Bharatiya Jyotihshastra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s3: Chapter 2: The Universe

Bhāratīya Jyotiḥśāstra (History of Indian Astronomy), translated by R. V. Vaidya v2.c12.s3

Language: English

Chapter 2: The Universe

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A general description of the Universe has been given in the INTRODUCTION (page XXIX of Part I) Now let the subject be discussed in greater details.

It has already been told (page XXXI of Part I) that the motion of planets in their orbits has been assumed to be the same, and it is about 11858¾ yojanas in one day. And it is also assumed that each planet travels in one Kalpa a length equal to the circumference of the celestial sphere. In other words, the length of the celestial orbit* is equal to the sum of the orbital lengths of the revolutions made by each planet in one Kalpa. Hence, the length of a planet’s orbit is obtained by dividing the length of the orbit of the celestial sphere by the number of revolutions made by each planet in one Kalpa. The Sūrya Siddhānta gives the measures of orbital lengths as follows :—

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*The orbit is really that path in the sky along which a planet appears to revolve round the Earth. The word “celestial orbit,” however, does not convey that meaning. As a matter of fact, the celestial orbit has no independent existence. It has been assumed only to facilitate the calculation of the lengths of planetary orbits.

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| Planet | Orbital length in Yojanas. | Planet | Orbital length in Yojanas. |
| Moon | 324000 | Jupiter | 51375764 |
| Mercury's mean elongation | 1043209 | Saturn | 127668255 |
| Venus's mean elongation | 2664637 | Starry belt | 259890012 |
| Sun | 4331500 | Celestial sphere (sky) | 18712080864000000 |
| Mars | 8146909 | | |

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MOON'S DISTANCE FROM THE EARTH

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All the siddhānta's except the first Ārya-Siddhānta, give the same daily motion, in yojanas, to the planets. Still, as the number of days in the Kalpa vary with each siddhānta, the length of the celestial orbit and those of the planets also differ by a small quantity. It would be of no use to give all these figures, since they contain very little truth. Most of the data are merely imaginary. The moon's orbit, however, has not been determined by imagination. There is a great degree of truth in it. The length of 1' arc of moon's orbit in its plane, has been assumed to be equal to 15 yojanas by all except Āryabhaṭa I. Hence the length of the complete orbit comes to (360 × 60 × 15) = 324000 yojanas; and the mean radius vector of the orbit comes to 51566 yojanas. This is the moon's distance from the Earth. The radius of the Earth, according to the Sūrya Siddhānta, is 800 yojanas; hence, the moon's distance from the Earth is 64.46 times earth's radius. According to modern discoveries it is 50.96 times earth's radius. The distance of the moon from the earth and its orbital length, as determined by the authors of our siddhāntas are so very near the truth that they deserve to be congratulated on having been able to establish such a correct measure.

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The motions of planets in their respective orbits have been assumed to be the same and the lengths of orbits are found by dividing the length of the celestial orbit by the number of revolutions of planets in the Kalpa. This amounts to assuming that the times required by planets to revolve in their orbits are proportional to their orbits, that is, to the distances of planets from the Earth. But this is not true according to the modern astronomical theory. The modern theory established by Kepler and confirmed by NEWTON and others is that the square of the revolution period of the planet is proportional to the cube of its distance from the Sun.

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DISTANCES OF PLANETS FROM THE EARTH

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The distance of the Sun from the Earth comes to 689430 yojanas according to the Sūrya Siddhānta, i.e. about 862 times the Earth's radius. But according to modern discoveries, it is about 23300 times the radius of the Earth. The distances of planets farther than the Sun, as given in our siddhāntas, are still more erroneous. Our astronomers have not determined the lengths of planetary orbits and the periods of their revolutions after first determining the length of the *Celestial orbit* and the mean daily motions of planets in their respective orbits in terms of Yojanas, by actually taking observations. It is quite clear that they first determined the periods of revolutions and the length of the moon's orbit by observation, and then, on the basis of these figures they fixed the lengths of the planetary orbits and that of the circumference of the celestial sphere. The Pañca Siddhāntikā does not mention the lengths of the orbits of planets and that of the circumference of the celestial sphere in terms of yojanas nor does it mention the length of the moon's orbit; it appears, therefore, that the original Sūrya Siddhānta never contained this information. The modern Sūrya Siddhānta does give these measures; and the author has arrived at the conclusion that the modern Sūrya Siddhānta existed before the Pañcasiddhāntikā. It is not, however, improbable that the lengths of planetary orbits may have been interpolated afterwards*. Secondly all the siddhāntas except the first Ārya Siddhānta have given 15 yojanas as the length of the moon's orbit subtended by one-minute angle. Thirdly the positions of planetary orbits are fixed, and the planets always move through them. The places of the orbits do not change. Hence, the statement that all planets make a revolution of the sky only once in a Kalpa has no meaning. Bhāskarācārya plainly remarks,

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ब्रह्मांडमेतन्मितमस्तु नो वा कल्पे ग्रहः कामति योजनानि ।। यावंति पूर्वेरिह तत्प्रमाणं प्रोक्तं खकक्षास्यमिदं मतं नः।।१३।।

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सि. शि. कक्षाध्याय

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"The Universe may or may not be limited by the *Celestial orbit*. It is the writer's opinion that earlier astronomers have defined "the circumference of the celestial orbit" as equal to the total length in yojanas moved through by a planet in a Kalpa."

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Our astronomers fixed the lengths of the planetary orbits from that of the moon and the time taken by each planet for making one revolution. The principle, assumed as basis, that the time of a revolution and the length of the orbit are in proportion, was not true, and hence, the measures of orbital lengths have proved to be wrong while the length of the circumference of the *celestial orbit* is found to be utterly fictitious.

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Although the measures of orbital lengths and consequently the planetary distances from the centre of the solar system are erroneous, as explained above, still the radii vectors of planets which are their distances from the centre of the solar system, mostly agree with their modern values, when calculated on the basis of the correction known as annual parallax whose values are given in our ancient works and which in fact are the variations affecting their true places as the result of these distances. This will be seen from the subjoined table in which the measures given by Ptolemy* are also noted :

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∗ An ancient work like the Brahma Siddhānta compiled in Śaka 550, does give these measures. Hence, if these measures have found an entry into the Sūrya Siddhānta later, they may have done so very soon after Śaka 550.

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| Planet | Sūrya Siddhānta | Ptolemy | Modern |
| | even quadrant | odd quadrant | | |
| S.n (or Earth) . . . . | 1 | 1 | 1 | 1 |
| Mercury . . . . | ·3694 | ·3667 | ·3750 | ·3871 |
| Venus . . . . | ·7278 | ·7222 | ·7194 | ·7233 |
| Mars . . . . | 1·5139 | 1·5517 | 1·5190 | 1·5237 |
| Jupiter . . . . | 5·1429 | 5·0000 | 5·2174 | 5·2028 |
| Saturn . . . . | 9·2308 | 9·0000 | 9·2308 | 9·5388 |

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The measures calculated from the Sūrya Siddhānta have been given above. The radii vectors of Mercury and venus are calculated by dividing the length of the circumference of the epicycles of conjunctions** by the lengths of the planetary orbit i.e. by 360 and those of the Superior planets by dividing 360 by the circumferences of the epicycles of conjunctions.

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It has been remarked above that the length of the moon's orbit according to Āryabhaṭa I is different . It is obtained as follows :

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शशिराशयष्ट १२ चक्रं तेंशकलायोजनानिय ३० व ६० अ १० गुणाः ।। ४ ।। दशगीतिकापाद.

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"The length of moon's orbit in yojanas is given by multiplying the total number of signs (i.e. 12) by 30, 60 and 10".

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In this it is stated that the number of minutes of arcs in the moon's orbit multiplied by 10 gives the length in yojanas. In other words, here 1' arc is taken to be equal to 10 yojanas, while other siddhāntas have taken it to be equal to 15. This appears to be contradictory to other siddhāntas at first sight, but in fact, there is no contradiction. The following table will show that just as the length of the moon's orbit given in other siddhāntas is 3/2 times that given by Āryabhaṭa I, so also are other measures :--

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| | Yojanas according to Āryasiddhānta | Yojanas according to Siddhānta Śiromaṇi |
| Diameter of the Earth | 1050 | 1581 |
| ,, Sun's disc. | 4410 | 6522 |
| ,, Moon's disc. | 315 | 480 |

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∗ Ptolemy's figures have been taken from the translation of the Sūrya Siddhānta by Burgess, and the modern measures from the work of Loomis.

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∗∗ The lengths of circumferences of the aphelia and the perihelia have been mentioned later on. The question will be discussed at greater length later on in the chapter on true places.

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The distance of the moon from the Earth comes to 34377 yojanas, according to the First Ārya Siddhānta. This figure is 65.5 times of 525, which is the Earth's radius according to that Siddhānta. Hence, from a relative point of view there is no cotradiction. The difference in the actual numerical quantities appears to be due to the difference in the measure of the unit, ‘Yojana’. Lalla is mostly the follower of Āryabhaṭa I, and hence, measures given by him are Similar to those given by Āryabhaṭa I. Those given by Āryabhaṭa II are like those of other siddhāntas.

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EARTH'S. RADIUS

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A reference to the term “earth's radius’. occurs in the above discussion. Let us give more cosideration to it. Different works mention the diameter of the Earth as follows :—

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WHAT DISTANCE DOES ‘YOJANA’ INDICATE ?

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It cannot be determined with certainty what distance was denoted by ‘yojana’. Hence, there is no reliable means to test the accuracy of these diameters of the earth mentioned in our works. According to the views of most of our works, one Yojana is equal to 32000 cubits, and if one cubit be assumed to be equal to 19.8 inches, the Yojana, in English miles, will be exactly 10 miles. Hence, the diameter of the earth, accepting the lowest figure, that given by the Pañca Siddhāntikā, would be 10186 miles. According to modern astronomy, the diameter of the earth, east to west, is 7925 miles. But the measure of yojana may have been somewhat smaller. The dictionaries known as Vācaspati and Śabdārṇava give 16000 cubits as the length of the yojana, which makes a yojana equal to 5 miles. In the middle of the 7th century A.D. Hjouen Thsang, a Chinese traveller, had visited India. He has described the whole of India. He has given distances of cities and places in terms of the Chinese measure, called Li*. He says, “According to the old system, a yojana consists of 40 Li ; according to the current practice of the administration in different parts, a yojana measures 30 Li ; but the scientific treatises regard a yojana as equal to 16 Li only.” According to St. Martin, the measure of one Li in China, in the 7th century A.D., was equal to 329 metres i.e. 1080 ft. by English measure**.

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∗ Julien's Memories de Hiouen Thsang I ; p. 59 ; Translation of the Sūrya Siddhānta by Burgess, p. 39.

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∗∗ Julien's Memoires de Hiouen Thsang II, p. 251 ; Translation of the Sūrya Siddhānta by Burgess p. 284.

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On this basis, the different values of yojana, mentioned by Hiouen Thsang come to 8⅙, 6⅛, and 3⅓ miles. In other words, the measure of a yojana according to current practice in his time was 6⅛ miles. General Cunningham also compared the distances between several well-known places, given by Hiouen Thsang, with the actual ones, and decided that Hiouen Thsang's 6 Li were equal to one mile.* But the distances given by Hiouen are those along roads by which he actually travelled; and the roads and paths are not always in a straight direction. Hence, in Cunningham's opinion, in order to get the direct distance between two places, distance given by Hiouen Thsang should be reduced by ⅙ and Cunningham has done so at several places; and if this rule be followed, the distance of 6 Li given by Hiouen is equal to (6-1=5) Li in a straight direction; and thus, the measures given by Cunningham and St. Martin agree with one another. On the whole, the author feels that the unit of yojana equal to 30 Li was in use in Hiouen's time, and that while converting his distances into miles, six Li must be assumed to be equal to one mile; in other words, according to the then prevalent system, one yojana was equal to 5 miles. And the diameter of the earth, 1581 yojanas, mentioned by Brahmagupta who lived just before or about the date when Hiouen came to this country in the 7th century A.D., comes to 7905 miles; and this figure is very nearly equal to that of the accurately established modern measure of the diameter.

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TO CALCULATE THE DISTANCE IN DEGREES

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In any case, the quotations like the following go to prove that our astronomers knew how to measure the distance on the globe equal to one degree and therefrom to calculate the circumference of the earth :—

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निरक्षदेशात् क्षितिषोडशांशे भवेद्वंती गणितेन यस्मात् ।।
तदंतरं षोडशगुणं स्याद्रुमानः ...... ।। १५ ।।
सि. शिरोमणि. भुवनकोश.

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“The city of Ujjayini** lies on the 16th degree (360/16=22½) of the earth's face, from the “zero-latitude” line. Hence, the circumference of the earth is 16 times the distance between the two places.”

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It is, however, true that attempts were not made in our country to find out the accurate value of one degree as they were made in Europe.

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SUPPORT FOR UNIVERSE

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It has been shown before that in the opinion of our astronomers, the earth is at the centre of the Universe; planets revolve round it and the earth stands suspended supportless in space. But we do not find anything in our works definitely to explain the support for planets. It is, however, their view that the planets and stars are set in motion by the ‘pravaha’ (provector) wind. It, therefore, seems to be the view of our astronomers that the planets remain in position in the sky, because of the ‘pravaha’ wind—

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निजनिजकर्मापकर्षैर्वैरुपभुज्यते फलं चित्रं । तद्योगस्थानानि स्वर्गादिकसंज्ञका लोकाः ।। ३ ।।
अनिलाधाराः केचित् के चिल्लोका वसुंधराधाराः ।। वसूधा नान्याधारा तिष्ठति गगने
स्वशक्त्यैव ।। ४ ।।
अध्या. १६

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∗ See general description given in the beginning in “Ancient-Geography” by Cunningham.

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∗∗ The latitude of Ujjayini has been determined to be 23°9′ by modern methods.

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It is stated in the verse that some worlds remain in position because of the support of the wind. But it is not clearly stated that planets and stars are worlds. It appears that our astronomers did not have the idea that planets and stars are very heavy and expansive globes like our earth.

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ATTRACTION

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Bhāskarācārya has assumed that the earth possesses the power of attraction. He says,

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आकृष्टिशक्तिश्च मही तया यत् खस्थं गुरु स्वाभिमुखं स्वशक्त्या ।।
आकृष्यते तततोव भाति ।। ६ ।।
गोलाध्याय, भुवनकोश,

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"The earth possesses the power of attraction. It pulls towards itself a heavy body in the sky and the latter appears to fall."

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The falling down of a body is here stated to be due to attraction. When Newton discovered the power of attraction possessed by the earth, what thing other than the falling down of a body to the earth, could have suggested to him the power of attraction possessed by the earth? He drew the inference that the planetary system revolves round the Sun on account of this power of attraction, and established the law of gravitation, after proving it mathematically; this further work (of research) was, however, not taken up in our country.

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DESCRIPTION OF THE EARTH

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All siddhānta works while describing the Universe, give a description of the seven oceans on the earth and seven continents along with mountains and rivers in them; but this, in fact, forms part of Geography, and hence, the author does not deal with it here for want of space.

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VIEWS OF THE SKY

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As regards the views of the heavenly bodies observed from different places on the earth, it may be stated that the Pole star appears to be in the horizon when observed from any place on the Equator, and the planets and other celestial bodies appear to rise and set at a vertical circle. As one goes northwards, the north pole appears to attain higher altitudes, and the diurnal circles along which planets and other bodies appear to move, are seen inclined to the horizon. For an observer at the pole, the Sun and other (celestial) bodies, appear to move along circles parallel to the Equator; all these things are discussed in the Siddhāntas. The author does not give quotations from the original here for want of space.* Again, most of the Siddhāntas discuss questions like these (i) what places in the northern hemisphere (i. e. places in what latitudes) can never see a particular portion of the ecliptic, (ii) what latitudes can never see certain signs of the ecliptic or (iii) what are the latitudes where the Sun is visible for 60 ghaṭis or even more, and for how many days. It is not necessary to repeat them here in detail.

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∗ The Marāṭhi book entitled Bhāskarācārya aṇi tatkāta Jyotiṣa (Bhāskarācārya & his astronomy) by J. B. Modak contains such quotations from Bhāskarācārya in the original along with their translation.

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MERU AND SEVEN WORLDS

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They have imagined a mountain called Meru situated at the Pole. Bhāskarācārya has imagined the abodes of Brahmā, Viṣṇu, Maheśa and lords of other worlds, as situated on this very mountain. Similarly, he observes about the seven worlds including the ‘bhulok’ (earth) as follows :

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भूर्लोकाख्यो दक्षिणे व्यक्षदेशात् । तस्मात् सौम्ययं भुवः स्वश्च मेरुः ॥
लभ्यः पुण्यैः खे महः स्याज्जनोऽतो ऽनल्पानल्पैः स्वैस्तपः सत्यमंत्यः ॥ ४३ ॥

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It has been told in the verse that the world ‘bhūḥ’ lies to the south of the zero-latitude line (i.e. equator) and the world ‘bhuvar’ (on which we live) lies to its north. Meru is the “swar” world, and the worlds ‘Mahar’, ‘Jana’, ‘Tapas’ and ‘Satyam’ are situated in the sky. Of these Satya occupies the last position. Bhāskarācārya says that the description of continents, of seven oceans and of the seven worlds begining with ‘bhūḥ’ has been borrowed from the Purāṇas. All works do not agree completely as far as this kind of description is concerned.

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ATMOSPHERE ENVELOPING EARTH

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It is said that the earth is enveloped by a sphere of air which spreads to a distance of 12 Yojanas from the surface of the earth, and the clouds and lightning abide in it. Bhāskarācārya expresses this idea in the line.

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भूमेर्बहिर्द्वादशयोजनानि । भूवायुत्रांबुदविद्युदाद्यं ॥ २ ॥

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Both the ĀRYABHAṬAS and LALLA have assumed the same height for the atmospheric covering. 12 yojanas make 60 miles. According to modern discoveries, the height of the atmosphere varies from 45 to 100 miles. Śrīpati has expressed in the following verse the idea that this very atmospheric belt provides a place for whirlwind, meteors, clouds, rainbow, lightning, Gandharva’s city and the halo round the Sun and Moon :—

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निर्घातोलकाघनसुरधनुर्विद्युदतः कुवायोः संदृश्यते खनगरपरीवेषपूर्व* ॥

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PRAVAHA AND OTHER KINDS OF WINDS

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LALLA, SRIPATI and BHASKARACARYA have assumed other kinds of winds higher up than the ‘pravaha’ wind. Lalla observes :

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आवहः प्रवह उद्वहस्तथा संवहः सुपरिपूर्वको वहौ ॥
सप्तमस्तु पवनः परावहः कीर्तितः कुमारदावहो परैः ॥ १ ॥

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“There are seven kinds of winds, one above the other in height. They are āvaha, pravaha, udvaha, samvaha, suparivaha, purvakavaha and parāvaha, as the seventh ; others suppose Kumarudavaha, as the seventh wind.”

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∗ Siddhānta Śiromaṇi edited by Bāpūdeva, p. 267, foot note.

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REVOLUTIONS OF PLANETS

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The cause of mean motions of planets has already been discussed above. The numbers of the revolutions of planets in the Zodiac during the period of one Kalpa or one Mahāyuga have already been given as mentioned by different siddhāntas. There is a noteworthy point about Mercury and Venus; and it is that because they are always found near the Sun, their revolutions in the Zodiac are assumed to be always equal to those of the Sun; and hence, our authors of the works have assumed for them the same number of revolutions and mean motion as for the Sun. But having regarded the conjunction (Śīghra) of Mercury and Venus as some planets, the numbers of their revolutions given by them in the works happen to be equal to those of their actual revolutions round the Sun; and although our astronomers had no idea that planets revolve round the sun, it is worth remembering that they had given importance to the revolutions of the ‘Śīghras’ of Mercury and Venus.

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LIGHT OF PLANETS

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It is the view of our Astronomical science that planets have no light of their own, but that they receive it from the Sun. Āryabhaṭa I observes

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भूग्रहमानां गोलार्धानि स्वछायया विवर्णानि ।।
अर्धानि यथासारं सूर्याभिमुखानि दीप्यन्ते ।। ५ ।।

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“Half the spherical portions of the earth, the planets and the stars lose their light in their own shadows. Their faces which are towards the Sun are illuminated in the proportion in which they are so directed.”

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In this verse it is stated that even the stars receive light from the Sun, which is a mistake. A great deal of discussion is, found in our works regarding the increase or decrease of the Moons’ digits, and the elevation of Moon’s cusps.

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VIKṢEPAS OF PLANETS

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The vikṣepas of planets which are the inclinations of their orbits with the ecliptic are given in some siddhāntas in the chapter on mean motions. Therefore, the measures of different siddhāntas are given here— Ptolemy’s values and the modern ones are also given* in this table.

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| | Modern Sūrya Siddhānta | I Ārya S. Lalla’s S. | Brahma S. Sid.Śiro, | II Ārya Siddhānta | Ptolemy | Modern |
| | ° ′ | ° ′ | ° ′ | ° ′ | ° ′ | ° ′ ″ |
| Moon | 4 30 | 4 30 | 4 30 | 4 30 | 5 0 | 5 8 47.5 |
| Mars | 1 30 | 1 30 | 1 50 | 1 46 | 1 0 | 1 51 2 |
| Mercury | 2 0 | 2 0 | 2 32 | 2 18 | 7 0 | 7 0 7.7 |
| Jupiter | 1 0 | 1 0 | 1 16 | 1 14 | 1 30 | 1 18 41.4 |
| Venus | 2 0 | 2 0 | 2 16 | 2 16 | 3 30 | 3 23 34.9 |
| Saturn | 2 0 | 2 0 | 2 10 | 2 10 | 2 30 | 2 29 39.2 |

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∗ The author has taken Ptolemy’s values as given by Burgess in his translation of Sūrya Siddhānta and the modern values are taken from Leverrier’s Tables.

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It would not be reasonable to compare the values of vikṣepa as given in our works with the modern ones in all respects. If the comparison is made in a proper way, the discussion which follows will show that our values are very accurate.

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The values of vikṣepa are calculated from the latitudes. Latitude is the distance of a planet from the ecliptic measured along the great circle passing through its pole. The latitude is zero at the node ; *i.e.* the point of intersection of the ecliptic and the planet's orbit ; and it is maximum at the point which is 3 signs away from it. The planetary orbits are not exactly circular. So the planets are not always at the same distance from the centre of their orbits. The Moon revolves round the Earth. Hence, when it is at a distance of 3 signs from the node, although her linear distance from the Earth is not always the same, the angular distance is necessarily so. This is not the case with other planets. They revolve round the Sun. The observer at the Sun's centre will always see the maximum latitudes of the planets to be equal to those of the modern values of vikṣepa ; but these will appear to be smaller or greater to an observer on the Earth. The latitude will appear to vary with the values of their radii vectors, *i.e.* their distances from the Sun's centre and their distances from the Earth. The variation due to the first cause is less than that due to the second. Our astronomers have not taken into account the variation due to the first cause. They have only considered that due to the second. Bhāskarācārya remarks about the ‘vikṣepa’ as follows :—

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(यदा) त्रिज्यातुल्यः शीघ्रकर्णो भवति तस्मिन् दिने वेधवलये यावान् परमो विक्षेप उपलभ्यते तावान् ग्रहस्य परमो मध्यमविक्षेपः ।।

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ग्रहच्छायाविकार, श्लोक १ टीका.

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“The maximum value of a planet’s mean polar latitude is obtained from the value of the maximum latitude of the planet measured on the circle of observation on the day on which the geocentric radius of the planet is equal to its mean radius vector.”

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This can be said to be the definition of Vikṣepa according to our ancient works. When the planet is at its mean distance from the earth, it is not necessarily situated at a place 3 signs apart from the node. If the planet is situated 90° away from the node, and its distance from the earth also happens to be equal to its mean value, then the latitude of the planet is regarded by our astronomers as the planet’s mean maximum Vikṣepa. This does not take into consideration the factor of planet’s radius vector. The latitude of a “superior*” planet is not much affected by the variation in the value of its radius vector but they do cause a difference in the latitudes of “inferior*” planets. Hence, there is no harm if the Vikṣepa values of planets other than Mercury and Venus as given by our works are compared with modern values ; and such a comparison will reveal that our works, as far as the Vikṣepas of

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*Marcury and Venus are ‘inferior’ planets; others are ‘superior’ ones.

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Mars and Jupiter are concerned, have given values nearer to the modern ones than those of Ptolemy. The values given in Brahma Siddhānta and Second Ārya Siddhānta are almost equal to the modern ones. The value for Saturn is wrong by a few minutes only. As regards the Vikṣepa values of Mercury and Venus, it was found that if Mercury occupies at present a position of maximum heliocentric latitude, then, at one position its radius vector attains the value of 3382 and at the other, 4114*, and if in addition to this, its distance from the earth has attained the mean value, its true latitude respectively becomes 2°23'' and 2°53.' The mean of these values comes to 2°38' and it agrees very closely with that given in our works. The radius vector of Venus at its position of maximum heliocentric latitude attains the values of 7193 and 7293. If at both these positions its distance from the earth attains the mean value, its true latitude would be about 2°28'. This also agrees with the value given in our works. The values given in the table in the column for "modern" values, were calculated by the author from the data in the English Nautical Almanacs for the six years 1883 to 1888. It was only on two or three dates during these six years that when Mercury attained maximum value for its heliocentric latitude, its distance from the earth was also equal to or very nearly equal to its mean value ; and in the case of Venus no such dates could be found. It shows that these values cannot be accurately found unless observations are taken for several years ; and for this reason our astronomers deserve praise for having found out very accurate values. The nodes of planetary orbits have got some slight motion. Hence, if the maximum values of the latitudes of planets at the time of Brahmagupta and Āryabhaṭa be calculated in the light of modern mathematics' the above Siddhānta values may perhaps prove to be nearer to the true values** than what we found in this comparison. The above verses will show that the values given by the two Āryabhaṭas are different from those given by Brahmagupta and this shows that the values have been found by them independently. It cannot be argued that the figures are different in the three cases because they have been set down at random. If no other proofs be forthcoming, even the Vikṣepa values given in the above table would be enough to show that our astronomers determined the values for inclusion in their works, actually after taking observations.

Linked texts (7)

Sūryasiddhānta — Maya, Sanskrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Vāsiṣṭhasiddhānta — Sanskrit
Pañcasiddhāntikā — Varāhamihira, Sanskrit
Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit