for Mercury, as the maximum S'īghraphala is 21°-31'-43", the quadrants are of magnitude 3-21-31-43, 2-8-28-17, 2-8-28-17 and 3-21-31-43 respectively. Also in the commentary Bhāskara adds that a₁ which is the point of intersectiou of the eccentric with the hori- zontal diameter E₁b₁ the S'īghraphala is maximum and that at that point the mean motion is itself the true motion “कक्षामध्यगतिर्यम्रेखाप्रतिवृत्तसम्पाते, मध्यैव गतिः स्पष्टा परं फल तत्र खेटस्य”. That the S'īghraphala at a₁ and a₃ is maximum is clear from the figure 13, where it is equal to the arcs b₁c₁ and b₂c₂ whose H sine is equal to r. To prove that the mean motion is itself the true motion, we have the equation M₂ + E₂ = S where M₂ is the mean planet here or the Mandasphutagraha or planet rectified for the equation of centre, (by the equation M₁ + E₁ = M₂, M₁ being the original mean planet and E₁ the equation of centre) so that δM₂ + δE₂ = δS where δM₂ is the mean motion here, δS the true motion and δM₂ is the variation in the S'īghraphala; but at a₁ and a₃ E₂ the S'īghraphala being maximum δE₂ is zero. Hence δM₂ = δS which means that the mean motion is itself the true motion. Verse 34. Latter half, 35 and 36 former half. The mean planet rectified for the equation of centre or Manda-phala is called Mandasphuta. Then subtracting the longitude of the Mandasphuta from that of the respective S'īghroccha, the result will be the S'īghra anomaly from which the S'īghra-phala is to be obtained. Rectifying the Mandasphuta for this second equation namely S'īghraphala, again obtaining therefrom the equation of centre effecting 20

154 this in the original mean planet and again correcting for Sīghraphala and repeating the process till a constant value is obtained, the true planet is had with respect to the star- planets other than Mars. But with respect to Mars, let first the mean planet be corrected for half of the equation of centre. Then make half of the correction of Sīghraphala. Take the resulting planet to be the mean planet and again finding the equation of centre, make this whole correction in the original mean planet. Again taking the resulting planet to be the Mandasphuta effect the entire Sīghraphala. Then we have the true planet. Comm. The Suryasiddhānta stipulates the same kind of correction in the case of all the star-planets. "शैघ्र्यं मान्दं पुनः मान्दं शैघ्र्यं चेति चातुर्विधं, कर्तव्यं हि कुजादीनां स्फुटत्वे कर्म सूरिभिः। शैघ्र्यं फलार्धं प्रथमं ततो मन्दार्धमेव च, पश्चान्मान्दफलं सर्वं तद्वत् शीघ्रफलं ग्रहे ॥" i.e. In the first place half of the Sīghra- phala is to be effected in the mean planet; taking that to be the mean planet and computing the equation of centre half of it is administered; taking the resulting to be the mean planet and computing the equation of centre, the entire equation of centre is now to be administered in the original mean planet; taking the result to be the Manda- sphutagraha, and computing the Sīghraphala, it is to be administered in full in the Mandasphuta. Then we have the true planet. In modern astronomy, the equation of centre is first done and the result will be the planet in its heliocentric elliptic orbit. To reduce it to the geocentric position, the second correction is made which corresponds to the Hindu Sīghraphala. Thus the two corrections administered suc- cessively gives the apparent or true geocentric planet. Though the mutual relationship as conceived between the equation of centre and the Sīghraphala in Hindu Astro- nomy is deemed irrational by modern interpreters of Hindu Astronomy, there is some rationale in the process as ex-

155 plained by this author in his work ‘A critical study of Ancient Hindu Astronomy’ (published by the Karnatak University) page 98. Verse. Cited from Golādhyāya. The equation of centre is to be applied to the mean planet to obtain the centre of the Śīghraepicycle; then to obtain the true position the Śīghraphala is to applied to the Mandasphutagraha; hence the two equations are mutually related so that the true position is obtained after repeated application of the two equations. Comm. Explained above. Verse 36 latter half and 37. The true daily motion of the planet is the excess of the longitude of the true planet of the next day over that of the true planet of the previous day. The Kotiphala being multiplied by the daily motion of the Manda mean anomaly and divided by the radius, and the result being added to or subtracted from the mean motion, gives what is called Mandasphutagati. Comm. Already explained before. If M₁, M₂ and S be the mean planet, Mandasphutagraha, and the true planet respectively, and if E₁ and E₂ be respectively the two equa- tions, then M₁ + E₁ = M₂, M₂ + E₂ = S so that δM₁ + δE₁ = δM₂ and δM₂ + SE₂ = δS Here δM₁ = mean daily motion, δE₁ = daily variation in the Mandaphala, δM₂ = daily motion of the Mandasphuta graha or what is the same Mandasphutagati, δE₂ = daily variation in the second equation and δS = True daily motion.

156 Lallāchārya formulates the Mandasphutagati in a different manner which is equally correct (Ref. verse 45 Spaṣtādhikāra Śiṣyadhī Vriddhida “त्रिज्याहता ग्रहगतिः मृदुकर्णभक्ता मन्दस्फुटा भवति ” i. e. (δm × R) / K = Mandasphuta- gati where δm = Madhyagati or mean motion and K is the Manda Karṇa equal to √(R² + r² ± 2 R r cos m) ∴ Mandasphutagati = (δm × R) / √(1 ± 2 (r / R) cos m + r² / R²) = δm ( 1 ± (2r / R) cos m )⁻¹⁄₂ = δm ( 1 ∓ (r / R) cos m ) neglecting the smaller term − r² / (2R²) within brackets = δm ∓ (r H cos m δm) / R² = δm ∓ (r H cos m / R) × (δm / R) = δm ∓ (Kotiphala × Mandakendragati) / — as given by Bhāskara. Verse 38. In the case of the Moon, obtaining the true Moon for a particular moment and his daily motion for the day, the ending moment of the tithi near at hand is to be computed with that daily motion, and the method of suc- cessive approximations is to be used to rectify the ending moment. In the case of the ending moment of the tithi be- ing sufficiently far away, then it does not matter even if the above daily motion is applied to get the approximate ending moment. In as much as the Moon’s daily motion is great and varies from moment to moment the motion at the moment is to be used. Comm. Strictly speaking the ending moment of every tithi is to be computed by the method of successive approxi- mation. That is why in the computation of eclipses,

157 Moon's hourly motion is given in modern almanacs. Since it is very cumbrous to use the method of successive approxi- mation to determine the ending moment of every tithi, the Hindu almanac-makers generally compute the ending moment of a tithi using the daily motion of the moon computed for the moment of Sun-rise of the day. Only for ritual purposes, the method of successive approximation is used and also in the computation of eclipses. Verse 39. Computation of the Sīghragatiphala. δl − [H sin (90−E₂) × δm] / K = δS where δl is the daily motion of the Sīghrōcca, E₂ = Sīghraphala, δm = daily motion in the Sīghra mean anomaly, K = Sīghrakarṇa, and δS the true motion of the planet. If δS is negative the planet is retrograde. Comm. This is mathematically an important verse, and the proof given by Bhāskara really reflects his genius. Before we attend to his proof, we shall give this a modern treatment. (Ref. figs. 14, 15). SA and EA' are the Fig. 14 heliocentric and geocentric directions to Aswini the Hindu Zero-point of the Zodiac; S = Sun, E = Earth; V = In- ferior planet Venus or Mercury; J = Superior planet; E = Sīghra-phala, K = Sīghra Karṇa; M = Sīghra anomaly.

188 [चित्र: Fig. 15] Fig. 15 True motion of the planets = δ (A'EV) or δ (A'EJ) But δ (A'EV) = δ (A'EV'–n̂) and δ (A'EJ) = δ (A'ES–n̂) In the case of the Inferior planet δ (A'EV') = δ (ASV) = Sīghrōcchagati and δn is Sphutakendragati where n is called Sphutakendra, m being called Madhyakendra. In the case of the Superior planet δ (A'ES) = S'īghrōcchagati because the Sun plays the part of S'īghrōccha in the case of a Superior planet and δn = Sphutakendragati as before. Hence in both the cases, Sphutagati = S'īghragati – Sphuta- kendragati. We have now to find Sphutakendragati to obtain Sphutagati, as Bhāskara remarks rightly “महामति- द्भिः केन्द्रगतिरेव स्पष्टीकृता”. In other words we have to find δn. From the figures K cosn – R cos m = r (i) Differentia- ting this we have – K sin n δn + cos n δK + R sin m δm = 0 (ii). But K² = R² + r² + 2Rr cos m so that 2δK × K = – 2Rr sin mδ m (3) Eliminating δK between (2) and (3) – K sin nδ n – (Rr sin mδ m × cos n) / K + R sin mδm = 0 i.e. K sin nδn = R sin mδm ( 1 – r / K cos n ) = (R sin mδm / K) (K – r cos n)

159 But K — r cos n = R cos E. ∴ δn = (R sin mδm × R cos E) / (K² sin n) But R sin m = K sin n Cancelling δn = (R cos Eδm) / K = (H cos E δm) / K as given by Bhāskara. In the formula H sin E = (r H sin m) / K, Bhāskara perceived the variability of both H sin m and K on the right hand side and he exclaims “ न हि केन्द्रगतिजमेव फलयोरन्तरं स्यात्, किन्त्वन्यथाऽपि अद्यतनभुजफलश्वस्तनभुजफलान्तरे त्रिज्यागुणे अद्यतनकर्णहृते यादृशं फलं न तादृशं श्वस्तनकर्णहृते, स्वल्पा- न्तरेऽपि कर्णे भाज्यस्य बहुत्वात् बह्वन्तरं स्यादित्येतदानयनं हित्वा अन्यत् महामतिमद्भिः कल्पितम्, तद्यथा केन्द्रगतिरेव स्पष्टीकृता ” i.e. “ The variation in the Sīghraphala is not entirely constituted by the variation in m but also by that in K......So leaving the method of seeking δE through the formula H sin E = (r H sin m) / K the great intelligent astronomers used the formula. Sphutabhukti = Sīghra Bhukti — Sphuta Kendra- bhukti (Bhukti means gati) wherein it was'sought to obtain the variation in Sphutakendra i.e. n̂ in the figures. We have given a proof of Bhāskara’s formula, which circumvented finding Sīghragatiphala but which sought directly Sphutabhukti, by using the modern heliocentric figures. We shall now see how Bhāskara could deal with such a tough problem. Refer fig 16. Let P₁, P₂ be two positions of the planet on two consecutive days relative to the Sīghra A₂, so that P₁E₁P₂ is the Sphuta Kendragati spoken of. It will be noted that it is not Sphutagati be- cause P₁, P₂ are positions of the planet relative to A, which is itself moving (as rightly remarked by Bhāskara). It is Sphuta Kendragati because A₂E₁P₁ and A₂E₁P₂ are the

160 Kendras on two consecutive days whereas Madhyakendragati ^ is P₁E₂P₂. Also, we have the equation. Sīghra — Sphutagraha = Sphutakendra so that Sīghra- gati — Sphutagati = Sphuta- kendragati. Hence Sphutagati = Sīghragati — Sphutakendra- Fig. 16 gati. So we have now to seek the value of P₁Ê₂P₂. Let P₁a stand for the Sīghra- phala of the first day which is equal to e f, f being the true planet of the first day. E₂ d will be parallel to E₁ f because P₁ e being parallel to E₁ E₂ and e f being equal to P₁ d, d f 11 P₁ e (parallels to E₁ E₂ cut off equal arcs on the two circles). This may be seen also as follows. Since P₁ d is taken to be equal to e f, e being the mean planet and f the true on the first day e Ê₁ f = P₁ Ê₂ d. But E₁ e 11 E₂ P₁ ∴ e E₁ f = E₁ P̂₁ E₂ ∴ P₁ Ê₂ d = E₁ P̂₁ E₂ and alter- nate angles being equal E₁ P₁ 11 E₂d. P₁ a is the H sine of P₁ d where P₂ b is the H sine P₂ d. Looking upon P₁P₂ as an increment in P₁ d i.e. looking upon the Kendragati P₁ Ê₂ P₂ as an increment in Sīghraphala, Bhāskara uses the method of Bhogyakhanda sphuti Karana to obtain the Sphutakendragati. From the figure. P₂c = P₂b — P₁a = H sin (G + δm) — H sin E H sin E H cos δm + H cos E H sin δm = ————————————————————————————————————— — H sin E R

161 Taking H cos δ M = R and H sin δm = δm P₂c = (H cos E δm) / R This is at the end of E₁ P₂ i.e. at the end of K; so to get the corresponding chord in the deferent we do Karnānupāta so that the result is (H cos E δm) / R × R / K = (H cos E δm) / K as given by Bhāskara. We have cut short Bhāskara’s method of Bhōgyakhanda Sphutīkaraṇa to make it clear to a modern student. Since P₂c is small, passing on from the H sine to the arc is not necessary, for, the H sine of a small arc is equal to the arc itself. Bhāskara’s argument, however, is as follows:— “If for 225, we have Bhōgya Khanda, what for δm?” The result is (B × δm) / 225; Then B is rectified as follows:— “When the H cosine E is equal to the radius, i.e. initially in the H sine table, the Bhōgyakhanda is 225, then what is it for H cos E?” The result is (H cos E × 225) / R which we have to substitute for B. Then if this be at the end of K what is it at the end of R?” The result is (H cos E × 225) / R × δm / 225 × R / K = (H cos E × δm) / K as given. Before we proceed to explain ‘शेषं च वक्रा विपरीतशुद्धौ’, we shall explain what Bhāskara pointed out as a mistake in Lallācharya. Verse 40. Let Mathematicians understand that what formula was given by Lallācharya for Sīghragatiphala is not correct. When the anomaly is 90° or 270°, the gati- phala vanishes and there will be gatiphala at the points where it ought to be Zero according to his formula. 21

162 Comm. Ref. verse 45, Spaṣṭādhikāra, Śiṣya Dhī- vṛddhida तद्रहिताऽऽशुभुक्तिः, त्रिज्ज्याहता स्वचलकर्णहताऽऽशुचापभोग्य ज्यया विगुणिता विहृताऽऽद्यमौर्व्या, लब्धं त्यजेत्, स्वचलतुङ्गगतेः सदैव शेषं स्फुटा भवति च ग्रहभुक्तिरेवम्" i.e. ( शीघ्रगति-मन्दस्फुटगति ) × R Śīghraphala Bhōgyakanda --------------------------- × ----------------------- K 225 = Sīghragatiphala. Here the quantity within the brackets is δm. What Lallācharya had in his mind is as follows. ‘If for the Ādya Khanda 225 we have H cos E = R, what shall we have for the Bhōgya Khanda of the Sīghraphala ?’ The result would be [ H sin (E + 225) - H sin E ] × R / 225 = ( H sin E H cos 225 + H cos E H sin 225 / R - H sin E ) × R / 225 Taking H cos 225 = R and H sin 225 = 225 it would be H cos E × 225 / R × R / 225 = H cos E. So Lallācharya’s formula would become (δm / K) × H cos E as given by Bhās- kara. The charge levelled at Lallācharya is due to the fact that by the word ‘Āsu-chāpa’ by which Lallācharya meant ‘आशुफलचाप’ Bhāskara meant ‘आशुकेन्द्रचाप’. When the Kendra = 90 or 270, the Bhōgyakhanda being Zero, the Sīghragatiphala would be Zero. Also where it ought to be zero namely (90 + H sin⁻¹r) & (270 - H sin⁻¹r) it would not be zero. If Lallācharya had really meant what Bhāskara allributed to him, the formula would be (H cos mδm / K) and it is very unlikely that Lallācharya would have meant this wrong formula, for, even if K were taken by him to be steady, δ (r H sin m / K) = (r H cos mδm / K) and Lallāchar- ya’s formula does not contain ‘r’. The only non-rigorous part in Lallācharya’s formula is at the point where he took