Aryabhata

“ There is no more reason for questioning the authenticity of the paribhāsā stanza than for questioning that of the invocation and colophon. Kaye would like to eliminate it since it seems to furnish evidence for Āryabhata's knowledge of place-value. Nothing is gained by doing so since Lalla gives in numerical words the most important numerical elements of Āryabhata without change, and even without this paribhāsā stanza the rationale of the alphabetical notation in general could be worked out and just as satisfactory evidence of place-value furnished. ”

“ The Āryabhatīya is divided into four sections which contain in all only 123 stanzas. It is not a complete and detailed working manual of mathematics and astronomy. It seems rather to be a brief descriptive work intended to supplement matters and processes which were generally known and agreed upon, to give only the most distinctive features of Āryabhata's own system. Many commonplaces and many simple processes are taken for granted. For instance, there are no rules to indicate the method of calculating the ahargana and of finding the mean places of the planets. ”

“ No distinction is made between long and short vowels.
From Aryabhata's usage it is clear that the letters k to m have the values of 1-25. The letters y to h would have the values of 3-10, but since a short a is regarded as inherent in a consonant when no other vowel sign is attached and when the virama is not used, and since short a refers the avarga letters to the place of ten's, the signs ya, etc., really have the values of 30-100. [30] The vowels themselves have no numerical values. They merely serve to refer the consonants (which do have numerical values) to certain places. ”

“ I feel justified in assuming that the Āryahhaṭīya on the whole is genuine. It is, of course, possible that at a later period some few stanzas may have been changed in wording or even supplanted by other stanzas. Noteworthy is I, 4, of which the true reading bhūḥ, as preserved in a quotation of Brahmagupta, has been changed by Parameśvara or by some preceding commentator to bham in order to eliminate Āryabhaṭa's theory of the rotation of the Earth. ”

“ If nava is to be separated from antyavarge it is possible to take it with what precedes and to translate, "The vowels (are to be used) in two nine's of places, nine in varga places and nine in avarga places," but antyavarge va remains enigmatical.
The translation must remain uncertain until further evidence bearing on the meaning of antya can be produced. Whatever the meaning may be, the passage is of no consequence for the numbers actually dealt with by Aryabhata in this treatise. The largest number used by Aryabhata himself (1, 1) runs to only ten places. ”

“ The earth-sine is the distance in the plane of the day-circle between the observer's horizon and the 72 ARYABHATIYA
horizon of Lanka (the unmandala) . When trans- formed to the plane of a great circle it becorrfes the ascensional difference.^
27. The first and fourth quadrants of the ecliptic rise in a quarter of a day (15 ghatikds) minus the ascensional difference; The second and third quadrants rise in a quarter of a day plus the ascensional difference, with regular increase and decrease. ”

“ In a yuga the revolutions of the Sun are 4,320,000, of the Moon 57,753,336, of the Earth eastward 1,582,237,500, of Saturn 146,564, of Jupiter 364,224, of Mars 2,296,824, of Mercury and Venus the same as those of the Sun. ”

“ There is no warrant for treating the revolutions of the Earth given here as based on false knowledge (mithyajnana) , which causes the Earth to seem to move eastward because of the actual westward movement of the planets (see note to I, 4) . ”

“ Further, Brahmagupta (Brāhmasphutasiddhānta, XI, 8) names the Dasagītika as the work of Āryabhata, gives direct quotations (XI, 5; I, 12 and XI, 4; XI, 17) of stanzas 1, 3, and 4 of our Dasagītika, and XI, 15 (although corrupt) almost certainly contains a quotation of stanza 5 of our Dasagītika. Other stanzas are clearly referred to but without direct quotations. Most of the Dasagītika as we have it can be proved to be earlier than Brahmagupta (628 A.D.) .
The second section in 33 stanzas deals with mathematics. The third section in 25 stanzas is called Kālakriyā, or "The Reckoning of Time." ”

“ It is impossible as a translation of the Sanskrit.
13. The circle is made by turning, and the triangle and the quadrilateral by means of a karna; the horizontal is determined by water, and the perpendicular by the plumb-line. Tribhuja denotes triangle in general and caturbhuja denotes quadrilateral in general. The word karna regularly denotes the hypotenuse of a right angle triangle and the diagonal of a square or rectangle. ”

“ The last phrase means that the values for signs 1, 2, 3 are equal, respectively, to those of signs 6, 5, 4 and that the values of 7, 8, 9 are equal, respectively, to those of 12, 11, 10. They increase in the first quadrant, decrease in the second, increase in the third, and decrease in the fourth. There are, there- fore, only three numerical values involved, those cal- culated for the first three signs. ”

“ Page:Aryabhatiya of Aryabhata, English translation.djvu/106 Page:Aryabhatiya of Aryabhata, English translation.djvu/107 Page:Aryabhatiya of Aryabhata, English translation.djvu/108 Page:Aryabhatiya of Aryabhata, English translation.djvu/109 Page:Aryabhatiya of Aryabhata, English translation.djvu/110 translated it must deal with the so-called ayanavalana or "deflection due to the deviation of the ecliptic from the equator." ”