Aryabhatta (476-550 A.D.)

Aryabhatta , one of the world’s greatest mathematician-astronomer, was born in Patliputra in Magadha, modern Patna in Bihar.

He wrote his famous treatise the “Aryabhatta-siddhanta” but more famously the “Aryabhatiya”, the only work to have survived.The surviving text is Aryabhata’s masterpiece the Aryabhatiya which is a small astronomical treatise written in 108 verses plus an introductory 13 giving a summary of Hindu mathematics up to that time.

the whole being divided into four pAdas or chapters:

gitikApAda: (13 verses) large units of time – kalpa, manvantra, yuga, which present a cosmology that differs from earlier texts such as Lagadha’s Vedanga Jyotisha(ca. 1st c. BC). Also includes the table of sines (jya), given in a single verse. For the planetary revolutions during a mahayuga, the number of 4.32mn years is given. 

gaNitapAda (33 verses), covering mensuration (kShetra vyAvahAra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka) 

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

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. 

It was composed 3,600 years into the KaliYuga, when he was 23 years old. It contains mathematical and astronomical theories that have been revealed to be quite accurate in modern mathematics. For instance he wrote that if 4 is added to 100 and then multiplied by 8 then added to 62,000 then divided by 20,000 the answer will be equal to the circumference of a circle of diameter twenty thousand. This calculates to 3.1416 close to the actual value Pi (3.14159). But his greatest contribution has to be zero. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines..

At the trigonometry contained in Aryabhata’s treatise. He gave a table of sines calculating the approximate values at intervals of 90°/24 = 3° 45′. In order to do this he used a formula for sin(n+1)x – sin nx in terms of sin nx and sin (n-1)x. He also introduced the versine (versin = 1 – cosine) into trigonometry.

Other rules given by Aryabhata include that for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are correct. He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses.

In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes:

1^2 + 2^2 + \cdots + n^2 = {n(n + 1)(2n + 1) \over 6}


1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2

He already knew that the earth spins on its axis, the earth moves round the sun and the moon rotates round the earth. He talks about the position of the planets in relation to its movement around the sun. He refers to the light of the planets and the moon as reflection from the sun. He goes as far as to explain the eclipse of the moon and the sun, day and night, the contours of the earth, the length of the year exactly as 365 days. He even computed the circumference of the earth as 24835 miles which is close to modern day calculation of 24900 miles.

 Aryabhata described 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.  The order of the planets in terms of distance from earth are taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.

The positions and periods of the planets were calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn move around the Earth at specific speeds representing each planet’s motion through the zodiac. Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Another element in Aryabhata’s model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.

Calendric calculations worked out by Aryabhata and followers have been in continuous use in India for the practical purposes of fixing the Panchanga, or Hindu calendar, These were also transmitted to the Islamic world, and formed the basis for the Jalali calendar introduced 1073 by a group of astronomers including Omar Khayyam, versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The Jalali calendar determines its dates based on actual solar transit, as in Aryabhata (and earlier Siddhanta calendars). This type of calendar requires an Ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were lower in the Jalali calendar than in the Gregorian calendar.

A collection by Upendra kumar.

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