Aryabhata the Elder 

Born: 476 in Kusumapura (now Patna), India
Died: 550 in India Aryabhata is also known as Aryabhata I to
distinguish him from the later mathematician of the same name who
lived about 400 years later. AlBiruni has not helped in
understanding Aryabhata's life, for he seemed to believe that there
were two different mathematicians called Aryabhata living at the same
time. He therefore created a confusion of two different Aryabhatas
which was not clarified until 1926 when B Datta showed that
alBiruni's two Aryabhatas were one and the same person. We
know the year of Aryabhata's birth since he tells us that he was
twentythree years of age when he wrote Aryabhatiya which he finished
in 499. We have given Kusumapura, thought to be close to Pataliputra
(which was refounded as Patna in Bihar in 1541), as the place of
Aryabhata's birth but this is far from certain, as is even the
location of Kusumapura itself. As Parameswaran writes in [26]:
... no final verdict can be given regarding the locations of
Asmakajanapada and Kusumapura. We do know that Aryabhata
wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the
capital of the Gupta empire and a major centre of learning, but there
have been numerous other places proposed by historians as his
birthplace. Some conjecture that he was born in south India, perhaps
Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he
was born in the northeast of India, perhaps in Bengal. In [8] it is
claimed that Aryabhata was born in the Asmaka region of the Vakataka
dynasty in South India although the author accepted that he lived
most of his life in Kusumapura in the Gupta empire of the north.
However, giving Asmaka as Aryabhata's birthplace rests on a comment
made by Nilakantha Somayaji in the late 15th century. It is now
thought by most historians that Nilakantha confused Aryabhata with
Bhaskara I who was a later commentator on the Aryabhatiya.
We should note that Kusumapura became one of the two major
mathematical centres of India, the other being Ujjain. Both are in
the north but Kusumapura (assuming it to be close to Pataliputra) is
on the Ganges and is the more northerly. Pataliputra, being the
capital of the Gupta empire at the time of Aryabhata, was the centre
of a communications network which allowed learning from other parts
of the world to reach it easily, and also allowed the mathematical
and astronomical advances made by Aryabhata and his school to reach
across India and also eventually into the Islamic world. As
to the texts written by Aryabhata only one has survived. However Jha
claims in [21] that: ... Aryabhata was an author of at
least three astronomical texts and wrote some free stanzas as well.
The surviving text is Aryabhata's masterpiece the Aryabhatiya which
is a small astronomical treatise written in 118 verses giving a
summary of Hindu mathematics up to that time. Its mathematical
section contains 33 verses giving 66 mathematical rules without
proof. The Aryabhatiya contains an introduction of 10 verses,
followed by a section on mathematics with, as we just mentioned, 33
verses, then a section of 25 verses on the reckoning of time and
planetary models, with the final section of 50 verses being on the
sphere and eclipses. There is a difficulty with this layout
which is discussed in detail by van der Waerden in [35]. Van der
Waerden suggests that in fact the 10 verse Introduction was written
later than the other three sections. One reason for believing that
the two parts were not intended as a whole is that the first section
has a different meter to the remaining three sections. However, the
problems do not stop there. We said that the first section had ten
verses and indeed Aryabhata titles the section Set of ten giti
stanzas. But it in fact contains eleven giti stanzas and two arya
stanzas. Van der Waerden suggests that three verses have been added
and he identifies a small number of verses in the remaining sections
which he argues have also been added by a member of Aryabhata's
school at Kusumapura. 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. Let
us examine some of these in a little more detail. First we
look at the system for representing numbers which Aryabhata invented
and used in the Aryabhatiya. 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. Ifrah in [3] argues that Aryabhata was also
familiar with numeral symbols and the placevalue system. He writes
in [3]: ... it is extremely likely that Aryabhata knew the
sign for zero and the numerals of the place value system. 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 placevalue 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 placevalue
system and zero. Next we look briefly at some algebra
contained in the Aryabhatiya. This work is the first we are aware of
which examines integer solutions to equations of the form by = ax + c
and by = ax  c, where a, b, c are integers. The problem arose from
studying the problem in astronomy of determining the periods of the
planets. Aryabhata uses the kuttaka method to solve problems of this
type. The word kuttaka means "to pulverise" and the method
consisted of breaking the problem down into new problems where the
coefficients became smaller and smaller with each step. The method
here is essentially the use of the Euclidean algorithm to find the
highest common factor of a and b but is also related to continued
fractions. Aryabhata gave an accurate approximation for p.
He wrote in the Aryabhatiya the following: Add four to one
hundred, multiply by eight and then add sixtytwo 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 gives p = 62832/20000 = 3.1416 which
is a surprisingly accurate value. In fact p = 3.14159265 correct to 8
places. If obtaining a value this accurate is surprising, it is
perhaps even more surprising that Aryabhata does not use his accurate
value for p but prefers to use v10 = 3.1622 in practice. Aryabhata
does not explain how he found this accurate value but, for example,
Ahmad [5] considers this value as an approximation to half the
perimeter of a regular polygon of 256 sides inscribed in the unit
circle. However, in [9] Bruins shows that this result cannot be
obtained from the doubling of the number of sides. Another
interesting paper discussing this accurate value of p by Aryabhata is
[22] where Jha writes: Aryabhata I's value of p is a very
close approximation to the modern value and the most accurate among
those of the ancients. There are reasons to believe that Aryabhata
devised a particular method for finding this value. It is shown with
sufficient grounds that Aryabhata himself used it, and several later
Indian mathematicians and even the Arabs adopted it. The conjecture
that Aryabhata's value of p is of Greek origin is critically examined
and is found to be without foundation. Aryabhata discovered this
value independently and also realised that p is an irrational number.
He had the Indian background, no doubt, but excelled all his
predecessors in evaluating p. Thus the credit of discovering this
exact value of p may be ascribed to the celebrated mathematician,
Aryabhata I. We now look 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
(n1)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, but the formulae for the
volumes of a sphere and of a pyramid are claimed to be wrong by most
historians. For example Ganitanand in [15] describes as
"mathematical lapses" the fact that Aryabhata gives the
incorrect formula V = Ah/2 for the volume of a pyramid with height h
and triangular base of area A. He also appears to give an incorrect
expression for the volume of a sphere. However, as is often the case,
nothing is as straightforward as it appears and Elfering (see for
example [13]) argues that this is not an error but rather the result
of an incorrect translation. This relates to verses 6, 7,
and 10 of the second section of the Aryabhatiya and in [13] Elfering
produces a translation which yields the correct answer for both the
volume of a pyramid and for a sphere. However, in his translation
Elfering translates two technical terms in a different way to the
meaning which they usually have. Without some supporting evidence
that these technical terms have been used with these different
meanings in other places it would still appear that Aryabhata did
indeed give the incorrect formulae for these volumes. We
have looked at the mathematics contained in the Aryabhatiya but this
is an astronomy text so we should say a little regarding the
astronomy which it contains. Aryabhata gives a systematic treatment
of the position of the planets in space. He gave the circumference of
the earth as 4 967 yojanas and its diameter as 1 5811/24 yojanas.
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. He believed that the apparent rotation of the
heavens was due to the axial rotation of the Earth. This is a quite
remarkable view of the nature of the solar system which later
commentators could not bring themselves to follow and most changed
the text to save Aryabhata from what they thought were stupid errors!
Aryabhata gives the radius of the planetary orbits in terms
of the radius of the Earth/Sun orbit as essentially their periods of
rotation around the Sun. He believes that the Moon and planets shine
by reflected sunlight, incredibly he believes that the orbits of the
planets are ellipses. He correctly explains the causes of eclipses of
the Sun and the Moon. The Indian belief up to that time was that
eclipses were caused by a demon called Rahu. His value for the length
of the year at 365 days 6 hours 12 minutes 30 seconds is an
overestimate since the true value is less than 365 days 6 hours.
Bhaskara I who wrote a commentary on the Aryabhatiya about 100 years
later wrote of Aryabhata: Aryabhata is the master who,
after reaching the furthest shores and plumbing the inmost depths of
the sea of ultimate knowledge of mathematics, kinematics and
spherics, handed over the three sciences to the learned world.
Article by: J J O'Connor and E F Robertson Source:
www.history.mcs.standrews.ac.uk/Mathematicians



