Āryabhaṭīya

Āryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only surviving work of the 5th century Indian mathematician, Āryabhaṭa.

Structure and style

The text is written in Sanskrit and divided into four sections, covering a total of 121 verses that describe different results using a mnemonic style typical for such works in India.

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha(ca. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.

2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)

3. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.

4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

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 coherence makes it extremely difficult for a casual reader to follow.

Indian mathematical works often used word numerals before Aryabhata, but the Aryabhatiya is the 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.

Contents

Crowning glory of Aryabhatiya is the decimal place value notation without which modern, mathematics, science and commerce would be impossible. Prior to Aryabhata, 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 c. 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.

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.

It has been suggested by some commentators, most notably B. L. van der Waerden, that certain aspects of Aryabhata's geocentric model suggest the influence of an unknown, underlying heliocentric model. This view has been contradicted by others and, in particular, strongly criticized by Noel Swerdlow, who characterized it as flatly contradicting Aryabhata's own words.

Aryabhata asserted that the Moon, planets, and asterisms shine by reflected sunlight. 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 Eratosthenes (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.

Significant verses

aryabhatta predicted the line numbers 1-9 inspired from the nine planets in the solar system considering sun as 0

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 diameter is 20,000.

i.e. $\pi \approx \frac{62,832}{20,000} = 3.1416$

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 $\pi \approx 3.1416$, 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 a carefully chosen radius of 3,438 these values are successive differences of $3438\times\sin \theta$ to within one digit;

for example,

$3438\times \sin 15{^\circ} = 225 + 224 + 222 + 219 = 890$

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.

Influence

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 Bhāskara I and Brahmagupta among other notables.

The estimate of the diameter of the Earth in the Tarkīb al‐aflāk of Yaqūb ibn Tāriq, of 2,100 farsakhs, appears to be derived from the estimate of the diameter of the Earth in the Aryabhatiya of 1,050 yojanas.

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.

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

See also

• Aryabhata's sine table
• Indian astronomy

References

• William J. Gongol. 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
• O'Connor, John J.; Robertson, Edmund F., "Aryabhata", MacTutor History of Mathematics archive, University of St Andrews . University of St Andrews.

External links

• The Āryabhaṭīya by Āryabhaṭa (translated into English by E. C. Clark, 1930) hosted online by the Internet Archive