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

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

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.