Aryabhata – The Master of Universe
Aryabhata was born in 476 C.E. CE represents the alternate name of the traditional Anno Domini (A.D). It is believed that the astronomer Aryabhata was from Ashmaka. Ashmaka was one of the solasa (sixteen) mahajanapadas or Kingdoms in the 6th century BCE, a region of ancient India (700–300 BCE) as mentioned in the Buddhist text Anguttara Nikaya. The region was located on the banks of the Godavari river, between the rivers Godavari and Manjira.
Aryabhata lived in the late 5th and the early 6th centuries at Kusumapura (Patliputra a village near the city of modern Patna) coincited with the kingdom of buddhagupta, the last great ruler of Gupta Dynasty, where he wrote Āryabhaṭīya or Āryabhaṭīyaṃ, the magnum opus and only surviving work of the 5th century. The Gupta age is supposed to be the golden age of ancient Indian learning. Patliputra was a great center of learning as the university of “Nalinda” was situated there.”
There are three known works of Aryabhata, which are Aryabhatiya, Arya-siddhanta and third one in Arabic translation, is Al ntf or Al-nanf mentioned by Iranian scholar Abū Rayhān al-Bīrūnī.
1. Aryabhatiya :He composed Aryabhatiya in the year 3600 (499 C.E) in kaliyuga as according to Hindu calendar kaliyuga started in 18th February 3101 B.C.E. i.e (3600-3101)=499 C.E when he was 23 year old. It is a compendium of mathematics and astronomy , written in Sanskrit consisting of 123 verses or sutras . It is divided into four chapters that describe different results. The mathematical part of the Aryabhatiya covers Arithmetic, algebra, plane trigonometry and spherical trigonometry in continuation with fractions, quadratic equations, sums of power series and a table of sines. Let us put light on some of his extraordinary work-
Dasagitika Or Ten Giti Stanza (13 verses):
- Represents the sytem of expressing number by using letters of alphabet; It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, … , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, …. In fact the system allows numbers up to 1018to be represented with an alphabetical notation.
- It talks about revolution of sun, moon , Earth and planets in one yuga
- Division of circle; Circumference of sky and orbits of planets in yojanas; Earth moves in one kala in a prana; orbit of sun one sixtieth that of asterisms
- Aryabhata gave a systematic treatment of the position of the planets in space. Diameter of sun, moon, Earth and planets; number of years in a yuga; He gave the circumference of the earth as 4967 yojanas and its diameter as 1 5811/24 Since 1 yojana = 5 miles this gives the circumference as 24,835 miles, which is an excellent approximation to the currently accepted value of 24,902 miles.
- Greatest deviation of moon and planets from ecliptic
- Position of ascending nodes of planets, and of Apsides of sun and planets
- Dimensions of epicycles of apsides and conjunction of planets; circumference of Earth wind
- Table of sine differences
Ganitapada ( Maths ) (33 verses):
- Definition of square (Varga) and cube (Ghana) , Square root and cube root, volume of pyramid, Area of circle, volume of sphere
- Area of a triangle as” tribhujasya phalashariram samadalakoti bhujardhasamvargah” meaning is “for a triangle, the result of a perpendicular with the half-side is the area.”
- Area of Trapezium; Length of perpendiculars from intersection of diagonals to parallel sides
- Arithmetical progression; Aryabhata provided elegant results for the summation of series of squares and cubes
- Interest calculation
- Inverse method
- Names and values of classes of numbers increasing by power of 10
- Method of constructing sines by forming triangles and quadrilaterals in quadrant of circles
- Calculation of Sin-Difference Table. He gave a table of sines calculating the approximate values at intervals of 90°/24 = 3° 45′ by using 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.
- His definitions of sine(jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trigonometry.
- He was the first to specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.
- Construction of circles, triangles and quadrilateral; how to determine the horizontal and perpendicular
- Hypotenuse of Right Angle triangle formed by Gnomon ( it is the part of a sundialthat casts the shadow.) and shadow
- Hypotenuse of Right-Angle triangle; Relation of half-chord to segments of diameter which bisects chord
- Intersection of two circles
- Product of two factors half the difference between square of their sum and sum of their squares
- To find the two factors when product and difference are known
Kalakriya or The reckoning of Time (25 verses): Division of time
- Conjuction of two planets in a yuga;
- Number of revolutions of epicycles of planets; years of Jupiter
- Definition of solar year, lunar month, civil day and sidereal day
- Yuga, year, month and day began at first of caitra; Endless time measured by movements of planets and Asterism; He calculated the length of the year at 365 days 6 hours 12 minutes 30 seconds which is a bit overestimated since the true value is less than 365 days 6 hours.
- Periods of revolution differ because orbits differ in size and for the same reason signs, degrees and minutes differ in length
- Calculation of true place of planets from mean positions
- Calculation of true distance between planets and earth
- Order in which planets are arranged around the Earth
Gola or Sphere
- Zodical signs in northern and southern halves of ecliptic, even deviation of ecliptic from equator
- Sun, nodes of moon and planets, and Earth’s shadow move along ecliptic
- Distance from sun at which moon and planets become visible
- Sun illumines one half of Earth, planets and asterisms; other half is dark
- Spherical earth, surrounded by orbit of planets and by Asterisms, situated in center of space; consit of Earth, water, fire and Air
- Radius of Earth increases and decreases by yojnas during day and night of brahman
- (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 signson horizon, etc.
2. The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata’s contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.
A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.
Place value system and zero
It is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. The concept of zero as a number and not merely a symbol or an empty space for separation was first given by Aryabhata and by the 9th century AD, practical calculations were carried out using zero. Zero was treated like any other number, even in case of division. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero. In 498 AD, Aryabhata stated that “sthānāt sthānaṁ daśaguņaṁ syāt” i.e. “from place to place each is ten times the preceding,” which is the origin of the modern decimal-based place value notation.
Approximation of π
Aryabhata gave an accurate approximation for π. He wrote in the second part of Aryabhatiya (Ganitapada) the following:-
“caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ”.
“ It says “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. This implies that the ratio of the circumference to the diameter is…
=((4 + 100) × 8 + 62000)/20000 = 62832/20000 =3.14159265 ,
i.e π = 62832/20000 = 3.14159265 which is a surprisingly correct up to 8 places.
But Aryabhata does not use the accurate value for π but prefers to use √10 = 3.1622 in practice. Though Aryabhata did not explain how he found this accurate value.
Diopphantine or Indeterminate equations
Aryabhata gave the systematic methods for finding the integer solutions of Diophantine equation. The first explicit description of general integral solution of the linear Diphantine equation . ay-bx = c occured in his texts, where a and b are positive integers for the first degree equation. Aryabhata introduced the method called “ Kuttuka” (pulverization) . It means breaking the bigger problems in to small ones and the method involves a recursive algorithm for writing the original factors in smaller numbers.
Diphantus of Alexandria (250AD) first introduced the equations with integer coefficients whose solutions are to be found in integers are called Diphontine equations. He investigated the soluton of equation in rationale numbers not integers. Aryabhatta was the first to systematically investigate the methods for the determination of integral solution with integers.
Afterwards, this algorithm further elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in crytology.
- India’s first satellite Aryabhata and the lunar crater Aryabhata are named in his honour.
- Aryabhatta Knowledge University(AKU), Patna has been established by Government of Bihar for the development and management of educational infrastructure related to technical, medical, management and allied professional education in his honour. The university is governed by Bihar State University Act 2008.
- An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute of Observational Sciences (ARIES) near Nainital, India.
- The inter-school Aryabhata Maths Competition is also named after him, as isBacillus aryabhata, a species of bacteria discovered by ISRO scientists in 2009.