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Bharatiya Jyotihshastra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2.t16: First Āryabhaṭa

Bhāratīya Jyotiḥśāstra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s2.t16

Language
English

Contents

First Āryabhaṭa

v2.c12.s2.t16.1

Name

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He has compiled the well known work *Āryabhaṭīya*. There is no other more ancient work from amongst the available ‘pauruṣa’ (human) astronomical works. He calls his work as *Āryabhaṭīya*, but many other astronomers have named it as *Āryasiddhānta* and it is quite proper to name it thus. A second Āryabhaṭa flourished after him and he has also written an *Āryasiddhānta*. It is, therefore, convenient to mention this Āryabhaṭa as Āryabhaṭa I, and his work as the First *Āryasiddhānta*, and the author has followed this principle all throughout.*

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This Siddhānta is divided into two main parts. The first part consists of 10 verses in the ‘gīti’ metre ; and almost all the topics mentioned in the Madhyamādhikāra of other Siddhāntas ; for instance, the elements of revolutions etc. in a particular period of time, have been mentioned in these 10 ‘gītis’. This part is called the ‘daśagītikā’.

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The second part has three chapters, which contain the remaining questions mentioned in the other Siddhāntas. It contains 108 āryās and hence, it is called the ‘āryāṣṭaśata’. Some people regard the two parts as separate works. Sūryayajvana, one of the commentators of this work, calls the two parts as two ‘prabandhas’ (compositions). Each of the two parts begins with a verse which is an auspicious prayer and hence, they might have been regarded by some as two separate works. But they are interdependent and one will be of no use without the other. It is, therefore, proper to regard the two together as one work. Āryabhaṭa himself appears to hold the same view. He has not given any separate name for the first part, nor has he closed it by a conclusion. He has no doubt added a ‘conclusion’ at the end of the whole work together and has mentioned *Āryabhaṭīya* as the name only at that place. Similarly, the whole work together has four chapters; the author himself does not call them as ‘pādas’, still it has been customary with others to call them so. If the ‘Daśagītikā’ be supposed to be a separate work, it will be said to have one ‘pāda’ and the other work three and these cannot justly be called ‘pādas’ (fourth parts). Considering, therefore, from all points of view, it will be proper to regard the ‘Daśagītikā’ and ‘Āryāṣṭaśata’ together as one Siddhānta. The Daśagītikā contains two more ‘gītis’ in addition to the Ten ; one of them is the auspicious prayer and the second describes its numerical code. The whole work, therefore, contains 120 verses. The word ‘Āryāṣṭaśata’ is misleading, and it seems that because of this, some European scholars thought that it contained 800 verses. This *Āryasiddhānta*, along with the *Bhaṭadīpikā* commentary by Paramādīśvara has been recently printed by Dr. Kern at Leden in Holland in 1874. It was not much known to European scholars before this.

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Three Schools

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The astronomical works in our country are at present believed to belong to one of the three main pakṣas or schools ; the Saurapakṣa, the Āryapakṣa and the Brahmapakṣa. The basic work for the first school is the Sūrya Siddhānta, that for the second is the Ārya Siddhānta, and that for the third is the Brahma Siddhānta. The reason for the formation of the three different schools is that the length of the year according to each school is somewhat different and the motions of planets in a particular period of time, say, a Kalpa or a Mahāyuga, are different. The remaining subjects in the case of all schools of all works following each of these schools are practically the same. It will be shown at the proper place when the party spirit relating to different schools raised its head.

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∗ If any reference without the epithet ‘first or second’ occurs anywhere later on, it should be taken to refer to this, (tha 1is) the first Āryabhaṭa.

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The words Āryasiddhānta and Āryapakṣa are well known in our country but the Āryasiddhānta itself is not much known to anyone. It is considered that no orthodox astronomer in Mahārāṣṭra possesses its copy. The Āryapakṣa (school) is still flourishing and it has a number of followers; but very few of them understand its correct form from the original Āryasiddhānta.

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Numerical Code

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Other astronomical works are found to be using ‘bhū’ for 1, ‘rāma’ for 3, and such other words to represent digits and numbers. But the First Āryabhaṭa, instead of following this system has adopted letters to denote numbers, as shown below :—

v2.c12.s2.t16.11

| a—1 | | ḷ—100000000 | |
| i—100 | | e—10000000000 | |
| u—10000 | | o—1000000000000 | |
| ṛ—1000000 | | au—100000000000000 | |

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| Ka— 1 | Ca— 6 | ṭa—11 | ta—16 | pa—21 | ya—30 | ṣa —80 | |
| kha— 2 | cha— 7 | ṭha—12 | tha—17 | pha—22 | ra—40 | sa —90 | |
| ga— 3 | ja— 8 | ḍa—13 | da—18 | ba—23 | la—50 | ha —100 | |
| gha— 4 | jha— 9 | ḍha—14 | dha—19 | bha—24 | va—60 | | |
| ṅa— 5 | ña—10 | ṇa—15 | na—25 | ma—25 | śa—70 | | |

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Varāhamihira has followed in the Pañcasiddhāntikā the system adopted by other Siddhāntas while using words to denote numbers; and this shows that the system was in vogue before Āryabhaṭa; and it must have been so. Hence, Āryabhaṭa must have adopted it to represent numbers briefly and as it is not found in other works, it appears to have originated with him. This system ensures brevity. Other Siddhāntas generally require about 9 or 10 verses to mention the number of revolutions of all the planets. But it has been done only in two couplets by this system. Similarly, other Siddhāntas generally require 50 to 70 verses to be devoted to ‘Madhyamādhikāra’. This system could describe almost all the subjects in 10 verses (gītis) only; and hence, it is very easy to commit to memory the Daśagītika aphorisms written according to this system. But while this system has some advantage, it has at the same time a very serious drawback. To explain briefly the nature of the system and its inconvenience, an illustration is given below :—

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The first half of the first couplet describing revolutions of planets, runs thus :— युगरविभगणाः ख्युबृ शशि चयगियिदुशृछलृ कु डिशिबुन्लृरष्प्राक्

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The number of revolutions* of ‘Ku’ (the earth), according to this verse comes to be 1582237500 in a Mahāyuga.

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∗ Āryabhaṭa holds the view that the earth has the diurnal motion and that is why he has given the number of rotations of the earth. Other Siddhāntas give revolutions of stars instead.

v2.c12.s2.t16.17

Dr. Kern's book gives 'ṣu' in place of 'bu' in this line. 'ṣu' denotes 800000, that is, a number greater by 570000. The misprint of 'ṣu' for 'bu' has caused so much error.

v2.c12.s2.t16.18

dhi = 500 śi = 7000 bu = 230000 nḷ = 1500000000 khṣr = 82000000

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This is an error* occurring in a printed book which is very carefully checked and printed ; what then are the chances of errors having crept in in a manuscript and of these errors getting aggravated by traditional use, can be understood only by those who have occasions to go through manuscripts. Such a work is bound to go out of use in course of time, if there be no means like traditional interpretations, and checks of agreement with other works.

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Motions and Revolutions of Planets

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After first quoting the two couplets which mention the numbers of revolutions and other elements of planets, the author gives the numbers derived from them. The first part of the first couplet has already been given above. The remaining couplets are given below :—

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शनिदुङ् ध्व गुरुखिच्युभ कुजभद्लिझनुक् भृगुबुधसतेराः ।। १ ।।
चंद्रोच्चज् ष्खिघ बुधसुगुशियथ भृगुजबिखुछ शेषाः ।।
बुफिनच पातविलोमा बुधान्त्रजाकोदयाच्च लंकायां ।। २ ।।

v2.c12.s2.t16.23

The revolutions in a mahāyuga as derived from these verses are :—

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| Rotations of the earth | 1582237500 | Revolutions of Jupiter | 364224 |
| Revolutions of the Sun | 4320000 | Revolutions of Venus | 7022388 |
| Number of sāvana days | 1577917500 | Revolutions of Saturn | 146564 |
| Revolutions of the Moon | 57753336 | Solar months | 51840000 |
| Revolutions of the Moon's Apogee | 488219 | Intercalary months | 1593336 |
| Revolutions of the Moon's node | 232226 | Lunar months | 53433336 |
| Revolutions of Mars | 2296824 | Tithis | 1603000080 |
| Revolutions of Mercury | 17937020 | Suppressed tithis | 25082580 |

v2.c12.s2.t16.25

Length of the year=365d 15gh 31p 15vp.

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The numbers of revolutions given by the original Sūrya Siddhānta on page 23..do not include those for Rāhu, but comparison of the remaining numbers with the above figures from the Āryasiddhānta shows that the revolutions of Jupiter and Mercury are different in the two, while the remaining numbers are the same ; and it has already been shown before, that the original Sūrya Siddhānta existed before Āryabhaṭa. This shows that Āryabhaṭa borrowed the elements of all planets except those for Mercury and Jupiter from the original Sūrya Siddhānta. The elements for the revolutions of Jupiter and Mercury appear to have been adopted by him by experience after testing their accuracy with their observed positions.

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∗ This error cannot be detected from the commentary, but it can be easily found after considering the theory and agreement with other works. However, this error of Dr. Kern will throw several scholars into confusion.

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It has been remarked above that Āryabhaṭa’s system is somewhat different from that of the other Siddhāntas. It is as follows. He says in his ‘daśagītika’ :—

v2.c12.s2.t16.29

काहो मनवो ढ १४ मनुयुगश्च ७२ गतास्तेच ६ मनुयुगछना २७ च ।।
कल्पादेर्युगपादा ग ३ च गुरुदिवसाच्च भारतात् पूर्वं ।। ३ ।।

v2.c12.s2.t16.30

“The number of Manus in a Brahmā’s day is ‘ḍha’ (=14). A manu consists of śkha (=72) yugas. The number of manus elapsed from the beginning of Kalpa is ‘ca’ (=6) and that of yugas elapsed is ‘chnā’ (=27) ; and the number of yugapādas elapsed, prior to *Thursday*, the beginning day of Bhārata is ‘ga’ (=3)”.

v2.c12.s2.t16.31

In this verse a Manu is said to consist of 72 yugas and not 71 as in others A ‘sandhi’(transition or twilight period) has not been mentioned as to occurring before the commencement of each ‘manvantara’. This verse also tells us what period has elapsed from the commmencement of Kalpa up to *Thursday*, the beginning day of Bhārata*. This verse and the 2nd couplet quoted on page53, show that according to Āryabhaṭa the Kaliyuga started on Friday, and the previous day was a Thursday. But the second couplet quoted above shows that the Mahāyuga** began on Wednesday at sunrise. This shows that Āryabhaṭa does not accept the definitions of time units, e.g. a Dvāpara equal to twice Kali, and so on. If these definitions be accepted, the Kaliyuga would not be found to have commenced on Friday, after supposing the yuga to begin on Wednesday. We get the result if all ‘pādas’ or quarters of a yuga be sup posed to be equal in length .This shows that he regarded Kṛta and other ‘uygapādas’ as equal in length, and from this assumption it appears that the number of years elapsed, from the commencement of Kalpa to that of the present Kaliyuga, comes to be1986120000 and the Kalpa appears to have commenced on Thursday. The number of years elapsed from the commencement of Kalpa to that of the present Kaliyuga comes to 1972944000† according to other Siddhāntas; and according to some, who suppose that some years were spent at the beginning of the Kalpa or over Creation, it was a Sunday when the planets started to move. Brahmagupta has criticised†† Āryabhaṭa for having entertained this kind of view so different from others.

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न समा युगमनुकल्पाः कल्पादिगतं कृतादियातं च ।। स्मृत्युक्तौतार्थभटो नातो जानाति
मध्यगतिं ।। १० ।।
ब्रह्मगु सि. अ. ११.

v2.c12.s2.t16.33

“The measures of Yuga, Manu, and Kalpa, and the number of Kṛta and other yugas elapsed after the commencement of the Kalpa, are not equal to those mentioned by the Smṛti. This shows that Āryabhaṭa does not know mean motions of planets.”

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*The word ‘Bhārata’ stands for ‘Bhārata war’. The word is here used in the se of commencement of Kaliyuga. **Although the word mahāyuga has not been explicitly mentioned, it was evide ly so, as may be seen from context and theory. †After including the years supposed to have been spent over creation. ††Most of the above facts have been mentioned by Brahmagupta; but instead of placing implicit faith in them, I have actually found them out for myself.

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In this verse, Āryabhaṭa has been accused by Brahmagupta of not having given the Yugas, Manu and Kalpa according to the Smṛti. The length of a mahāyuga adopted by him is equal to that given by others. The number of revolutions mentioned above will show that they are divisible by four, and the second couplet says that all the planets were together in the beginning of the mahāyuga. Similarly, according to Āryabhata the ‘yugapādas’ (i.e. parts of a mahāyuga) are all equal ; and in his opinion the number of mahāyugas elapsed between the commencement of Kalpa up to the present Kṛta and other yugas is a whole number. Hence, according to his view, all planets come together at the commencement of each Kalpa, each mahāyuga and also at each Pāda or fourth part of a yuga. He has not at all given the number of revolutions of aphelia and nodes of other planets in a Kalpa. He had no reason to consider, if any years were spent over Creation ; but in his opinion, all planets come together at the beginning of Kalpa. This shows that he did not at all make the assumption that some years were spent in Creation. If he were required to give the number of nodes and aphelia of planets he would have given them on the assumption that the beginning of a Kalpa coincided with the first start of the planets for their movement.

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Date

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Āryabhaṭa has recorded his date in the following couplet :

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षष्ट्यब्दानां षष्टिर्यदा व्यतीतास्त्रयश्च युगपादाः ।।
व्यधिका विंशतिरब्दास्तदेह ममजन्मनोतीताः ।।

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Āryabhaṭa says that he was 23 when sixty 60-year cycles (i.e. 3600 years) had elapsed after the three ‘yugapādas,’ that is, in the 3600th ‘Kali-elapsed’ year, which is the same as Śaka 421. This shows that his birth year was Śaka 398.

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Length of a Year

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The length of the year according to the Sūrya Siddhānta of the Pañcasi-dhāntikā is 365ᵈ—15ᵍʰ—31ᵖ—30ᵛⁱᵖ, and that calculated from the elements given by Āryabhaṭa above comes to be 365ᵈ—15ᵍʰ—31ᵖ—15ᵛⁱᵖ, that is 15 vipalas less. But according to the original Sūrya Siddhānta of the Pañca-siddhāntikā, the Kali yuga commenced at midnight on Thursday; and Āryabhaṭa has assumed it to begin it at sunrise on Friday, that is 15 ghaṭis later ; but because the length of this year is less by 15 vipalas, the cumulative difference in 3600 years would be 15 ghaṭis less ; and hence, the moment of the mean Sun’s entry into Meṣa i.e. the moment of the beginning of the elapsed year 3600 after Kali i.e. in Śaka 421, according to the original Sūrya Siddhānta and Āryasiddhānta, comes to be the same; and this shows that he assumed the length of the year to be less by 15 vipalas in order to avoid the discrepancy which would occur, if the yuga be assumed to commence from the sunrise.

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If anyone entertains any doubt about his date, the length of the year as given above will leave no room for such doubt. His date of birth is definitely Śaka 398.

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Āryabhaṭa observes in the first couplet of his ‘Gaṇitapāda’

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आर्यभटस्त्वहनिगदति कुसुमपुरेऽभ्यर्चितं ज्ञानं ।।

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“Āryabhaṭa, however, imparts the sacred knowledge in this (town of) Kusumapura.”

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From this, his place of residence seems to be Kusumapura, which is believed to be Patna in Bengal.

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Subject Matter

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The ‘daśagītikāpāda’ of Āryabhaṭa’s Siddhānta contains the numbers of revolutions and other elements of planets. The next three pādas are devoted to ‘gaṇita’ (Mathematics), Kālakriyā (time units) and ‘Gola’ (celestial sphere). The gaṇitapāda includes some subjects from arithmetic, algebra, geometry and trigonometry from among the branches of pure mathematics; and the remaining two pādas are devoted to astronomical questions alone. As a matter of fact, according to modern conception, astronomy is a branch of applied mathematics and hence, it need not deal with arithmetic and other branches of pure mathematics; but astronomy often requires the help of pure mathematics; and hence, it is natural for such ancient works to include both types; but such a combination is found in very few works. We have no means to know if it existed in the original Sūrya Siddhānta; but it is not found in the Pañcasiddhāntikā and also in modern Sūrya and Soma Siddhāntas. This Āryasiddhānta, the Brahmaguptasiddhānta and the second Āryasiddhānta, however, do contain pure mathematics also. Bhāskarācārya’s definition of a siddhānta has been given above (page xxviii of part 1), according to which he has included in the siddhāntas both the branches of mathematics, viz. ‘vyakta’ or known (arithmetic) and ‘avyakta’ or unknown (algebra), and accordingly he calls his two works Līlāvatī and Bījagaṇita, as parts of his work, Siddhānta Śiromaṇi; still he has computed them as independent works; and some references in it show that independent works on algebra were already compiled before Bhāskarācārya’s time. The two Āryabhaṭas and Brahmagupta have included algebra and other branches of mathematics in the siddhānta itself; but these subjects have been treated in separate chapters.

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The contents of the ‘gaṇitapāda’ will be briefly described in Āryabhaṭa’s work. This pāda consists of 32 couplets in addition to the benedictory verse. It contains the following subjects:—

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Place names of digits of numbers, squares and cubes; square root and cube root; triangles, circles, and other figures and their areas; volumes of cubes and spheres; calculation of sines and their brief treatment; progressions, rule of three, fractions; an interesting type or two of problems solvable by rule of three or by algebra, and a section of mathematics known as ‘kuṭṭaka’ (problems of multipliers). These are the topics dealt with in the ‘gaṇitapāda’. Ptolemy and the Greek astronomers before his time had no knowledge of the sines. They used to make use of chords. The Europeans, before they studied Indian astronomy, held the view that it was Al Battani, the Arab astronomer* (who ved in the latter half of the 9th century A.D.) first made use of sines in place of chords. But this work of Āryabhaṭa shows that we knew the use of sines in Śaka 421. Even the modern Sūrya-Siddhānta gives sines. One more thing worth being specially mentioned is that Āryabhaṭa has very accurately given the ratio of the circumference of the circle to its diameter by the following verse :—

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∗ See page 56, translation of Sūryasiddhānta by Burgess.

v2.c12.s2.t16.52

चतुरधिकं शतअष्टगुणं द्वाषष्टिस्तथा सहस्राणां ।। अयुतद्वयविष्कंभस्यासन्नो
वृत्तपरिणाहः ।। १० ।।
गणितपाद.

v2.c12.s2.t16.53

The approximate length of the circumference of a circle whose diameter is 2000 is 62000 increased by "104 multiplied by 8". This gives 62832 as the length of the circumference corresponding to 2000 as diameter, which gives 1:3.1416 as the ratio. Even this has been given by him as only approximate.

v2.c12.s2.t16.54

DIURNAL MOTION OF THE EARTH

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Āryabhaṭa is the only astronomer in our country who holds that the earth rotates round itself; in other words, he holds that the earth has a diurnal motion. He remarks :—

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अनुलोमगतिर्नौस्थः पश्यत्यचलं विलोमगं यद्वत् ।। अचलां भानि तद्वत् समपश्चिमगानि
लंकायां ।।
गोलपाद.

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"Just as one, sitting in a boat, observes stationary objects moving backwards; similarly, to an observer in Laṅkā (i.e. equator) the stationary stars appear to be moving towards the West".

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The commentator of Bhaṭaprakāśikā has attempted to make out in the following lines that even Āryabhaṭa holds the view that the earth is stationary.

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"भानि कर्तृ भूतानि चलानि भूमिगतानि वस्तूनि कर्मभूतानि विलोमानीव
प्राचीं दिशं गच्छंतीव पश्यति"*

v2.c12.s2.t16.60

But Āryabhaṭa instead of giving the revolutions of stars in the list of elements has given the rotations of the earth. He has also remarked at another place (4th couplet of daśagitika) that the earth revolves through one minute of arc in a unit of time known as 'prāṇa' (i.e. 1/6 of a pala). Similarly, Brahmagupta and others have criticised Āryabhaṭa for holding the view that the earth rotates. Brahmagupta says :—

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प्राणैर्नति कलां भूर्यदि तर्हि कृतो ब्रजेत् कमध्वानं ।। आवर्तनमुव्यां चेन्न पतंति
समुच्छ्रयाः कस्मात् ।।
ब्र. सि. अ. ११.

v2.c12.s2.t16.62

If it be assumed that the earth does rotate one minute arc in one 'Prāṇa' unit of time, where, then, does it go and by what track, and how is it that objects situated at elevated places do not fall off ?

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∗ Meaning :—The stars, (when they take the place of a 'subject') observe the earthly objects (which became an 'object') as moving towards the east.

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The compiler of Bhaṭaprakāśikā commentary has offered his comments by taking ‘bha’ (i.e. bhamandalam) in place of ‘bhūḥ’ in the couplet ‘prāṇenaiti kalāṃ bhūḥ’. The Āryabhaṭa’s couplet just followirg the Āryā “anuloma etc.”, runs thus :—

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उदयस्तमयनिमित्तं नित्यं प्रवहेग वायुना क्षिप्तः ।। लंकासमपश्चिमगोभपचरः सग्रहो
भ्रमति ।। १० ।।
गोलपाद.

v2.c12.s2.t16.66

“The starry cage, being daily tossed by the ‘pravaha’ wind, with the object of causing rises and sets, is seen at the equator to revolve from east to west.”

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Considering all things, however, it appears that it was Āryabhaṭa’s definite view that the earth rotates. He only accepts the earth’s diurnal rotational movement*. It does not appear that he held the view of the earth’s revolution round the sun.

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This Siddhānta by Āryabhaṭa does not contain chapters like other Siddhāntas; but it deals with almost all the subjects in them excepting those in the chapters on “elevation of the moon’s cusps” and “conjunctions of planets with stars”. Brahmagupta has criticised him that the Āryabhaṭīya will not be helpful in obtaining the knowledge of elevation of the moon’s cusps, shadow, etc. This Siddhānta does not give the longitudes and latitudes of Junction-Stars like other Siddhāntas, and this is another drawback. Had these been given, it would have been a great help to the history of astronomy, since the date of Āryabhaṭa is definitely known; not that this subject was unknown in his time or before, for the Pañcasiddhāntikā gives some information about the latitudes and longitudes of Junction-Stars of nakṣatras. This Siddhānta mentions nothing about the precession of equinoxes which is again a very important subject.

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KARAṆA WORK OF ĀRYABHAṬA

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This Āryasiddhānta is very brief; still, the subjects dealt with in it are so treated as to be clearly understood. Brevity has not caused any lack of clarity. Still its general form shows that it has not been compiled with the object of being useful to astronomers for every day use, but for mentioning the important subjects which are the Siddhāntas or established truths. It is true that a Siddhānta work cannot be useful for every day purposes; a Karaṇa work is necessary for the purpose. The work, however, is not extensive like other Siddhāntas and does not deal with all the subjects. It is true that the calculator requires more time if he takes the help of the modern Sūrya Siddhānta, Brahma Siddhānta or Siddhānta Śiromaṇi instead of any Karaṇa work, but he will not be handicapped if he has only one Siddhānta available and not others. Such is not the case with this Siddhānta. For instance, it does not give the methods of calculating tithis, nakṣatras and karaṇa nor does it give the calculation of ‘mahāpāta’. It is not that Āryabhaṭa did not know what ‘mahāpāta’ is; the Āryasiddhānta does refer to it. Similarly, the terms tithi, nakṣatra and karaṇa must have been known in his time. There are also other subjects which are found in other siddhāntas. This leads me to think that Āryabhaṭa might have compiled some Karaṇa work. Āryabhaṭa’s conception of the day commencing with sunrise is given by the second couplet from Daśagītikā given above (page 53); but Varāhamihira remarks that according to Āryabhaṭa, the day is said to commence even from the midnight at Laṅkā (page 25). This statement of Āryabhaṭa is not found in the Āryabhaṭīya. and even Brahmagupta is not seen to criticise him for this. It proves that even at the time of Brahmagupta, the Āryabhaṭīya did not contain some such couplets. Brahmagupta has referred to the two parts of Āryasiddhānta by these very words, viz. ‘daśagītikā and āryaṣṭaśata’. From this it appears that no one has added anything to or taken away from the Āryasiddhānta which existed before Brahmagupta. The Varāhamihira’s statement, therefore, suggests that Āryabhaṭa must have compiled some other work, and Brahmagupta’s Khaṇḍakhādyaka and Varuna’s commentary on it, lead one to conjecture that there must have existed some Karaṇa work written by Āryabhaṭa. It is, however, not available at present.

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∗ It is stated on page 2 of Grant’s History of Physical Astronomy :—It is said that it was the view of Nicetas of Syracuse that the earth only rotates about its axis. It is also said that it was the opinion of Pythagoras, the Greek philosopher (6th century B.C.) that the sun is at the centre of the Universe and the earth revolves round him. But the auther does not think that these views were formed from the results of observations, and that accurate methods of calculating planetary places were established by them. Perha ps. these might have been simply notions.

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Criticism

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Brahmagupta has levelled a great deal of criticism against Āryabhaṭa. After enumerating different points of criticism, he further remarks :—

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स्वयमेवं नाम यत्कृतमार्यभटेन स्फुटं स्वगणितस्य ।। सिद्धं तदस्फुटत्वं ग्रहणादीनां
विसंवदति ।। ४२ ।।
जानात्येकम्पि यतो नार्थभटो गणितगोललानां ।। न मया प्रोक्तानि ततः पृथक्
पृथक् दूषण्येषां
।। ४३ ।। आर्यभटदूषणानां संख्या वक्तुं न शक्यते ।।
ब्र. गु. सि. अ. ११.

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“Aryabhaṭa himself has claimed the correctness of this calculation; but that calculation has been proved to be incorrect on account of its disagreement with the actual phenomena of eclipses, etc. Since Āryabhaṭa understands nothing of mathematics, celestial sphere or time, I have not mentioned separately his demerits concerning short-comings in respect of other subjects. It is impossible to enumerate all demerits of Āryabhaṭa.”

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The fact that calculations of eclipses, etc., made from Āryabhaṭa’s works showed disagreement with the observed results is worth-considering. Their correctness or otherwise can be judged from some of the points of criticism enumerated above. Although it is true that some of the points are correct. still, Brahmagupta’s statement betrays a great deal of prejudice.

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Loss of Works

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Brahmagupta says :—

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कालांतरेण दोषा येन्यैः प्रोक्ता न ते मयाभिहिताः ।।

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“I have not repeated the demerits which have been stated by others as time elapsed.”

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But, of the available works compiled before Brahmagupta, it is only the Pañcasiddhāntikā which mentions Āryabhaṭa’s name only, and makes no mention of any of his faults. This shows that some works of pre-Brahmagupta period must have been lost. The works of authors belonging to the period before Śaka 421 and mentioned above are not at present available.

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HIS CALIBRE

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The main criterion of testing the capability of the author of an astronomical work, is the agreement of its calculation with observation, and this criterion will reveal a high degree of capability in the case of Āryabhaṭa. It has been noted above that he has found the numbers of revolutions of Jupiter and Mercury to be different from those of earlier works ; it, however, appears that he improved the earlier works as far as the calculation of true places of planets was concerned. According to Brahmagupta, Śrīṣeṇa and Viṣṇucandra borrowed the method of calculating planets’ places, the aphelia, nodes and epicycles from Āryabhaṭa’s works. The very fact, that although the original Sūrya Siddhānta of the Pañca-siddhāntikā, the works of Lāṭa, and others, and the Āryasiddhānta existed before them, they chose to borrow the calculation methods only from Āryabhaṭa’s works, easily shows that he possessed greater capability than others as far as agreement with observation was concerned. The table of ‘paridhyamśa’ (i.e. epicycles in degrees) of all the authors, given later on, in the chapter on true places, will show that Āryabhaṭa’s ‘paridhyamśas’ of the cycles of apsis and conjunctions, which form the main item in calculating true places of planets, were different from those of the Pañcasiddhāntikā. This shows that he improved the methods of calculating true places. Even though Brahmagupta, an adept in fault finding, has remarked that the demerits of Āryabhaṭa cannot be enumerated, he observes in the first couplet of Khaṇḍakhādyaka.

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वक्ष्यामि खंडखाद्यकमाचार्यभटतुल्यफलं ।।

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“I will compile the work Khaṇḍakhādyaka which would give results equal* to those of the learned Āryabhaṭa.”

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Brahmagupta had to set aside the vanity about his own Siddhānta, and to say that he has compiled a work which would be a match for that of his greatest rival. This clearly reveals Āryabhaṭa’s capability and it becomes further enhanced in our eyes because his work attained prominence, even when original Sūrya Siddhānta and other works compiled before his time were in existence and this fact is further confirmed from the following verse :—

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सिद्धांतपंचकाविधवापिट्टविरुद्धमौढयोपरागमुखखेचरचारकल्पतौ ।। सूर्यः स्वयं कुसुमपुर्यभवत् कलौ तु भूगोलवित् कुलप आर्यभटाभिधानः ।।

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It is not known who wrote this and when. Dr. Kern has quoted it in the Introduction. The compiler of this verse states that because the calculation of setting of planets and eclipses based on the existing methods of the five Siddhāntas did not tally with the observed results, the Sun god himself was born in Kusumapura, under the name of Āryabhaṭa to find out correct motions of planets. This states that the five Siddhāntas did not give results agreeing with observations. It, therefore, shows that some one has compiled the verse soon after Āryabhaṭa. This also clearly shows that Āryabhaṭa was regarded as very capable, and the capability was really very great considering the age in which he lived. Āryabhaṭa himself observes:—

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क्षितिरवियोगाहितकुद्रवींदुयोगात् प्रसाधितश्चंद्रः ।। शशिताराग्रहयोगात्तथैव ताराग्रहा सर्वे ।। ४८ ।। सदसज्ज्ञानसमुद्रात् समुद्धृतं देवताप्रसादेन ।। सज्ञानोत्तमरत्नं मया निमग्नं स्वमतिना वा ।। ४९ ।। “I have finally corrected* the ‘Sun’ from the yoga (conjunction) of the sun with the earth, the ‘moon’ from the conjunction of the moon and the sun ; and all the ‘planets’ from the conjunctions of the moon and stars with the planets. I took out the jewel, in the form of true knowledge, through God’s favour or with my own intellectual power, from the ocean of real and false knowledge.”

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∗ This comparison is not complete. To what extent it holds good will be shown later in Brahmagupta’s account.

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Eclipses and conjunctions can lead us to find even the mean motions; but it is the true place which is chiefly found from them. This verse and the one referred to before this, will show that Āryabhaṭa has made an improvement in the calculation of true places. Similarly, his high capability can be seen from the fact that he made researches by means of observation and intelligence after critically studying the old works with common sense.

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ITS INFLUENCE AND FOLLOWERS

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Utpala has extracted a number of couplets from Āryabhaṭīya in his commentary on Bṛhatsaṃhitā; and extracts from it are also found in a number of works compiled later on. LALLA, the famous astronomer, was a follower of Āryabhaṭa. He has suggested a correction to the planetary motions given by Āryabhaṭa. The Karaṇa work, named Karaṇaprakāśa which belonged to the Ārya-Pakṣa and was compiled in Śaka 1014, has been compiled after applying Lalla’s corrections to the planetary places and motions obtained from the elements given by Āryabhaṭa. (This will be explained in detail later on). Similarly, DĀMODARA’S Karaṇa work, named Bhaṭatulya, which was compiled in Śaka 1339, follows the same method. Many people use Karaṇaprakāśa even now for calculation and many are its followers. The Grahalāghava has adopted the positions of Jupiter, Mars and Rāhu from the Karaṇaprakāśa and the Grahalāghava is followed in more than one third part of India.

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PLACE

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It shows that the Āryasiddhānta is even now followed, if not in its original form, at least with the application of corrections to it. Quotations from Āryasiddhānta are not found in astronomical works which were compiled in Mahārāṣṭra and Vārānasī after Śaka 1400. It has already been pointed out above that the Ārya siddhānta is not available on our side in its original form. Dr. Kern has published an Āryabhaṭīya on the basis of three manuscripts obtained by him. All these manuscripts are written in the Malayalam script. This shows that the Āryasiddhānta is still known in South India and specially in the Malabar province. The provinces which speak the Tamil and Malayalam dialects follow the almanac computed on solar basis, and it belongs to the Āryapakṣa, since the year adopted in it is according to the first Āryasiddhānta. The Vaiṣṇavas are adherents of the Āryapakṣa. They form a large part of the population in Karnataka and Mysore. Patna in Bengal (at that time) is believed to be Āryabhaṭa’s place; but there is some doubt about it ; because, the Āryasiddhānta is not at all in use in Bengal. It appears from this that the Kusumapura mentioned by Āryabhaṭa might be some place in the south; nothing can be, however, said about it for certain.

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PLANETARY CORRECTIONS

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It has already been pointed out that the places of planets given by the Āryasiddhānta, sometimes tally exactly with those calculated from European tables; but for a clearer understanding and consideration of them, the mean positions of planets true for the mean Sun’s entry into Aries of Śaka 421 (i.e. 499 A.D.) as calculated from the Āryabhaṭīya and also from the European tables have been given together in a tabular form, on page 62-63

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∗ The first sentence refers to the lunar eclipse and the second to the solar eclipse.

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Mean places of planets for 15 Ghaṭikās after sunrise on sunday, Caitra Kṛṣṇa 9, śaka 421.

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| Planets | Original Sūrya Siddhānta | Variation+ or -from column 12 | First Ārya Siddhānta | Variation+ or -from column 12 | Five Modern Sūrya & other Siddhāntas | Variation+ or -from column 12 |
| | s ° ' " | | s ° ' " | | s ° ' " | |
| Sun | 0 0 0 0 | 0 | 0 0 0 0 | 0 | 11 29 58 37 | -0 1 23 |
| Moon | 9 10 48 0 | -0 4 48 | 9 10 48 0 | -0 4 48 | 9 10 29 33 | -0 23 15 |
| Moon's Apogee | 1 5 42 0 | -0 28 30 | 1 5 42 0 | -0 28 30 | 1 0 53 51 | -4 16 39 |
| Rāhu | 11 22 12 0 | -0 42 18 | 11 22 12 0 | -0 42 18 | 11 18 36 4 | -4 18 14 |
| Mars | 0 7 12 0 | +0 7 0 | 0 7 12 0 | +0 7 0 | 0 9 23 16 | +2 18 16 |
| Mercury | 6 0 0 0 | -3 22 12 | 6 6 0 0 | -2 37 48 | 6 17 54 16 | +14 32 4 |
| Jupiter | 6 6 0 0 | -1 29 54 | 6 7 12 0 | -0 17 54 | 6 5 59 53 | -1 30 1 |
| Venus | 11 26 24 0 | +0 6 24 | 11 26 24 0 | +0 6 24 | 11 22 45 45 | -3 31 51 |
| Saturt | 1 19 12 0 | +0 51 54 | 1 19 12 0 | +0 51 54 | 1 20 23 57 | +2 3 51 |

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| Planets | Variation in Solar distance as compared with Col. 12 | Brahmagupta Siddhānta | Variation+ or-from Col. 12 | Variation in Solar distance as compared with Col. 12 | Col. 11 | Col. 12 | Sāyaṇa places calculated from Keropant's planetary tables | Nirayaṇa places obtained from Col. 11 by applying ayan-āṃśa correction of +16' 54" |
| | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " |
| Sun | 0 | 0 0 51 45 | 0 | 0 | 11 29 43 6 | 0 0 0 0 | 0 0 0 0 | 0 0 0 0 |
| Moon | -0 21 52 | 9 11 31 46 | +0 51 45 | -0 12 47 | 9 10 35 54 | 9 10 52 48 | 9 10 52 48 | 9 10 52 48 |
| Moon's Apogee | -4 15 16 | 1 7 21 3 | +0 38 58 | +0 18 48 | 1 5 53 36 | 1 6 10 30 | 1 6 10 30 | 1 6 10 30 |
| Rāhu | -4 16 51 | 11 23 23 14 | +1 10 33 | -0 22 49 | 11 22 37 24 | 11 22 54 18 | 11 22 54 18 | 11 22 54 18 |
| Mars | +2 19 39 | 0 8 4 45 | +0 28 56 | +0 8 0 | 0 6 48 6 | 0 7 5 0 | 0 7 5 0 | 0 7 5 0 |
| Mercury | +14 33 27 | 6 0 41 2 | +0 59 45 | -3 32 51 | 6 3 5 18 | 6 3 22 12 | 6 3 22 12 | 6 3 22 12 |
| Jupiter | -1 28 38 | 6 7 28 9 | -0 1 45 | -0 53 30 | 6 7 13 0 | 6 7 29 54 | 6 7 29 54 | 6 7 29 54 |
| Venus | -3 30 28 | 11 26 57 12 | +0 39 36 | -0 12 9 | 11 26 0 42 | 11 26 17 36 | 11 26 17 36 | 11 26 17 36 |
| Saturn | +2 5 14 | 1 19 0 1 | +0 39 55 | -0 11 50 | 1 18 3 12 | 1 18 20 6 | 1 18 20 6 | 1 18 20 6 |

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For a simultaneous comparison of all the works, the places of planets have been calculated from the original Sūrya Siddhānta, the modern Sūrya Siddhānta, and the Brahma Siddhānta for the same moment and have been noted in the same table.

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*The moment of mean Sun's entry into Aries for the (elapsed year) Śaka 421 works*

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| | gh. | pal. | |
| Original Sūrya Siddhānta | 15 | 0 | } Time elapsed after mean sunrise at Ujjayinī on Sunday, 9th lunar day of the dark half of 'amānta' Caitra, i.e. the 21st of March. |
| First Ārya Siddhānta | 15 | 0 | |
| The modern five Siddhāntas (Sūrya and others) | 16 | 24 | |
| Brahma Siddhānta (Caitra Kṛṣṇa 8, Saturday). | 22 | 30 | |

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The 11th column on page 63 above, gives places of planets as calculated from Keropant's planetary tables; they are, therefore, as accurate as those calculated from European tables. They are Sāyana. Out of these, the secular equation is applied to the moon, the moon's apogee and the moon's node only. After applying the ayanāmśa correction of 'plus 16'-54"' in Śaka 421 to the planets in this column, the nirayana positions so found have been given in the 12th column, and the places given by works, like the original Sūrya Siddhānta and others, have been compared with these figures. The precessional motion in 20 years comes to be 16'-54"; and taking this figure as the ayanāmśa for Śaka 421, therefore, is equivalent to taking zero as the ayanāmśa for Śaka 441. This year is very near to Śaka 444. It is true that the equinox was near the junction-star of Revati about Śaka 496, and it is suggested that the ayanāmśa for that year should be taken as zero. But it has been pointed out thereafter in the study of the precession of equinoxes that the Indians were right in supposing Śaka 445 as the zero-precession year according to their system. The object of assuming 16'-54" as the ayanāmśa, while comparing the figures, is that it would facilitate the comparison with reséct to the sun. It is not that there will be much error resulting from this. The error, at the most, would be four minutes of arc.

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There is no harm if seconds of arc be neglected while making a comparisons since, they can be said to be carrying no value in the comparison. The sun' longitude as given in columns 1 and 3 is zero; and hence, the results of comparison of the planets' places in them with those of the 12th column which are noted in columns 2 and 4 give the differences obtained by comparing the planets' positions independently with those given in column 12. Similarly, the figures in columns 2 and 4 indicate also the differences obtained by comparing the figures, showing positions of planets in column 12 ahead of the sun's position given in it and the figures indicating the advance of planetary positions in columns 1 and 3 over the positions of the sun given in them. The sun's longitude given in cols. 5 and 8, is not zero; and hence, the differences in the planets' places in these columns, as independently compared with those in the 12th column, have been shown in the 6th and 9th columns; and their differences, when compared with respect to the sun, have been given in the 7th and 10th columns.

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The difference in the case of only Jupiter and Mercury out of other planets given by the original Sūrya-Siddhānta, is greater than one degree, while that of others is less than that. Only Mercury, as given by First Āryasiddhānta differs by more than 2 degrees, while other planets do not differ by more than 51 minutes. Almost all the planets, except the moon, obtained from the modern Sūrya-Siddhānta differ by a considerable quantity. As for the differences in the positions of planets (with respect to the sun) in the case of Brahmagupta Siddhānta and shown in column 10, that of Mercury only is really considerable, that of Jupiter is 53 minutes while that of others is less than 22 minutes of arc.

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The whole discussion shows that there is no harm if it be said that the places of planets of all Siddhāntas, except those of modern Sūrya Siddhānta, for Śaka 421, used to prove fairly correct. The position of the moon is no doubt given very accurately by all. All except Brahmagupta, have given the same number of revolutions for the moon; but the moon's place, as calculated from the modern Sūrya Siddhānta has come to be somewhat different from that of others because of different lengths adopted for a year. The discrepancy in the case of Mercury, according to all Siddhāntas, is considerable. The reason for this appears to be that its constant proximity to the sun allows rare chances for its observation.

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The method of judging the accuracy or otherwise of astronomical works by comparing the mean places of planets as found from the European and Indian works has been shown to be not without risk in all cases and all circumstances, as explained in the course of the discussion of Bentley's method of determining the date of old works (page 30). But in the absence of any better way of judging how far our works agreed with observations in respect of the results of their calculations, the writer has followed the same method.

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The numbers of revolutions and other elements mentioned by our different works have been given before and some more will be given later on. But, the periods of one sidereal revolution as found from European works and our works, have been given further (page 66) in order to facilitate the comparison of the two.

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Ptolemy's measures given in it have been taken from the translation of the Sūrya Siddhānta by Burgess, and those for the Sūrya Siddhānta and Brahma Siddhānta (or Siddhānta Śiromaṇi) have also been adopted from the same book. There is, however, probably no error* in them. The figures derived from the Practical Astronomy by Loomis have been adopted as the modern European measures.

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The length of the year, as adopted by our Sūrya Siddhānta, when compared with that of the modern European works, appears to be greater by about 8 palas—34.5 vipalas and that of the Brahma Siddhānta by about 7 palas—25.6 vipalas. Even though the motion of the moon is considerable, there is almost no error in it. The time taken by the moon's node for one revolution is longer by 4 days and that in the case of Saturn is different by 6 days; the discrepancies in the case of others are less than a day.

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PTOLEMY

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Prof. Whitney observes that he has calculated Ptolemy's elements from the daily motions given by him, after taking into account the precessional motion (of 36" per year) as adopted by him. They do not at all resemble the elements given by our Siddhāntas. This proves that our Siddhānta works have not borrowed the places and motions of planets from Ptolemy's works.

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∗ Even if there be one, no calculation of the author given in this book has been made on the basis of these elements.

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Times required for one sidereal revolution

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| Planets | Modern Sūrya Siddhānta | Brahmagupta Siddhānta | Ptolemy's work | Modern European works |
| | Days gh. p. v. | Days gh. p. v. | Days gh. p. v. | Days gh. p. v. |
| Sun | 365 15 31 31.4 | 365 15 30 22.5 | 365 15 24 31.5 | 365 15 22 56.87 |
| Moon | 27 19 18 1.6 | 27 19 18 0.25 | 27 19 18 0.2 | 27 19 17 58.866 |
| Moon's apogee | 3232 5 37 13.6 | 3232 44 2 45 | 3232 24 40 34 | 3232 34 31 14.088 |
| Moon's node | 6794 23 59 23.5 | 6792. 15 14 14.7 | 6799 58 16 38.5 | 6798 16 44 24.000 |
| Mercury | 87 58 10 55.7 | 87 58 11 43.7 | 87 58 11 47.2 | 87 58 9 24.998 |
| Venus | 224 41 54 50.6 | 224 41 52 34.7 | 224 42 9 52 | 224 42 2 47.486 |
| Māra | 686 59 50 5.87 | 686 52 52 33.7 | 686 58 49 50.2 | 686 58 46 2.518 |
| Jupiter | 4332 19 14 20.9 | 4332 14 24 19.2 | 4332 45 22 56.2 | 4332 35 5 17.49 |
| Saturn | 10765 46 23 4.1 | 10765 48 54 51.2 | 10758 44 30 37.2 | 10759 13 10 57.49 |

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| Planets | Modern five (Sūrya & other) Siddhāntas | Brahmagupta Siddhānta | Second Ārya Siddhānta | Parāśara Siddhānta | Sāyaṇa position calculated from Keropant's tables |
| **Aphelia** | s ° ' " | s ° ' " | s ° ' " | s ° ' " | s ° ' " |
| Sun | 2 17 7 48 | 2 17 45 36 | 2 17 45 36 | 2 17 45 36 | 0 15 12 6 |
| Mars | 4 9 57 36 | 4 8 18 14 | 4 3 50 24 | 4 2 43 36 | 2 1 11 0 |
| Mercury | 7 10 19 12 | 7 14 47 2 | 7 0 14 24 | 7 0 40 19 | 5 27 46 18 |
| Jupiter | 5 21 0 0 | 5 22 15 36 | 5 22 48 0 | 5 22 35 2 | 3 23 54 42 |
| Venus | 2 19 39 0 | 2 21 2 10 | 2 20 38 24 | 2 20 42 43 | 8 2 26 30 |
| Saturn | 7 26 36 36 | 8 20 53 31 | 7 23 9 36 | 7 28 14 53 | 5 29 7 36 |
| **Nodes** | | | | | |
| Mars | 1 10 8 24 | 0 21 59 46 | 1 10 19 12 | 1 9 3 36 | 0 10 49 0 |
| Mercury | 0 20 52 48 | 0 21 20 53 | 0 20 9 36 | 0 21 1 26 | 11 17 0 0 |
| Jupiter | 2 19 44 24 | 2 22 2 38 | 2 20 38 24 | 2 21 43 12 | 1 19 48 6 |
| Venus | 2 0 1 48 | 2 0 5 2 | 2 0 28 48 | 2 0 5 2 | 1 2 39 54 |
| Saturn | 3 10 37 12 | 3 13 23 31 | 3 10 48 0 | 3 10 26 24 | 2 0 9 0 |

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*Aphelia and Nodes in Śaka 421 (i.e. Kali elapsed year 3600)*

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| Planets | From Keropant's planetary tables | First Ārya Siddhānta | | Modern Sūrya Siddhānta | | Brahmagupta Siddhānta | |
| | 1 | 2 | Place | Variation from Keropant | Place | Variation from Keropant | Place | Variation from Keropant |
| | | | 3 | 4 | 5 | 6 | 7 | 8 |
| **Apsides** | | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " | ° ' " |
| Sun | . . . | 2 17 7 | 2 18 | + 0 53 | 2 17 15 | + 0 8 | 2 17 54 | + 0 47 |
| Mars | . . . | 4 8 11 | 3 28 | -10 11 | 4 10 1 | + 1 50 | 4 8 23 | + 0 12 |
| Mercury | . . . | 7 24 1 | 7 0 | -24 0 | 7 10 26 | - 13 35 | 7 14 53 | - 9 8 |
| Jupiter | . . . | 5 20 38 | 6 0 | + 9 22 | 5 21 16 | + 0 38 | 5 22 31 | + 1 53 |
| Venus | . . . | 9 21 3 | 3 0 | -201 3 | 2 19 49 | -211 14 | 2 21 14 | -209 59 |
| Saturn | . . . | 8 5 12 | 7 26 | - 9 12 | 7 26 37 | - 8 35 | 8 20 54 | + 15 42 |
| | | | | | | | | |
| **Nodes** | | | | | | | | |
| Mars | . . . | 1 8 9 | 1 10 | + 1 51 | 1 10 5 | + 1 56 | 0 21 55 | - 16 14 |
| Mercury | . . . | 1 0 18 | 0 20 | - 10 18 | 0 20 44 | - 9 34 | 0 21 12 | - 9 6 |
| Jupiter | . . . | 2 25 30 | 2 20 | - 5 30 | 2 19 41 | - 5 49 | 2 22 2 | - 3 28 |
| Venus | . . . | 2 3 40 | 2 0 | - 3 40 | 1 29 46 | - 3 54 | 1 29 49 | - 3 51 |
| Saturn | . . . | 3 10 13 | 3 10 | - 0 13 | 3 10 25 | + 0 12 | 3 13 13 | + 3 0 |

v2.c12.s2.t16.121

APHELIA & NODES

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The positions of aphelia and nodes of all planets according to different authors at the commencement of the Kaliyuga and for the year Śaka 421 (i.e. Kali elapsed year 3600) have been given respectively in tables on pages 67 and 68. Prof. Whitney, after giving the aphelia and nodes according to Ptolemy and the Sūrya Siddhānta together, has suggested that the Hindus must have taken them either from Ptolemy or from other earlier Greek works. But the following comparative table shows that this statement is incorrect, as can be seen from the positions of aphelia and nodes according to Ptolemy and also the figures for the same for Ptolemy's time i.e. the year 148 A. D. (Śaka year 70) as calculated from Keropant's planetary tables which give modern European figures.

v2.c12.s2.t16.123

A comparison of Ptolemy's figures for the Apsides and Nodes (in Śaka 70) with those calculated from Keropant's tables :—

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| Planet | Apsides | | | Nodes | | |
| | (Sāyana) from Keropant's tables | Ptolemy's work | | (Sāyana) from Keropant's tables | Ptolemy's work | |
| | | Position | Diff. with Keropant's place | | Position | Diff. with Keropant's place |
| | 1 | 2 | 3 | 4 | 2 | 3 | 4 |
| | s ° ' | s ° ' | ° ' | s ° ' | s ° ' | ° ' |
| Sun | 2 11 5 | 2 5 30 | — 5 35 | | | |
| Mars | 4 1 39 | 3 25 30 | — 6 9 | 1 5 29 | 0 25 30 | — 9 59 |
| Mercury | 7 18 32 | 6 10 0 | — 38 32 | 0 26 5 | 0 10 0 | —16 5 |
| Jupiter | 5 15 7 | 5 11 0 | — 4 7 | 2 22 1 | 1 21 0 | —31 1 |
| Venus | 9 16 18 | 1 25 0 | —231 18 | 2 0 39 | 1 25 0 | — 5 39 |
| Saturn | 7 28 45 | 7 23 0 | — 5 45 | 3 7 28 | 6 30 0 | +85 32 |

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The positions for apsides and nodes according to our old works (as shown on pages 67 and 68) at the commencement of Kaliyuga and in the Kali elapsed year 3600 will show that the variation during 3600 years is very small; and the reason for this is their very slow motion. None of our Siddhāntas mention the motion of the apsides and nodes as greater than 1 degree in 13000 years. The figures calculated from Keropant's tables and shown in the two tables above show that, if the equinox be taken to be the initial point, in other words, if the sāyana system be followed, the motion appears to be considerable; but the motions appear to be negligibly small if sidereal i.e. the nirayaṇa basis be adopted.

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The table on page 70, gives the annual motions of apsides and nodes very accurately calculated by modern European methods following the sāyana system and also the actual yearly motion* according to the nirayaṇa system.

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*These have been taken from Practical Astronomy by Loomis.

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Yearly motions of Apsides and Nodes.

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| Planets | Sāyana | True Nirayaṇa | That which must be adopted according to our Nirayaṇa system | According to the Sūrya-Siddhānta |
| **1** | **2** | **3** | **4** | **5** |
| *Aphelia* . . . . | Seconds | Seconds | Seconds | Seconds |
| Sun . . . . | + 61.5 | + 11.24 | + 1.5 | + 0.1161 |
| Mars . . . . | + 65.7 | + 15.46 | + 5.7 | + .0612 |
| Mercury . . . . | + 56.1 | + 5.81 | − 3.9 | + .1104 |
| Jupiter . . . . | + 56.9 | + 6.65 | − 3.1 | + .27 |
| Venus . . . . | + 47.0 | − 3.24 | −13.0 | + .1605 |
| Saturn . . . . | + 69.6 | + 19.31 | + 9.6 | + .0117 |
| *Nodes* . . . . | | | | |
| Mars . . . . | + 25.0 | −25.22 | −35.0 | −0.0642 |
| Mercury . . . . | + 40.2 | −10.07 | −19.8 | − .1464 |
| Jupiter . . . . | + 34.3 | −15.90 | −25.7 | −.0522 |
| Venus . . . . | + 29.7 | −20.50 | −30.3 | − .2709 |
| Saturn . . . . | + 30.7 | −19.54 | −29.3 | − .1986 |

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This motion has been calculated after assuming 50.2″ as the equinoctial motion; but our works have assumed 60″ as the equinoctial motion. Hence. the motion shown in column 4 is the actual annual motion according to our works; and if the figures denoting motions according to European works are to be compared at all, they should be compared with this motion. Even when the comparison is so made, the motions according to the Sūrya Siddhānta are found to be very erroneous. The motions according to other Siddhāntas can also be said to be equally erroneous. None of the other works gives for the annual motions of an aphelion or node, a figure greater than one third of a second. It is greater than 1 second according to the European system. It is very easy to criticise our works regarding this discrepancy, simply by seeing the figures on papers. But he who knows how difficult it is to observe an arc of one second in the sky even with the help of very accurate modern instruments will not blame them in that way. It has been observed by the author with naked eyes some conjuctions of planets with stars, and it has been our experience that those two planets etc. which are actually apart from each other by 5 minutes (300 seconds) of arc or more, as observed through a telescope appear to be in close contact with each other as seen with naked eyes, in other words, there appears to be no distance between them. Hence, this fact should be borne in mind while comparing the figures of our works with accurate measures of European works and we must praise our works instead of condemning them regarding the apsides and nodes. We should appreciate the authors of our works, in as much as they recognized the fact that the motions of apsides and nodes are very small. It should only be seen as to how much accurate the longitudes of the apsides and nodes calculated from their works, prove to be. The postions of the apsides and nodes in Śaka 421, have been given above (page 68) and their differences obtained after comparison with the figures calculated from Keropant’s tables, have also been given. It shows that the positions given by our Siddhāntas are very near to actual positions. Keropant-calculation gives sāyana results, but since the ayanāmśa in Śaka 421 was about 20 minutes only, there is no harm in comparing the figures taking them as nirayana. The Sun’s apogee, in fact, shows only a very little error; but the aphelion of Venus, however, shows a considerable error. Its cause is not known; it is a problem worth considering. But looking to the remaining aphelia, it appears that the aphelion of Mercury, according to First Āryabhaṭa, is less by 24 degrees, while others differ within 10 degrees. Those of Sūrya Siddhānta are more accurate than these, that of Mercury being less by 13 degrees and that of Saturn less by 8 degrees. The aphelia of Mars and Jupiter show only a small difference. Those mentioned by Brahamagupta are as accurate as in the Sūrya Siddhānta or even more so. The table given on page 69 compares Ptolemy’s aphelia with those derived by Keropant’s calculation. These positions given by Ptolemy were presumably sāyana like his other positions and the fact is corroborated from the positions of the sun’s apogee and hence, there is no harm, if the figures are compared with Keropant’s sāyana calculations; and the comparison reveals that even his position of the aphelion of Venus is considerably wrong and his aphelia on the whole show greater errors than those in the Sūrya Siddhānta and Brahma Siddhānta.

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