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Bharatiya Jyotihshastra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s6: (2) Adhikāra on True Places

Bhāratīya Jyotiḥśāstra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s6

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English

Contents

(2) Adhikāra on True Places

Chapter 1: True places and motions of planets

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A planet is not found moving every day at its mean daily rate of motion which is obtained from the time it takes to make a complete revolution through the Zodiac, but at a greater or smaller rate; and hence, on a given day it is not seen actually occupying the place in the sky which is found from its mean motion by calculation. The position and motion of a planet as actually seen in the sky are called its true (spaṣṭa) place and motion. It is the subject of study for the chapter on true places to find the true position and motion of a planet from its calculated mean position and motion. [It is a convention with our works to speak of the true position of a planet as “true planet” (spaṣṭa graha) and hence, this term is so used in some places in the following discussion.]

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The chief reason why the true motions of the Sun and Moon differ from their mean motions is that according to the laws of planetary motions, which are now almost universally recognized, and which were first discovered by Copernicus, and verified and firmly established by Kepler and Newton, the earth revolves round the Sun, and the Moon round the earth in an elliptical orbit; and there are two reasons why the true motions the remaining five planets differ from their mean motions; Mercury and the other four planets revolve round the Sun in an elliptical path, which gives them in their orbits a position different from their mean positions and the other reason is that this position of planets relative to the Sun, appears still different to us (observers on the earth), since the earth constantly changes its position in the sky while revolving round the Sun.

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Although our ancient astronomers did not know these reasons in their real perspective—two reasons in the case of five planets and one in the case of the Sun and Moon, they have unknowingly assumed the same principles to start with, while finding the places of planets; and the true positions of planets which we obtain by following our works agree to a considerable extent, if not to the fullest extent, with those obtained by following the methods which the Westerners have established after understanding the theory. In other words, the mean position of a planet being the same, if it be found by Western methods of calculation occupying a particulars place in the sky, we too get the same or very nearly the same place by following our own works; and the difference, if any, in the two results is due to some slight defects or approximations of the elements assumed in our calculations, and also to the fact that some other elements, in addition to the above two reasons, which have been lately discovered, were not known to our people. The following discussion will confirm the truth of the above statement that the two main reasons were indirectly known to our astronomers.

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The theory underlying the method of calculating the true place of a planet from its mean place is explained in our works by means of a diagram. Here the theory is given, as it will help one to understand what the reasons are which cause a difference between the true and the mean place of a planet, and what the ideas of our astronomers were regarding this question. They draw a circle to represent the planetary orbit, having for its centre, the centre of the earth. They draw another circle, equal to the first, such that its centre will be at a distance from the earth’s centre. This circle is termed a ‘Prativṛtta’ (eccentric circle) ; and the mean planet is supposed to move in that circular orbit, and the point of the planetary orbit which the mean planet would appear to occupy would be said to be its true place.

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In the adjoining figure the circle whose centre is E is the planet’s orbit. Similarly, the ‘eccentric’ circle has been drawn with C as its centre ; and point ‘M.P.’ in it represents the position of the mean planet and M, the corresponding position of the mean planet in its orbit. To an observer on the earth, the mean planet appears to lie in the line joining the point ‘M.P.’ to E. This line is called the ‘Karṇa’ (radius vector). This radius vector cuts the orbit circle at T, and the true planet appears to be in the orbit circle at that point. The difference between the true and the mean place, viz. the arc MT (i.e. MET))is termed ‘phalasaṃskār’, (i.e. equation of centre). The maximum value of this correction is called the ‘parama’ or ‘antya’-phalasaṃskār (max. equ. of centre). The centre C of the eccentric circle is marked away from E, the centre of the orbit circle, at a distance equal to the sine of the maximum correction. The ‘phala’ (i.e. correction) mentioned above, is called ‘mandaphala’ i.e. equation of centre. The position obtained by applying this correction, or equation of centre, to the mean place of a planet (i.e. adding the correction to or subtracting it from the mean place, according as it is positive or negative) is termed the ‘manda-spaṣṭagraha’ (true heliocentric place of a planet). The true places of the Sun and Moon are obtained by applying only one correction, that of the equation of centre. But, the positions so obtained in the case of other five planets are not the positions at which they would be visible to observers on the earth ; (according to modern theory they will represent the true heliocentric position, that is the position seen from the centre of the Sun). They require another correction to be applied ; it is called the ‘Śīghra-phala-saṃskār’ (i.e. the annual parallax). When it is applied to the heliocentric position, one gets the position where the planet is observed from the earth. In order to find it, they assume another circle called the ‘Śīghra-prati-vṛtta’ (śīghra eccentric circle) and they find the required correction by regarding the position of the heliocentric planet as that of the mean planet. The operation of obtaining the ‘mandaphala’ (equation of centre) is known as the ‘manda-karma’ and that of obtaining ‘śīghra-phala’ (annual parallax), as ‘śīghra-karma’. The method of finding the ‘śīghra phala’ is expalined below :—

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The orbit circle drawn in the operation of “manda karma” becomes the “śīghra-prativṛtta” in the operation of “śīghra-karma”, then taking a point away from the centre of this circle, at a distance equal to the sine of the maximum value of ‘śīghraphala’, another orbit circle is drawn with this point as centre.

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The earth itself is supposed to be at the centre of this orbit circle drawn in the operation of ‘śīghra-karma’. The place where the heliocentric planet, while moving at its own rate of motion in the “śīghra-prati-vṛtta”, appears to be in the “śīghra-kakṣāvṛtta” is its geocentric position. The planet appears to be at that place to an observer on the earth. Some people treat the “manda-kakṣāvṛtta” itself as the “śīghra-kakṣāvṛtta”, and then they draw an equal circle called the “śīghra prativṛtta”, having for its centre a point which is marked at a distance equal to the sine of ‘śīghrāntya-phala’ from the original centre. Then the heliocentric position of the planet from the ‘mandakakṣā-vṛtta’ is transferred to the “śīghra-prativṛtta”, and its place as seen in the “kakṣāvṛtta” is taken to be its geocentric position. Both the methods lead to the same result.

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The above diagram will show that the distance of the planet moving in the eccentric circle is not the same from the point E. The distance is greatest when the planet is at A (aphelion) and the least when it is at P (perihelion); in other words, the path of the moving planet is as it were assumed to be, elliptical. E is one focus of this ellipse.

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Parameśvara, the commentator of Āryabhaṭai, has described in simple words, the method of drawing the figure for finding the equation of centre The author has not come across an equally good explanation in any other work. The author, therefore, quotes the verses here :—

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विज्याकृतं कुमध्यं कक्षावृत्तं भवेत् तु तच्छ्रधयं ।।
शीघ्रदिशि तस्य केंद्रात् शीघ्रान्त्यफलांतरे पुनः केंद्रं ।। २ ।।
कृत्वा विलिखेत् वृत्तं शीघ्रप्रतिमंडलाख्यमुदितमिदं ।।
इदमेव भवेन्मांंद कक्षावृत्तं पुनस्तु तत्केंद्रात् ।। ३ ।।
केंद्रं कृत्वा मंदांत्यफलांतरे वृत्तमापि च मंददिशि ।।
कुर्यात्प्रतिमंडलमिदमुदितं मांंद शनीदयभूपुत्राः ।। ४ ।।
मंदप्रतिमंडलगास्तत्कक्षायां तु यत्र लक्ष्यंते ।।
तत्र हि तेषां मंदस्फुटाः प्रदिष्टास्तथैव शीघ्रं ते ।। ५ ।।
प्रतिमंडले स्थिताः स्युस्ते लक्ष्यंते पुनस्तु शीघ्राह्वये ।।
कक्षावृत्ते यस्मिन् भागे तत्र स्फुटग्रहास्ते स्युः ।। ६ ।।
मांंद कक्षावृत्तं प्रथमं बुधशुक्रयोः कुमध्यं स्यात् ।।
तत्केंद्रान्मंददिशि मंदांत्यफलांतरे तु मध्यं स्यात् ।। ६ ।।
मांंदप्रतिमंडलस्य तस्मिन् यत्र स्थितो रविसतव ।।
प्रतिमंडलस्य मध्यं शीघ्रस्य तस्य मानमपि च गदितं ।। १० ।।
शीघ्रस्ववृत्ततुल्यं तस्मिंश्चरत: सदा ज्ञशुक्रौच ।।

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“The orbit circle, at the centre of which is the earth, and whose radius is equal to ‘trijyā’ *is called a ‘śīghra’ circle (i.e. the orbit circle necessary in the geocentric calculation or ‘śīghra-karma’). Another circle should be drawn with a point for its centre, which should be away from the centre (of the orbit circle) at a distance equal to maximum ‘śīghra phala’ and in the direction of the ‘śīghra’ position. This circle is termed as ‘śīghra’-eccentric circle. The same becomes a ‘kakṣāvṛtta’ (orbit circle) in the heliocentric operation. Again, draw another circle, in the direction of the heliocentric position, having its centre at a distance of maximum equation of centre. This is termed ‘manda’-eccentric circle. Saturn, Jupiter and Mars, while moving in the ‘manda eccentric circle’ are seen occupying certain places in the heliocentric orbit circle, and they are termed ‘heliocentric’. (These are the heliocentric positions of Saturn, Jupiter, and Mars). They should similarly be treated as moving in the ‘śīghra’-eccentric circle and their corresponding positions in the ‘śīghra’-orbit circle would be called their ‘true position’ (These should be taken to be their geocentric positions). The circle having the earth for its centre, becomes the ‘Manda’ orbit circle for Mercury and Venus. The centres of their ‘manda’ eccentric circles lie at a distance, equal to their maximum equation of centre, from the centre of the orbit. The point of this circle occupied by the Sun should be taken to be the centre of the geocentric eccentric circle. The size of this circle has been mentioned as being equal to that of his ‘own-geocentric-circle’. Mercury and Venus always move in that circle.

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∗ The word ‘trijyā’ is now-a-days used as a technical term signifying semi-diameter. But originally it stood for ‘jyā’ (sine) of ‘tribha’ (i.e. 3 signs or 90 degrees); and our astronomical works are generally found using it in that sense. When the perimeter of a circle is supposed to consist of 360°, (or 21600 minutes), the semi-diameter becomes equal to 3438’; or the sine of 90° is equal to the semi-diameter. Hence, by “trijyā” is generally meant a line whose length is 3438’.

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The underlying theory of the equation of centre is explained in another way by assuming a circle known as the ‘nicocca-vṛtta’ (apsidal circle). BHĀSKARĀCĀRYA observes in this connexion :-

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कक्षास्थमध्यग्रहचिह्नतोथ वृत्तं लिखे दंत्यफलज्यया तत् ॥ नीचोच्चसंज्ञं रचयेच्च रेखां कुमध्यतो मध्यखगोपरिस्थां ॥ २४ ॥ कुमध्यतो दूरतरे प्रदेशे रेखायुते तुंगमिह प्रकल्प्यं ॥ नीचं तथासन्नतरेऽथ तिर्यङ् नीचोच्चमध्ये रचयेच्चरेखां ॥ २५ ॥ नीचोच्चवृत्ते भगणांकितेस्मिन् मांदे विलामं निजकेंद्रगत्या ॥ शैघ्रयेनुलोमं भ्रमति स्वतूंगादारभ्य मध्यद्युचरो हि यस्मात् ॥ २६ ॥ अतो यथोक्तं मृदुशीघ्रकेंद्रं देयं निजोच्चाद्युचरस्तदग्रे ॥

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“ Draw a circle with the mean planet in the orbit, as centre, and the sine of maximum equation of centre as the radius. It is called ‘the apsidal’ circle. Draw a line from the centre of the earth passing through the mean planet. It intersects the circumference of the epicycle in two points—the farthest point is called the ‘ucca’ (aphelion) and the nearest one the ‘nica’ (perihelion). Draw a horizontal line in between the aphelion and perihelion. Mark the signs and degrees on the circumference of the ‘nicocca’- circle. The mean planet moves from the aphelion in the mānda (heliocentric) epicycle in a regular direction, and in the śaighra (geocentric) epicycle in a reverse direction, at the same rate of motion as its anomaly (i.e. according to the motion of its mean anomaly or of the angle of commutation). Hence, the centres of the mean and true circles are to be marked in relation to the aphelion.

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The planet is seen at the end of the direction line (The heliocentric place at the end of the heliocentric apse line and geocentric at the end of the geocentric apse line.) ”

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In the adjoining figure, the circle whose centre is E is the orbit circle. M is the mean place of the planet in the heliocentric operation and its heliocentric place in the geocentric operation, and the same point is the centre of the heliocentric or geocentric epicycle. The epicycle has been drawn with that point as the centre and the sine of the maximum equation of centre (or annual parallax) as radius. The point P in it stands for the planet, and the line joining it to the centre (E) cuts the orbit circle in the point T. It is the position of the ‘ true planet ’ (i.e. heliocentric or geocentric planet)’. Bhāskarācārya observes in regard to this theory :

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ग्रह: पूर्वगत्या प्रतिमंडलेनैव भ्रमति । यदेतन्नीचोच्चवृत्तं तत् प्राज्ञैर्गणकै: फलार्थं कल्पितं ।।
गोलाध्याय, छेद्यकाधिकार.

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“ As a matter of fact, a planet moves in the eccentric circle itself with its original rate of motion. The epicycle has been invented by astronomers only to calculate the equation of centre.”

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In the above figure, of all points on the eccentric circle, A, is at the greatest distance from the centre of the earth ; it is called aphelion. B is at the nearest distance ; it is called Perihelion. The Aphelion belonging to the Manda eccentric circle, is called ‘ mandocca ’ or an ‘ apsis ’ ; and that of the śīghra eccentric circle is called the ‘ śīghrocca ’ or the apex of swift motion. The longitudes of the mandoccas of planets and their motions have been discussed in the account of Āryabhaṭa I (page 69). The mandoccas have got a very slow* motion. The Sun itself is taken to be the ‘ śīghrocca ’ for Mars and other superior planets, and our works have assumed the same motion for the śīghroccas of Mercury and Venus which these planets, according to modern theory actually have in their orbits. The above figure will show that when a planet comes to the position of aphelion or perihelion, the mean place of the planet and its true place coincide. In other words the equation of centre is Zero. As the planet advances three signs or 90° from the aphelion, the value of the equation of centre gradually increases and then it continually decreases till it reaches the perihelion. It increases again for three signs more and finally decreases till it reaches the aphelion. In short, whatever change occurs in the mean motion of the planet is in relation to the aphelion. The Sūrya Siddhānta makes the following observation :—

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∗ Grant says (History of Physical Astronomy P. 97) that it was al-Buṭṭānī, an Arab astronomer (880 A.D.) who first discovered that the Sun’s apogee has motion. This means that it was not known to Ptolemy and Hipparchus that the Sun’s apogee and the aphelia of the planets have motion. But our astronomer BRAHMAGUPTA has mentioned (628 A.D.) the motion of aphelia of planets. Moreover it is met with even in the modern S.S. Prof. whitney has simply ridiculed the fact that the motion given to the apsides of planets is very slow according to our astronomers. But the only reason of this is that his prejudiced mind could not tolerate the idea of bringing down the Westerners to an inferior position by admitting that while Ptolemy never knew that a planet’s aphelion has got motion, the Hindus knew it. But the writer has already explained in his account of ĀryabhaṭaI that the motion of aphelion mentioned in our old works is not as slow as Whitney considers it to be.

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अदृश्यरूपाः कालस्य मूर्तयो भगणाश्रिताः । शीघ्रंमदोच्च पातास्था ग्रहाणां गतिहेतवः ॥ १ ॥ तद्वातरश्मिभिर्बद्धास्तैः सव्येतरपाणिभिः । प्राकपरचादपकृष्यन्ते यथास्वं स्वदिक्मुखं ॥ २ ॥ स्पष्टाधिकार.

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“ The three invisible forms of time, viz. śīghrocca, mandocca and pāta which are supported by ‘bhagaṇas’(revolutions) are the generators of the motions*of planets. These forms of time drag towards themselves by tossing and fro**, the planets which are tied by the reins of wind held by them.”

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No other siddhānta has given so much importance to the aphelia as the S. S. has done by regarding them as some animate objects. Brahmagupta simply remarks,

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प्रतिपादनार्थमुच्चा : प्रकल्पिता*** : ग्रहतेस्तथा पाताः ॥ २६ ॥ गोलाध्याय.

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“ The aphelia and nodes have been imagined as points simply to explain the phenomena of planetary motions.”

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Nowhere does the S. S. explicitly state that the planets move in the epicyclic orbits ; hence, it appears that the aphelia have been supposed to be some objects having forms. But when the planets are assumed to be moving in epicyclic orbits, their mean positions naturally undergo a change which simply depends upon the distance of the planet from the aphelion.

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It has been mentioned above that the epicycles are supposed to be drawn at a distance from the centre of the orbit circle equal to the sine of the maximum value of the equation of centre or the annual parallax. The value of the equation of centre or of the annual parallax concerning each planet is given in our treatises, and it is the convention to give it in terms of the circumference of a circle, having the sine of maximum correction (viz. equation of centre) as semi-diameter ; in other words it is what the length of the circumference, expressed in degrees, would be, if a circle be drawn with the sine of maximum correction as semi-diameter ; and it is generally termed ‘ Paridhi ’ or ‘ circumference ’. The circumference of the circle drawn with respect to the equation of centre is called the “ mandaparidhi ” i.e. the dimension of the epicycle of the apsis, and that with respect to the annual parallax is called the ‘ śīghra-paridhi ’ or the dimension of the epicycle of the apex. The reason for expressing the equations in terms of epicycles seems to be the above system of drawing the epicycles of apsides. Considered independently, the circumference of the apsidal circle no doubt represents 360° ; but as the value of the equation is to be reckoned in terms of degrees of the orbit circle, the length of the perimeter of an epicycle is also expressed by the same system of degrees.

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*Here, the word ‘ motion ’ is to be taken to mean true motion. **By “ to and fro ” is meant the actual position in advance of or behind the mean position of a planet. (Ranganātha has interpreted this in a somewhat different way). The position of a planet in the north or south direction changes because of the nodes. ***By ‘ motion ’ here is meant the true motion of the planet.

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The next table gives the dimensions of the epicycles related to the apsis and the apex as given by different authors; similarly, their radii also have been calculated and given. The radii themselves represent the maximum values of their equations. While calculating the radius, the ratio of the circumference to its radius as mentioned by Āryabhaṭa I and Bhāskarācārya, viz. 62832 : 10000, has been adopted.

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The arc of anomaly containing 3 signs or 90° is termed *pada* (quadrant). The first and third are termed *oja* (odd) and the second and fourth *Yugma* (even). The authors of some siddhāntas hold the view that the length of the circumference related to the odd quadrants is different from that of the circumference of the even quadrants; and that it varies in the intermediate portions proportionately. In the following table, the *paridhis* of some planets according to the Pañcasiddhāntikā have not been given, because the figures as given in that work are not known for certainty. In the case of other siddhāntas, wherever the *paridhis* related to the even quadrants are not mentioned, they are to be taken to be equal to those related to the odd quadrants.

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MANDAPARIDHI

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Dimensions of the Epicycles of Apsis and their radii or max. value of the equation of centre

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| Planet | The S. S. from Paca Siddhān/tikā | Modern Sūrya Siddhānta | | | First Ārya siddhānta |
| | | At the end of odd quadrant | At the end of even quadrant | At the end of odd quadrant |
| | Circum Radius | Circum Radius | Circum Radius | Circum Radius |
| | ° ' " | ° ' " | ° ' " | ° ' " |
| Sun | 14 2 13 41 | 13 40 2 10 30 | 14 2 13 41 | 13 30 2 8 55 |
| Moon | 31 4 56 2 | 31 40 5 2 24 | 32 5 5 35 | 31 30 5 0 48 |
| Mars | 70 11 8 27 | 72 11 27 33 | 75 11 56 12 | 63 0 10 1 36 |
| Mercury | 28 4 27 23 | 28 4 27 23 | 30 4 46 29 | 31 30 5 0 48 |
| Jupiter | 32 5 5 35 | 32 5 5 35 | 33 5 15 8 | 31 30 5 0 48 |
| Venus | 11 1 45 2 | 12 1 54 35 | 8 0 2 5 53 |
| Saturn | 48 7 38 22 | 49 7 47 55 | 40 30 6 26 45 |

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| Planet | First Ārya Siddhānta | Brahma Siddhānta | Second Ārya Siddhānta |
| | At the end of even quadrant | At the end of odd quadrant | At the end of even quadrant | |
| | Circum Radius | Circum Radius | Circum Radius | Circum Radius |
| | ° ' " | ° ' " | ° ' " | ° ' " |
| Sun | 13 40 2 10 30 | 13 40 2 10 30 |
| Moon | 31 36 5 1 45 | 31 34 5 1 26 |
| Mars | 81 0 12 53 29 | 70 0 11 8 27 | 65 30 10 25 29 |
| Mercury | 22 30 3 34 51 | 38 0 6 2 52 | 27 36 4 23 34 |
| Jupiter | 36 0 5 43 46 | 33 0 5 15 8 | 28 15 4 29 46 |
| Venus | 9 0 1 25 57 | 9 0 1 25 57 | 11 1 45 2 | 9 35 1 31 31 |
| Saturn | 58 30 9 18 38 | 30 0 4 46 29 | 52 42 8 23 15 |

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Dimensions of the EPICYCLES OF APEX and their radii or max. value of the annual parallax.

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| Planet | The S. S. from Pañca Siddhāntikā | Modern Sūrya Siddhānta | First Ārya Siddhānta |
| | | At the end of odd quadrant | At end end of even quadrant | At end end of odd quadrant |
| | Circum Radius | Circum Radius | Circum Radius | Circum Radius |
| | ° ' " | ° ' " | ° ' " | ° ' " |
| Mars | 234 37 14 32 | 232 36 55 26 | 235 37 24 5 | 238 30 37 57 30 |
| Mercury | 142 21 0 30 | 132 21 0 30 | 133 21 10 3 | 139 30 22 12 7 |
| Jupiter | 72 11 27 33 | 72 11 27 33 | 70 11 8 27 | 72 0 11 27 33 |
| Venus | 260 41 22 49 | 260 41 22 49 | 262 41 41 55 | 265 30 42 15 20 |
| Saturn | 40 6 21 58 | 40 6 21 58 | 39 6 12 25 | 40 30 6 26 45 |

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| Planet | First Ārya Siddhānta | Brahma Siddhānta | Second Ārya Siddhānta |
| | At the end of even quadrant | At the end of odd quadrant | At the end of even quadrant | At the end of odd quadrant |
| | Circum Radius | Circum Radius | Circum Radius | Circum Radius |
| | ° ' " | ° ' " | ° ' " | ° ' " |
| Mars | 229 30 36 31 33 | 243 40 38 46 50 | | 230 59 36 45 43 |
| Marcury | 130 30 20 46 11 | 132 21 0 30 | | 134 30 21 24 23 |
| Jupiter | 67 30 10 44 35 | 68 10 49 21 | | 69 30 11 3 40 |
| Venus | 256 30 40 49 23 | 263 41 51 28 | 258 41 3 43 | 261 30 41 37 8 |
| Saturn | 36 0 5 43 46 | 35 5 34 13 | | 40 40 6 28 20 |

v2.c12.s6.t1.42

The maximum values of the equation of centre according to Ptolemy and those according to modern European astoronomers have been given below in a table.* One can, of course, compare them with the above values given in our old treatises. But in order to facilitate such comparison, the maximum values of the equation of centre at the end of the odd quadrants as given by Āryabhaṭa I, out of our siddhāntas, have again been given below :—

v2.c12.s6.t1.43

MAXIMUM VALUES OF EQUATION OF CENTRE

v2.c12.s6.t1.44

| | First Ārya Siddhānta | Ptolemy | Modern |
| | ° ' " | ° ' " | ° ' " |
| Sun | 2 8 55 | 2 23 | 1 25 27 |
| Moon | 5 0 48 | 5 1 | 6 17 13 |
| Mars | 10 1 36 | 11 32 | 10 41 33 |
| Mercury | 5 0 48 | 2 52 | 23 40 43 |
| Jupiter | 5 0 48 | 5 16 | 5 31 14 |
| Venus | 2 51 53 | 2 23 | 0 47 11 |
| Saturn | 6 26 45 | 6 32 | 6 26 12 |

v2.c12.s6.t1.45

∗ These have been adopted from the translation of the Sūrya Siddhānta by Burgess (p. 76).

v2.c12.s6.t1.46

The modern values of Mercury and Venus cannot properly be compared with those given in our works, because the modern values are true only with respect to an observer on the Sun's disc while our works have given them with regard to the observer on the earth. It would not be wrong, however, if the values of other planets from both the systems are compared; and when they are so compared, it will be seen that the values given in our works agree with the modern ones to a great extent. The orbits of the moon and the planets are elliptical according to the modern astronomical theory. The value of their equation of centre varies with the changing values of the eccentricity of their orbits; and these values of the equation of centre as given in our works agree with the values given in modern works. The form of planetary motions as given in our works has been shown above which will show that although the writers of our works have not assumed an elliptical orbit for the movement of planets, still they have assumed that their distances from the centre of the orbit never remain the same, and that the equations vary with their position with respect to the points of apices known as aphelion and perihelion, in the orbit; from this it would appear that ancient authors indirectly knew the main cause of the difference between the true and mean places of planets, that is the phenomenon of the movement of these planets (or of the Moon) in elliptical orbits. The annual parallax of a planet depends upon the variable distance of the planet's heliocentric position in its orbit from the earth. The figures showing the annual parallax as given in our works have been given in the above table (page 243) and it has been pointed out before (page 197) that the radii vectors of planets calculated from them agree with the corresponding modern values. This fact coupled with the trend of the above discussion will show that our astronomers indirectly knew the second factor which was responsible for the difference between the true and mean position of a planet,—that the position of a planet with respect to the Sun, which is known as its heliocentric position, differs from the positions seen by an observer on the earth, because the earth also revolves round the Sun.

v2.c12.s6.t1.47

That none of the values given by PTOLEMY agree with the corresponding values given in our siddhāntas is one out of the many proofs* to show that Ptolemy had no concern with any of the siddhāntas.

v2.c12.s6.t1.48

Now some more noteworthy facts about the epicycles of apsides and apices may be mentioned. According to some siddhāntas, the dimensions of epicycles are different in the odd and even quadrants. ĀRYABHAṬA I has shown much variation in these dimensions; the Sūrya Siddhānta does not mention so much variation. BRAHMAGUPTA has assumed different dimensions for the epicycles in the odd and even quadrants only in the case of Venus. The modern Romaśa, Soma, Śakalya Brahmasiddhānta and Vasiṣṭha siddhānta are almost similar to the modern Sūryasiddhānta; still the measures of epicycles, as given in the Romaśa and Soma have been assumed to be the same throughout, and they agree with those of the even quadrants mentioned by the S. S. The dimension of the apsidal epicycle of mercury, given by Soma siddhānta as 34, however, does not agree (with S. S.). The Vasiṣṭha siddhānta does not mention* the apsidal epicycles at all. The dimensions of the epicycles of the apices have been given, but they do not agree with those of the Sūryasiddhānta, and hence they are given below :—

v2.c12.s6.t1.49

∗ The Romaka siddhānta of the Pañcasiddhāntikā gives 4°57′ as the maximum equation of centre for the Moon (See 8.6. Pañc S.). This does not agree with that given by Ptolemy. This is one of the proofs to show that the Romaka Siddhānta of the Pañca Siddhāntikā was not written by Ptolemy.

v2.c12.s6.t1.50

Mars 234 ; Mercury 133 ; Jupiter 71 ; Venus 261 ; Satur 39.

v2.c12.s6.t1.51

Theses are the same for both the quadrants, and though they do not agree with the Sūrya Siddhānta, they are easily seen to be approximately mean measures of the two kinds of quadrants. The copy of the Śakalya Brahma siddhānta in the author's possession does not mention any epicycles at all ; but there undoubtedly appears** to be in this copy, a break at the place where they are expected to have been mentioned. They must have been given in the original work. Lalla being the follower of Āryabhata I, the measures of epicycles given by both of them are identical. Similarly BHĀSKARĀCĀRYA was the follower of Brahmagupta and both have therefore, given the same measures. But Bhāskarācārya has mentioned 50° as the measure of the apsidal epicycle of Saturn and 40° as that of its epicycles of apex. The Sundursiddhānta of JNĀNARĀJA has given measures of epicycles similar to those of the modern Surya Siddhānta. According to Muniswara,, the author of Siddhānta Sārvabhauma, it is illogical to asume different measures for the epicycles in the odd and even quadrants. He has given the mean values of the measures of the epicycles in the odd and even quadrants given in the modern Surya Siddhānta. Different Karaṇa works show some variations the measures of epicycles ; but it appears to be due to the fact that sufficient attention was not paid to their accuracy. There is nothing specially worth mentioning about them.

v2.c12.s6.t1.52

The modern values of the equations of centre are given above ; but they are not always the same. They very with the lapse of time. The following table *** gives an idea of the long-period variation affecting the value of the Sun's equation of centre.

v2.c12.s6.t1.53

| Years before Śaka era | Maximum value of eqn. of centre | Years after Śaka era | Maximum value of eqn of centre |
| | ° ' | | ° ' |
| 10000 | 2 31 | 0 | 2 1 |
| 9000 | 2 28 | 1000 | 1 58 |
| 8000 | 2 25 | 2000 | 1 55 |
| 7000 | 2 22 | 3000 | 1 52 |
| 6000 | 2 19 | 4000 | 1 49 |
| 5000 | 2 16 | 5000 | 1 46 |
| 4000 | 2 13 | 6000 | 1 43 |
| 3000 | 2 10 | 7000 | 1 40 |
| 2000 | 2 7 | 8000 | 1 37 |
| 1000 | 2 4 | 9000 | 1 34 |
| 0 | 2 1 | 10000 | 1 31 |

v2.c12.s6.t1.54

∗ It is found neither in the edition printed at Varanasi nor in the version in the Deccan College collection.

v2.c12.s6.t1.55

∗∗ The second chapter begins abruptly after an incomplete line of the verse : "Maurvya Catuṣke", which is given after 111 verses of the first chapter. The second chapter opens with an unexpected question. It seems that the epicycles may have been mentioned in between. It is surprising that the break was found precisely at the same place in the copies seen at Gwalior, Asta, and later on in the Anandāśrama copy (No. 4341).

v2.c12.s6.t1.56

∗∗∗ This table has been taken from the Planetary Tables of Keropant.

v2.c12.s6.t1.57

The table shows that the correction due to the equation of centre, in the case of the Sun, is gradually diminishing. This correction, according to our works, ranges from 2° 13' 41" to 2° 8' 55". It may be noticed that the more ancient treatises have given a greater value and the modern works a smaller one, as can be seen from the above table (page 242). It is easily seen that the value was determined after actual observations were taken at different times. Our astronomers have determined the correction figures for the Sun and Moon from the observations of their eclipses, that is from their positions at the moments of lunations. The modern European method of finding the true place of the Moon from that of its mean place, requires the application of five main corrections. It has been shown further that our people had determined a value as the maximum equation of centre for the moon, which is equal to the sum of the four (out of five) corrections applicable at the moment of lunation. The maximum value of the fifth correction of the equation of centre is 11 minutes. (Planetary Tables, by Keropant, p. 105). As its argument is Sun's anomaly, it was taken to be applicable to the Sun with minus or plus sign, where it was actually to be applied to the Moon with plus or minus sign. This has not affected the result of the eclipse. If 11' be subtracted from 2° 14', which is the value of the sun's maximum equation of centre as given in our ancient works, our works can, in fact, be said to have given 2° 3' as the correction for the Sun, and that was actually the value in the year 500 B.S. Hence, our people appear to have found out the correction for the Sun at that early period or at least two or three centuries before Śaka era. The equation of centre for the Sun, according to Ptolemy, is 2° 23', which means that he had nothing to do with our works. Although PTOLEMY has given 2° 23' as the Sun's correction, it was actually 2° in his time (about Śaka 70). Obviously it was not found by Ptolemy himself, but probably adopted from some previous writer. The fact that no one except Hipparchus possessed, before him, the knowledge of calculating the true place of the Sun and the fact that the length of the year according to Ptolemy and Hipparchus was the same* measure (365ᵈ 14ˢ -48ᵖ) lead one to infer that Ptolemy had adopted the value of the Sun's equation of centre from Hipparchus. This inference is further confirmed by the fact that the Romaka siddhānta which was compiled on the basis of the work on Hipparchus has adopted 2° 23' 23" as the maximum equation of centre for the Sun. No one ever says that the Hindus borrowed astronomy from some work compiled after Ptolemy. No astronomer of equal capability flourished within 3 or 4 centuries after Ptolemy. None of our Siddhāntas contain the same equation of centre for the Sun as was given in the original Romaka Siddhānta. From all these facts, any impartial thinker will have to admit that it is proved beyond all doubt that our astronomers did not borrow the figures for the equation of centre for the Sun from any European work, but that they themselves determined it before the Śaka era.

v2.c12.s6.t1.58

The modern maximum values of the equation of centre that have been given above (page 242) show that the equation of centre for the moon is 6° 17'. But there are certain factors, other than the equation of centre, which cause a difference between the Moon's mean and true places; and these sometimes cause a variation of 8° to 8½° between the mean and true places of the Moon. For finding this, about 40 corrections are required to be applied. Of these, the correction known as the equation of centre, mentioned above, is a very large figure and the other four corrections are also appreciably large. Of these, the one known as ‘variation’ (pākṣika or taithik) has “moon minus true sun” as its argument. This argument will, of course, become 6 signs and zero at the moment of the full moon and the new moon respectively, and this correction too becomes zero* at these moments. Similarly, for the argument is [2 × (corrected moon—true sun) —moon’s anomaly].

v2.c12.s6.t1.59

∗ Grant's History of Physical Astronomy, Chap. XVIII.

v2.c12.s6.t1.60

The first term in this formula becomes zero at the moment of the full moon and new moon; and the argument is reduced to (zero minus moon’s anomaly) at that moment. When the value of the argument is three or nine signs, the correction attains its maximum value viz. 1° 20.2′. Hence, if the moon’s anomaly at the time of the full or new moon is 3 or 9 signs, the argument for calculating the correction due to evection becomes “zero minus three signs” i.e. 9 signs or ‘zero minus line signs’ i.e. 3 signs and the corresponding values of the correction due to evection become respectively +1° 20′ and —1° 20’**, and at that time, if the moon’s anomaly is 3 signs, the equ. of centre becomes —6° 17′ and if the moon’s anomaly is 9 signs, the equ. of centre becomes +6° 17’***.

v2.c12.s6.t1.61

Hence at the moments of full or new moon, the maximum value of the correction, as given by the equation of centre and the evection together, would not exceed ±1° 20′ ∓ 6° 17′ = ∓ 4° 57′.

v2.c12.s6.t1.62

One correction of 11′ out of the four has been applied to the Sun as already explained. The fourth correction is about 7 minutes.**** Applying it to the above figure of 4° 57′, we get 5° 4′. The remaining 35 out of 40 corrections are very small. In short, the maximum value of the moon’s equation of centre as given by our siddhāntas, which lies between 4° 56′ and 5° 6′, has proved to be very accurate†. The best means of testing the accuracy of the values of the equations of centre in the case of the sun and moon are the eclipses; and it has already been observed before (pages 60, 130 etc.), with the aid of the eclipses.

v2.c12.s6.t1.63

Sudhākara states that Munjāl has mentioned a correction similar to that of evection and another like variation, and that Nityānanda has mentioned those for variation and for the nodes.

v2.c12.s6.t1.64

No Western astronomer before Ptolemy knew how to find the true places of the five planets; even Hipparchus did not know it††. And the maximum values of the equation as given by Ptolemy do not agree with those given in our works. This shows that our astronomers have themselves found out the method of calculating the true places of the five planets. The calculation of the true places of the Sun, Moon and the five planets is the most important part of mathematical astronomy; in fact it is the quintessence of astronomy, and this we have decidedly not borrowed from Westerners.

v2.c12.s6.t1.65

∗ Keropant’s Planetary Tables, p.

v2.c12.s6.t1.66

¹¹⁰ ** Keropant’s Planetary Tables, p.

v2.c12.s6.t1.67

¹⁰⁶ *** Keropant’s Planetary Tables, p.

v2.c12.s6.t1.68

¹⁰⁹ **** Keropant’s Planatary Tables, pp. 105 & 111. † The above explanation about the equation of centre relating to the Sun and Moon was suggested by Venkateś Bāpūji Ketkar. ††Grant’s History of Physical Astronomy, Chap. XVIII.

v2.c12.s6.t1.69

The equation of centre is found by the formula epicycle X sine of anomaly of planet. semi-diameter

v2.c12.s6.t1.70

The ‘Kendra’ (anomaly) is the difference between the place of the planet and that of the aphelion. The Sun and Moon require only one correction, that of the equation of centre. Others require two corrections, the equation of centre and the annual parallax. The calculation requires the use of the planet’s distance from the earth; and in order to find the equation of centre accurately, one has to use the method of successive approximation.

v2.c12.s6.t1.71

Sines and Radius

v2.c12.s6.t1.72

The siddhānta works give sines of divisions of circle, each of which is equal to 3¾°. The Karaṇa works use divisions consisting of 10° to 15° each since they are not very particular about accuracy. Most of the siddhāntas have assumed 3438 as the value of ‘trijyā’ (radius) while calculating values of sines. Brahmagupta has assumed it to be 3270. Kamalākara, the author of Siddhāntatattva-viveka has mentioned sines of each degree of the quadrant assuming 60 as the value of the ‘trijyā’. The Karaṇa works generally take 120 as the value of the trijyā. According to Sudhākara, Munjāl has adopted 8° 8' as the ‘trijyā’ and Gangādhara, author of Cāndramāna (page 195) has adopted 191 as the measure. The work Yantrarāj has adopted 3600 as the ‘trijyā’ and given the sine of each degree of the quadrant. Keropant Nānā remarks* that the value of ‘trijyā’, 3438, which this Hindu astronomers adopt is an awkward figure entailing unnecessary multiplication and division. This is to a certain extent, true. But our astronomers have at places made use of artifices to avoid lengthy multiplications and divisions; and the reason for adopting that figure for the radius is a rational one, inasmuch as the circumference of the circle is equal to 21600 minutes and the radius computed from it comes to 3438. The most accurate ratio of the diameter to the circumference is 1 : 3.1415927. On the basis of this ratio, if the circumference be 21600, the semidiameter comes to 3437¾. Our astronomers have adopted 3438 as the value after leaving out the fraction. This will show that the value of ‘trijyā’ adopted by our astronomers is very accurate†. Our ancient authors have assumed different figures as the ratio of the diameter to the circumference. They are given below :‡

v2.c12.s6.t1.73

† The European mathematicians assume the value of a ‘trijyā’ as equal to the 10th or some other power of 10. They have ready made tables for the purpose, which give sines and other ratios of each minute of arc; and as the trijyā is a very large number greater accuracy is ensured.

v2.c12.s6.t1.74

| Sūrya Siddhānta, Brahmagupta, Āryabhaṭa II | .. 1 : √10 or 1 : 3.1623 |
| Āryabhaṭa I | .. 20000 : 62832 or 1:3.1416 |
| Āryabhaṭa II & Bhāskarācārya** | .. 7 : 22 or 1 : 3.1428 |
| Bhāskarācārya | .. 1250 : 3927 or 1 : 3.1416 |
| 3438 as the radius gives | .. 1 : 3.14136 |
| Precise Modern European value | .. 1 : 3.1415927 |

v2.c12.s6.t1.75

Evidently our people had a very accurate knowledge of the ratio of the diameter to the circumference. If they have, at places, adopted an approximate value, it is only with a view to simplifying calculations in practical work.

v2.c12.s6.t1.76

∗ See Planetary Tables, page 314.

v2.c12.s6.t1.77

∗∗ Āryabhaṭa II and Bhāskarācārya each has mentioned this ratio in two ways.

v2.c12.s6.t1.78

Brahmagupta has given in the following verse the reasons for adopting 3270 as the value of the semi-diameter :—

v2.c12.s6.t1.79

भगणकलाभ्यासार्धं भवति कलाभिर्यतो न सकलाभिः
ज्यार्धानि न स्फुटाति ततः कृतं व्यासदमन्त्यत् ॥ १६ ॥

v2.c12.s6.t1.80

गोलाध्याय.

v2.c12.s6.t1.81

Accurate calculation shows that the radius, corresponding to 21600 as the circumference, is not the whole number 3438, and it is true that because of this the "first sines" are not very accurate. But the author does not think that the value of the semi-diameter viz. 3270 which Brahmagupta has adopted on the basis of the ratio 1 : √10 of the diameter to the circumference or by some other method can be justified.

v2.c12.s6.t1.82

The question of calculating sines and their origin has been dealt with as great length by Bhāskarācārya. Even the author of Siddhāntatattvaviveka has given much thought to it. It is not necessary to deal with it here in detail. Playfair (1782 A.D.), an European scholar, observes about the origin of sines in our treatises, "The method devised by Hindu astronomers to find the sines implies the following proposition* :—'The ratio of the sum of the sines of the first and the last of three arcs in arithmetical progression to twice the sine of the middle one is equal to the ratio of the cosine of the difference of the arcs to the radius'—This proposition was apparently not known to European mathematicians till the beginning of the 17th century". It is really creditable to our people. Similarly the Greeks only knew what a chord is, but they did not know to make use of the first sines (jyārdha). Even the Arabs didn't know of it till the 9th century A.D. It has been pointed out in the account of Āryabhaṭa I that it was known to our astronomers from Śaka 421. Our people, however, did not have an idea of the tangent and the secant. Their purpose was, however, served by sines alone.

v2.c12.s6.t1.83

OTHER MATTERS

v2.c12.s6.t1.84

The question of finding when planets become direct or retrograde and when they rise and set, these and other like matters of secondary importance are dealt with in the chapter on true motions. It is not necessary to deal with them at length here.

v2.c12.s6.t1.85

DECLINATION

v2.c12.s6.t1.86

Our works assume 24° as the maximum declination of the Sun. The obliquity of the ecliptic had reached that value 2400 years before the aka era. It is 23° 27' 10" in the beginning of śaka 1818. This means that the maximum declination at the present time, as calculated from our works, is wrong by 32' 50". The obliquity about Śaka 400 was about 23° 39'. Ptolemy's work (SYNTAXIS, Part I) mentions it as a value lying between 23° 50' and 23° 52' 30", and it seems to be Prof. Whitney's opinion** that he borrowed this value from the works of Hipparchus. As this value of obliquity does not agree with the value given in our works, it is obvious that our people did not borrow their figure from the works of Hipparchus and Ptolemy. They must, of course, have found it independently and some time before the Śaka era. The work entitled, Yantrarāj, assumes 23° 35' to be the obliquity. (In fact, this was correct about Śaka 900). But no later writers accepted it nor did they attempt to find it for themselves.

v2.c12.s6.t1.87

∗ Asiatic Researches, Vol. IV.

v2.c12.s6.t1.88

∗∗ Transaltion of the Sūryasiddhānta by Burgess, p. 57.

Chapter II: Pañcānga

v2.c12.s6.t2.2

(Almanac)

v2.c12.s6.t2.3

The calculation of the five elements or parts (aṅgas) of the Almanac (Pañcāṅga) is usually given in the Spaṣṭādhikāra (the chapter on true Places); and hence, the question of the almanac is being taken up in this very chapter. Such matters as the Śaka era, the beginning of the year, the samvatsara (year), the “pūrṇimānta” and “amānta” systems etc. are but integral parts of the almanac; after considering these, therefore, the consideration of the five elements of the almanac, the defferent kinds of almanac etc., will be taken up.

v2.c12.s6.t2.4

Astronomical calculations require some moment of time to start with in order to predict future planetary positions. In accordance with this convention, the siddhānta works assume the commencing moment of Mahāyuga or that of some other Yuga and especially that of the Kaliyuga as the starting moment for calculation; and the Karaṇa works assume some particular year of the Śaka era as the commencing year. A Karaṇa works two however, can be found to have adopted Vikram Samvat along with the Śaka era. Thus the Karaṇa work Rāmavinod has adopted Akbar year along with the Śaka year, and the Phatteśāha Prakāśa has adopted the Phatteśāha year coupled with the Śaka year. The Vārṣiktantra (page 167), which is really a Karaṇa work has adopted the commencement of the Kaliyuga for the epoch for calculation, and the author has accordingly classed the work as ‘tantra’. Even than it has brought in some association with the Śaka era.

v2.c12.s6.t2.5

A study of different eras

v2.c12.s6.t2.6

Our almanacs, in their opening pages which are usually devoted to the study of the “samvatsaraphala” (forecast of the year), refer to six founders of eras for the Kaliyuga, like Yudhiṣṭhira, Vikrama, Sālivāhana and others. Of these, Yudhiṣṭhira and other to lived in the past and the remaining three are yet to be born. The word Śaka, in fact, denotes a certain tribe of people. Bhaṭotpala and others have stated that the era was introduced under the name of Śaka, since the time when Vikrama defeated the Śaka kings. But this does not appear to be reasonable. The Śaka kings themselves may have started the reckoning of the era in their own name. The word Śaka originally denoted a particular tribe, but in compound words like Yudhiṣṭhira Śaka, Vikrama Śaka etc., it signifies time, generally known as ‘Era’ in English and as ‘San’ in Arabic. The word ‘Kāla’ (time) is found to have been used in the sense of era in ancient copper plates. For instance, Śakanṛpa Kāla, Vikrama Kāla, Gupta Kāla (meaning—the era started in the name of Gupta Kings) are some examples of the usage of the word Hence, the word Kāla has been used in the sense of era in the discussion which follows.

v2.c12.s6.t2.7

Expired year and current year

v2.c12.s6.t2.8

A number of eras like Vikrama era and Śaka era had been and still continue to be, in use in our country. A brief description of them may be given here.

v2.c12.s6.t2.9.1

Before doing so, however, let some idea be given about two types of the year : the expired and the current. In the account of Brahmagupta (page 90) the date of compilation of the Uttara Purana has been stated to be Śaka 820. But the positions of planets, purporting to be true for the year 820, are found to agree with those for what would be Śaka 819, according to the present mode of reckoning. Hence, one begins to doubt whether the real date of compilation of the Purāṇa was Śaka 819 or 820. The Śaka year which is mentioned as "Śaka 1818" by the almanac-makers of this province and of most of the Provinces in our country is found to have been counted as Śaka 1819 in the Tamil and Telegu almanacs and in some of the Kannad ones printed in Mysore. The reason for this difference appears to be this : The positions of planets given in Siddhānta works are true for the initial moment of the first year of the Kaliyuga. Supposing the positions of planets in the beginning of the 11th year of the Kaliyuga are required, they must be found by adding to the original positions the motion of planets for 10 years, since 10 years would have been elapsed from the beginning of the Kaliyuga to the desired moment. In such calculations the number 10 has to be taken instead of 11. The two dates of compilation of the above Purāṇa, viz. 819 and 820 may possibly be accounted for in some such way. In other words 820 is the current year and 819 the expired year in this case.

v2.c12.s6.t2.9.2

Similar examples are found in copper plate and other inscriptions. It has been pointed out above that the Śaka year which is numbered 1818 in these parts is counted as 1819 in some almanacs of the Madras Presidency : But there is doubt if the people of that region are really aware of the difference between the expired and current years and if it is taken into account at all for practical purposes. Now-a-days the Tamil almanacs compiled by Anna Ayyangar of Tanjore District, are in general use in Tamil country of the Madras Presidency. The author has procured a number of such almanace for the last several years. Among these, the almanac for the last Sarvajit samvatsara mentions the corresponding Śaka year to be 1809, while the almanac for the very next year, known as Sarvadhāri, compiled by the same author gives 1811 as the Śaka year. These year were counted as 1809 and 1810 in other provinces. It seems that the almanac-makers themselves do not clearly understand the distinction between the expired year and current year ; how can we then expect others to understand it ? On enquiries made of the well-known Natesh Sastri of Madras and the eminent scholars Sundareshwar Shrotri and Vyankateshwar Dikshit, from Tiruvadi in Tanjhore District, the author has learnt that the year described as "current" above is not at all in vogue in those parts at present.

v2.c12.s6.t2.9.3

Hence, there is reason to believe that the distinction between the expired and current years must be merely imaginary and that it arose when two different numbers might have been applied by the same year by mistake. If any such distinction exists at all it can possibly be true only with respect to the Kali era and Śaka era which are in use in works on astronomy ; and of these two, the distinction can be applied with greater clarity to Kali era. The Vikrama and other eras are not in use in astronomical works and no such distinction can be found in their case. However, examples are found in which two different numbers are applied to the same year of the Vikrama era ; but it must be the result of an error. In short, on full consideration, the author is led to believe that, as a matter of fact, there is no such distinction as expired and current year. All years are only current years. For example, the present year (when the author is writing this chapter) named Durmukh (Śaka 1818) is actually 'current'. Later on, in the study of different eras, figures denoting corresponding years have been given —and compared, and in so doing the system used for reckoning the year number is the same as the one prevalent in most parts of India. In some places the terms ‘current’ and ‘expired’ have been used; but they have been used for differentiation in the case of only those years, to which two different numbers are likely to be applied. Let us now consider the different eras.

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