Aryabhatiya

"Āryabhatīya", an astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Aryabhata.

Structure and style

The text is written in Sanskrit and structured into four section, overall covering 121 verses that describe different results using a mnemonic style typical of the Indian tradition.

33 verses are concerned with mathematical rules.

The four chapters are:

(i) the astronomical constants and the sine table (ii) mathematics required for computations (gaNitapāda) (iii) division of time and rules for computing the longitudes of planets using eccentrics and ellipses (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses (golādhyaya).

It is highly likely that the study of the "Aryabhatiya" was meant to be accompanied by the teachings of a well-versed tutor. While some of the verses have a logical flow, some don't and its lack of coherance makes it extremely difficult for a casual reader to follow.

Indian mathematical works often used word numerals before Aryabhata, but the "Aryabhatiya" is oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form words with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it. At the same time, this system of numeration allows for poetic license even in the author's choice of numbers. "Cf. ĀryabhaIPA|ṭa numeration, the Sanskrit numerals."

Contents

Crowning glory of Aryabhatiya is the decimal place value notation without which mathematics, science and commerce would be impossible. Prior to Aryabhatta, Babylonians used 60 based place value notation which never gained momentum. Mathematics of Aryabhatta went to Europe through Arabs and was known as "Modus Indorum" or the method of the Indians. This method is none other than our arithmetic today. The "Aryabhatiya" begins with an introduction called the "Dasagitika" or "Ten Giti Stanzas." This begins by paying tribute to Brahman, the "Cosmic spirit" in Hinduism. Next, Aryabhata lays out the numeration system used in the work. It includes a listing of astronomical constants and the sine table. The book then goes on to give an overview of Aryabhata's astronomical findings.

Most of the mathematics is contained in the next part, the "Ganitapada" or "Mathematics."

The next section is the "Kalakriya" or "The Reckoning of Time." In it, he divides up days, months, and years according to the movement of celestial bodies. He divides up history astrologically - it is from this exposition that historians deduced that the "Aryabhatiya" was written in 522 C.E. It also contains rules for computing the longitudes of planets using eccentrics and epicycles.

In the final section, the "Gola" or "The Sphere," Aryabhata goes into great detail describing the celestial relationship between the Earth and the cosmos. This section is noted for describing the rotation of the earth on its axis. It further uses the armillary sphere and details rules relating to problems of trigonometry and the computation of eclipses.

Significance

The treatise uses a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the "Paitāmahasiddhānta" (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller "manda" (slow) epicycle and a larger "śīghra" (fast) epicycle. [David Pingree, "Astronomy in India", in Christopher Walker, ed., "Astronomy before the Telescope", (London: British Museum Press, 1996), pp. 127-9.]

It has also been interpreted as advocating Heliocentrism, where Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun (according to this view, it was heliocentric). [The concept of Indian heliocentrism has been advocated by B. L. van der Waerden, "Das heliozentrische System in der griechischen,persischen und indischen Astronomie." Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman AG, 1970. A detailed rebuttal to this heliocentric interpretation is in a review which describes van der Waerden's book as "show [ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata's description." Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," "Isis", 64 (1973): 239-243.] Aryabhata asserted that the Moon and planets shine by reflected sunlight and that the orbits of the planets are ellipses. He also correctly explained the causes of eclipses of the Sun and the Moon. His value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the true value of 365 days 6 hours 9 minutes 10 seconds. In this book, the day was reckoned from one sunrise to the next, whereas in his "Āryabhata-siddhānta" he took the day from one midnight to another. There was also difference in some astronomical parameters.

A close approximation to π is given as : "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, π ≈ 62832/20000 = 3.1416, correct to four rounded-off decimal places.

Aryabhata was the first astronomer to make an attempt at measuring the Earth's circumference since Erastosthenes (circa 200 BC). Aryabhata accurately calculated the Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation remained the most accurate for over a thousand years.

Aryabhata's methods of astronomical calculations have been in continuous use for practical purposes of fixing the Panchanga (Hindu calendar).

Significant verses

shulva-sUtras: form a shrauta part of kalpa vedAnga - nine texts -mathematically most imp - baudhAyana, Apastamba, and kAtyAyanashulvasUtra.

dIrghasyAkShaNayA rajjuH pArshvamAnI tiryaDaM mAnI. cha yatpr^thagbhUte kurutastadubhayAM karoti.

The diagonal of a rectangle produces both areas which its length andbread produce separately.

samasya dvikaraNI. pramANaM tritIyena vardhayettachchaturthAnAtma chatusastriMshenena savisheShaH.

sqrt(2) = 1 + 1/3 + 1/(3.4) - 1(3.4.34) -- correct to 5 decimals = 1.41421569

chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAmAyutadvayaviShkambhasyAsanno vr^ttapariNahaH. [gaNita pAda, 10]

Add 4 to 100, multiply by 8 and add to 62,000. This is approximatelythe circumference of a circle whose diamenter is 20,000.

i.e. PI = 62,832 / 20,000 = 3.1416

correct to four places. Even more important however is the word"Asanna" - approximate, indicating an awareness that even this is anapproximation.

tribhujasya falasharIraM samadalakoTI bhujArdhasaMvargaH

It depicts the area of a triangle.

jyA = sine, koTijyA = cosine

jyA tables : Circle circumference = minutes of arc = 360x60 = 21600. Gives radius R = radius of 3438; (exactly 21601.591) [ with pi = 3.1416, gives 21601.64]

The R sine-differences (at intervals of 225 minutes of arc = 3:45deg),are given in an alphabetic code as225,224,222,219.215,210,205,199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7which gives sines for 15 deg as sum of first four = 890 --> sin(15) = 890/3438 = 0.258871 vs. the correct value at 0.258819. sin(30) = 1719/3438 = 0.5

Expressed as the stanza, using the varga/avarga code:ka-M 1-5, ca-n~a: 6-10, Ta-Na 11-15, ta-na 16-20, pa-ma 21-25the avargiya vyanjanas are:y = 30, r = 40, l=50, v=60, sh=70, Sh=80, s =90 and h=100

makhi (ma=25 + khi=2x100) bhakhi (24+200) fakhi (22+200) dhakhi (219) Nakhi 215, N~akhi 210, M~akhi 205, hasjha (h=100 + s=90+ jha=9)skaki (90+ ki=1x00 + ka=1) kiShga (1x100+80+3), shghaki, 70+4+100kighva (100+4+60) ghlaki (4+50+100) kigra (100+3+40) hakya (100+1+30)dhaki (19+100) kicha (106) sga (93) shjha (79) Mva (5+60) kla (51)pta (21+16, could also have been chhya) fa (22) chha (7).

makhi bhakhi dhakhi Nakhi N~akhi M~akhi hasjha 225, 224 222 219 215 210 205skaki kiShga shghaki kighva ghlaki kigra hakya 199 191 183 174 164 154 143dhaki kicha sga shjha Mva kla pta fa chha 119 106 93 79 65 51 37 22 7

given radius R = radius of 3438, these values give the Rxsin(theta) within one integer value; e.g. sine (15deg) = 225+224+222+219= 890, modern value = 889.820.

Both the choice of the radius based on the angle, and the 225 minutesof arc interpolationinterval, are ideal for the table, better suited than the moderntables.

Translations

The "Aryabhatiya" was an extremely influential work as is exhibited by the fact that most notable Indian mathematicians after Aryabhata wrote commentaries on it. At least twelve notable commentaries were written for the "Aryabhatiya" ranging from the time he was still alive (c. 525) through 1900 ("Aryabhata I" 150-2). The commentators include Bhaskara and Brahmagupta among other notables.

The work was translated into Arabic around 820 by Al-Khwarizmi, whose "On the Calculation with Hindu Numerals" was in turn influential in the adoption of the Hindu-Arabic numerals in Europe from the 12th century.

Although the work was influential, there is no definitive English translation.

ee also

*Aryabhata
*Indian astronomy

Notes

References

*William J. Gongol. [http://www.gongol.com/research/math/aryabhatiya "The Aryabhatiya: Foundations of Indian Mathematics".] University of Northern Iowa.
*Hugh Thurston, "The Astronomy of Āryabhata" in his "Early Astronomy", New York: Springer, 1996, pp. 178-189. ISBN 0-387-94822-8
*MacTutor Biography|id=Aryabhata_I|title=Aryabhata University of St Andrews.