Aryabhatiya

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

1 Structure and style
2 Contents
3 Significance
4 Significant verses
5 Translations
6 References

[edit] 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 epicycles (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. Āryabhaṭa numeration, the Sanskrit numerals.

[edit] Contents

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 499 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.

[edit] Significance

It presents astronomical and mathematical theories in which the Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun (in other words, it was heliocentric).He believes that the Moon and planets shine by reflected sunlight and he believes that the orbits of the planets are ellipses. He correctly explains 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.

He also propounded the Heliocentric theory of the universe, thus predating Copernicus by almost one thousand years.

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

[edit] Significant verses

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

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

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

samasya dvikaraNI. pramANaM tritIyena vardhayet tachchaturthAnAtma chatusastriMshenena savisheShaH.

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

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

Add 4 to 100, multiply by 8 and add to 62,000. This is approximately the 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 an approximation.

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 as 225,224,222,219.215,210,205, 199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7 which 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 21-25 the 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+100 kighva (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    205

skaki kiShga shghaki kighva ghlaki kigra hakya

 199    191     183    174    164   154   143

dhaki 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 minutes of arc interpolation interval, are ideal for the table, better suited than the modern tables.

[edit] 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 8th century Arabic translation of the Aryabhatiya was translated into Latin in the 13th century, before the time of Copernicus. Through this translation, European mathematicians may have learned methods for calculating square and cube roots, and it is also possible that Aryabhata's work had an influence on European astronomy.

Although the work was influential, there is no definitive English translation. There are a number of translations but many are incomplete. The meanings of certain parts of the work are still disputed to this day.