Aryabhata
Biography
Name
While there is a tendency to misspell his name as
"Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical
text spells his name thus,[6] including Brahmagupta's references to him "in more than a
hundred places by name".[7] Furthermore, in most instances
"Aryabhatta" does not fit the metre either.[6]
Time and place of birth
Aryabhata mentions in the Aryabhatiya that it was composed
3,630 years into the Kali Yuga,
when he was 23 years old. This corresponds to 499 CE, and implies that he was
born in 476.[4]
Aryabhata's birthplace is uncertain, but it may have been in the
area known in ancient texts as Ashmaka India which may have beenMaharashtra or Dhaka.[6]
Education
It is fairly certain that, at some point, he went to Kusumapura
for advanced studies and lived there for some time.[8] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[6] A verse mentions that Aryabhata was the head of
an institution (kulapati) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an
astronomical observatory, it is speculated that Aryabhata might have been the
head of the Nalanda university as well.[6] Aryabhata is also reputed to have set up an
observatory at the Sun temple in Taregana,
Bihar.[9]
Other hypotheses
Some archeological evidence suggests that Aryabhata could have
originated from the present day Kodungallur which was the historical
capital city of Thiruvanchikkulam of ancient Kerala.[10] For instance, one hypothesis was that aśmaka (Sanskrit for "stone") may be the region in Kerala that
is now known as Koṭuṅṅallūr, based on the belief that it was earlier
known as Koṭum-Kal-l-ūr ("city of hard stones");
however, old records show that the city was actually Koṭum-kol-ūr
("city of strict governance"). Similarly, the fact that several
commentaries on the Aryabhatiya have come from Kerala were used to suggest that
it was Aryabhata's main place of life and activity; however, many commentaries
have come from outside Kerala.
Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a
point on the equator at the same longitude as his Ujjayini.[11]
Works
His major work, Aryabhatiya, a compendium of mathematics and astronomy, was
extensively referred to in the Indian mathematical literature and has survived
to modern times. 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.
The Arya-siddhanta, a lot 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.[12]
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
Chanakya
Chanakya 
pronunciation (help·info) (c.
370–283 BCE)[3] was
an Indian teacher, philosopher and royal advisor.
pronunciation (help·info) (c.
370–283 BCE)[3] was
an Indian teacher, philosopher and royal advisor.
Originally a professor of
economics and political science at the ancient Takshashila University, Chanakya
managed the first Maurya emperor Chandragupta's rise to power at a young age. He is widely
credited for having played an important role in the establishment of the Maurya
Empire, which was the first empire in archaeologically recorded
history to rule
most of the Indian subcontinent. Chanakya served as the chief advisor to both
Chandragupta and his sonBindusara.
Chanakya is traditionally
identified as Kautilya or Vishnu
Gupta, who authored the ancient Indian political treatise called Arthaśāstra.[4] As such, he is considered as the pioneer of the field of economics
and political
science in India,
and his work is thought of as an important precursor to Classical Economics.[5][6][7][8] Chanakya is often called the "Indian Machiavelli",[9]although
his works predate Machiavelli's by about 1,800 years.[10] His works were lost near the end of the Gupta
dynasty and not
rediscovered until 1915.[6]
Birth [edit]
Chanakya's birthplace is a matter
of controversy, and there are multiple theories about his origin.[1] According to the Buddhist textMahavamsa Tika, his
birthplace was Taxila.[12] The Jain scriptures, such as Adbidhana
Chintamani, mention him as a Dramila,
implying that he was a native of South
India.[12][13] According to the Jain writer Hemachandra's Parishishtaparva,
Chanakya was born in the Canaka village of the Golla region, to a Jain
Brahmin named
Canin and his wife Canesvari.[14] Other sources mention his father's name as Chanak and state that
Chanakaya's name derives from his father's name.
Early life [edit]
Chanakya was educated at Takshashila, an
ancient centre of learning located in north-western ancient India (present-day
Pakistan).[20] He later became a teacher (acharya) at the same place.
Chanakya's life was connected to two cities: Takshashila and Pataliputra (present-dayPatna in Bihar,
India). Pataliputra was the capital of the Magadha kingdom, which was connected to Takshashila by the
northern high road of commerce.
Death [edit]
The real cause of Chanakya's
death is unknown and disputed. According to one legend, he retired to the jungle and starved himself to death.[28] According to another legend mentioned by the Jain writer Hemachandra, Subandhu, one of Bindusara's ministers, did
not like Chanakya. One day he told Bindusara that Chanakya was responsible for
the murder of his mother. Bindusara asked the nurses, who confirmed the story
of his birth. Bindusara was horrified and enraged. Chanakya, who was an old man
by this time, learnt that the King was angry with him, he decided to end his
life. In accordance with the Jain tradition, he decided to starve himself to
death. By this time, the King learnt the full story: Chanakya was not directly
responsible for his mother's death, which was an accident. He asked Subandhu to
convince Chanakya to give up his plan to kill himself. However, Subandhu,
pretending to conduct a ceremony for Chanakya, burnt Chanakya alive.[29]
Literary works [edit]
The Arthashastra discusses monetary and fiscal policies, welfare, international relations, and war
strategies in detail.
The text also outlines the duties of a ruler.[30] Some scholars believe that Arthashastra is actually a compilation of a number
of earlier texts written by various authors, and Chanakya might have been one of these authors.[11]
Neetishastra is a treatise on the ideal way of life, and shows Chanakya's deep
study of the Indian
way of life. Chanakya also developedNeeti-Sutras (aphorisms – pithy sentences) that
tell people how they should behave. Of these well-known 455 sutras, about 216
refer to raja-neeti (the dos and don'ts of running a
kingdom). Apparently, Chanakya used these sutras to groom Chandragupta and
other selected disciples in the art of ruling a kingdom.
Srinivasa
Ramanujan
Srinivasa Ramanujan FRS ( pronunciation (help·info)) (Tamil: ஸ்ரீனிவாஸ ராமானுஜன்; 22 December 1887 – 26 April 1920) was
an Indian mathematician and autodidact who, with almost no formal training in pure
mathematics, made extraordinary contributions tomathematical analysis, number
theory, infinite
series, and continued fractions. Living in India with no access
to the larger mathematical community, which was centred in Europe at the time,
Ramanujan developed his own mathematical research in isolation. As a result, he
sometimes rediscovered known theorems in addition to producing new work.
Ramanujan was said to be a natural genius by the English mathematician G. H.
Hardy, in the same league as mathematicians such as Euler and Gauss.[1] He died at the age of 32.
Born at Erode, Madras
Presidency (now Tamil
Nadu) in a Tamil Brahmin family of ThenkalaiIyengar
sect[2][3][4] Ramanujan's introduction to formal mathematics began at age 10. He demonstrated a natural ability, and was given
books on advanced trigonometry written by S. L.
Loney that he
mastered by the age of 12; he even discovered theorems of his own, and re-discovered Euler's
identity independently.[5] He demonstrated unusual mathematical skills at school, winning
accolades and awards. By 17, Ramanujan had conducted his own mathematical
research on Bernoulli
numbers and the Euler–Mascheroni constant.
Ramanujan received a scholarship
to study at Government College in Kumbakonam, which
was later rescinded when he failed his non-mathematical coursework. He joined
another college to pursue independent mathematical research, working as a clerk
in the Accountant-General's office at the Madras Port Trust Office to support himself.[6] In 1912–1913, he sent samples of his theorems to three academics
at the University of Cambridge. G. H.
Hardy, recognizing the brilliance of his work, invited Ramanujan to
visit and work with him at Cambridge. He
became aFellow of the Royal Society and a Fellow of Trinity College, Cambridge.
Ramanujan died of illness, malnutrition, and possibly liver infection in 1920
at the age of 32.
During his short lifetime,
Ramanujan independently compiled nearly 3900 results (mostlyidentities and equations).[7] Nearly all his claims have now been proven correct, although a
small number of these results were actually false and some were already known.[8] He stated results that were both original and highly
unconventional, such as the Ramanujan
prime and theRamanujan theta function, and these have inspired a vast
amount of further research.[9]However,
the mathematical mainstream has been rather slow in absorbing some of his major
discoveries. The Ramanujan
Journal, an international publication, was launched to publish work in all
areas of mathematics influenced by his work.[10]
In December 2011, in recognition
of his contribution to mathematics, the Government of India declared that
Ramanujan's birthday (22 December) should be celebrated every year asNational Mathematics Day, and also declared 2012 the National Mathematics Year
Brahmagupta
Brahmagupta (Sanskrit: ब्रह्मगुप्त; listen (help·info)) (597–668
AD) was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His
best known work is the Brāhmasphuṭasiddhānta (Correctly Established Doctrine of Brahma), written in 628 in Bhinmal. Its 25
chapters contain several unprecedented mathematical results.
Brahmagupta was the first to use
zero as a number. He gave rules to compute with zero.
Brahmagupta used negative numbers and zero for computing. The modern rule that
two negative numbers multiplied together equals a positive number first appears
in Brahmasputa siddhanta. It is composed in elliptic verse, as was common
practice in Indian mathematics, and consequently has a poetic
ring to it. As no proofs are given, it is not known how Brahmagupta's
mathematics was derived. Life and work [edit]
Brahmagupta is believed to have
been born in 598 AD in Bhinmal city in the state of Rajasthan of Northwest India. In ancient times Bhillamala was the seat of
power of the Gurjars. His
father was Jisnugupta.[2] He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) during
the reign (and possibly under the patronage) of King Vyaghramukha.[3] As a result, Brahmagupta is often referred to as Bhillamalacharya,
that is, the teacher from Bhillamala. He was the head of the astronomical
observatory at Ujjain, and
during his tenure there wrote four texts on mathematics and astronomy: the Cadamekela in 624, the Brahmasphutasiddhanta in 628, the Khandakhadyaka in 665, and the Durkeamynarda in 672. The Brahmasphutasiddhanta (Corrected Treatise of Brahma)
is arguably his most famous work. The historian al-Biruni (c. 1050) in his book Tariq
al-Hind states that the Abbasid caliph al-Ma'mun had an embassy in India and from India a book was brought to
Baghdad which was translated into Arabic as Sindhind.
It is generally presumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta.[4]
Although Brahmagupta was familiar
with the works of astronomers following the tradition of Aryabhatiya, it is
not known if he was familiar with the work of Bhaskara
I, a contemporary.[3] Brahmagupta had a plethora of criticism directed towards the work
of rival astronomers, and in hisBrahmasphutasiddhanta is found one of the earliest attested
schisms among Indian mathematicians. The division was primarily about the
application of mathematics to the physical world, rather than about the
mathematics itself. In Brahmagupta's case, the disagreements stemmed largely
from the choice of astronomical parameters and theories.[3] Critiques of rival theories appear throughout the first ten
astronomical chapters and the eleventh chapter is entirely devoted to criticism
of these theories, although no criticisms appear in the twelfth and eighteenth
chapters.[3]
Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,
The difference between rupas, when inverted and divided by the difference of the unknowns, is
the unknown in the equation. The rupas are [subtracted on the side] below that from
which the square and the unknown are to be subtracted.[5]
which is a solution equivalent to 
, where rupas represents constants. He
further gave two equivalent solutions to the generalquadratic equation,
, where rupas represents constants. He
further gave two equivalent solutions to the generalquadratic equation,
18.44. Diminish by the
middle [number] the square-root of the rupas multiplied by four times
the square and increased by the square of the middle [number]; divide the
remainder by twice the square. [The result is] the middle [number].
18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.[5]
18.45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown.[5]
which are, respectively, solutions equivalent to,
and
He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then
the equation must be divided by the desired variable's coefficient. In particular, he recommended using "the
pulverizer" to solve equations with multiple unknowns.
18.51. Subtract the colors different from the first color. [The
remainder] divided by the first [color's coefficient] is the measure of the
first. [Terms] two by two [are] considered [when reduced to] similar divisors,
[and so on] repeatedly. If there are many [colors], the pulverizer [is to be used].[5]
Like the algebra of Diophantus,
the algebra of Brahmagupta was syncopated. Addition was indicated by placing
the numbers side by side, subtraction by placing a dot over the subtrahend, and
division by placing the divisor below the dividend, similar to our notation but
without the bar. Multiplication, evolution, and unknown quantities were
represented by abbreviations of appropriate terms.[6] The extent of Greek influence on this syncopation,
if any, is not known and it is possible that both Greek and Indian syncopation
may be derived from a common Babylonian source.[6]
Four fundamental operations (addition, subtraction, multiplication
and division) were known to many cultures before Brahmagupta. This current
system is based on the Hindu Arabic number system and first appeared in
Brahmasputa siddhanta. Brahmagupta describes the multiplication as thus “The
multiplicand is repeated like a string for cattle, as often as there are
integrant portions in the multiplier and is repeatedly multiplied by them and
the products are added together. It is multiplication. Or the multiplicand is
repeated as many times as there are component parts in the multiplier”. [7] Indian arithmetic was known in Medieval Europe
as "Modus Indoram" meaning method of the Indians. In BrahmasputhaSiddhanta,
Multiplication was named Gomutrika. In the beginning of chapter twelve of his Brahmasphutasiddhanta, entitled Calculation,
Brahmagupta details operations on fractions. The reader is expected to know the
basic arithmetic operations as far as taking the square root, although he
explains how to find the cube and cube-root of an integer and later gives rules
facilitating the computation of squares and square roots. He then gives rules
for dealing with five types of combinations of fractions, 
, 
, 
, 
, and 
.[8]
, , , , and .[8]
Brahmagupta then goes on to give the sum of the squares and cubes
of the first n integers.
12.20. The sum of the squares is that [sum] multiplied by twice
the [number of] step[s] increased by one [and] divided by three. The sum of the
cubes is the square of that [sum] Piles of these with identical balls [can also
be computed].[9]
Here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice.[10]
He gives the sum of the squares of the first n natural numbers as
n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as
(n(n+1)/2)².
Brahmagupta's Brahmasphuṭasiddhanta is the first book that
mentions zero as a number, hence Brahmagupta is considered the first to
formulate the concept of zero. He gave rules of using zero with negative and
positive numbers. Zero plus a positive number is the positive number and negative
number plus zero is a negative number etc. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own
right, rather than as simply a placeholder digit in representing another number
as was done by the Babylonians or as a symbol for a
lack of quantity as was done by Ptolemy and the Romans. In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative
numbers. He first describes addition and subtraction,
18.30. [The sum] of two positives is positives, of two negatives
negative; of a positive and a negative [the sum] is their difference; if they
are equal it is zero. The sum of a negative and zero is negative, [that] of a
positive and zero positive, [and that] of two zeros zero.
[...]
18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.[5]
[...]
18.32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added.[5]
He goes on to describe multiplication,
18.33. The product of a negative and a positive is negative, of
two negatives positive, and of positives positive; the product of zero and a
negative, of zero and a positive, or of two zeros is zero.[5]
18.34. A positive divided by a positive or a negative divided by a
negative is positive; a zero divided by a zero is zero; a positive divided by a
negative is negative; a negative divided by a positive is [also] negative.
18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.[5]
18.35. A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root.[5]
Here Brahmagupta states that 
and as for the question of 
where 
he did not commit himself.[11] His rules for arithmetic onnegative numbers and zero are quite close to the modern
understanding, except that in modern mathematics division by zero is leftundefined.
and as for the question of where he did not commit himself.[11] His rules for arithmetic onnegative numbers and zero are quite close to the modern
understanding, except that in modern mathematics division by zero is leftundefined.
12.39. The height of a mountain multiplied by a given multiplier
is the distance to a city; it is not erased. When it is divided by the
multiplier increased by two it is the leap of one of the two who make the same
journey.[9]
or in other words, for a given length m and an arbitrary multiplier x, let a = mx and b = m + mx/(x + 2). Then m, a, and b form a Pythagorean triple.[9]
Brahmagupta went on to give a recurrence relation for generating
solutions to certain instances of Diophantine equations of the second degree
such as 
(called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the
"pulverizer" since it breaks numbers down into ever smaller pieces.[12]
(called Pell's equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the
"pulverizer" since it breaks numbers down into ever smaller pieces.[12]
The nature of squares:
18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.[5]
18.64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed.
18.65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas.[5]
The key to his solution was the identity,[13]
Using his identity and the fact that if 
and 
are solutions to the equations 
and 
, respectively, then 
is a solution to 
, he was able to find integral solutions to the
Pell's equation through a series of equations of the form 
. Unfortunately, Brahmagupta was not able to
apply his solution uniformly for all possible values of N, rather he was only able to show that if 
has an integral solution for k = ±1, ±2, or ±4,
then 
has a solution. The solution of the general
Pell's equation would have to wait for Bhaskara II in c. 1150 CE.[13]
and are solutions to the equations and , respectively, then is a solution to , he was able to find integral solutions to the
Pell's equation through a series of equations of the form . Unfortunately, Brahmagupta was not able to
apply his solution uniformly for all possible values of N, rather he was only able to show that if has an integral solution for k = ±1, ±2, or ±4,
then has a solution. The solution of the general
Pell's equation would have to wait for Bhaskara II in c. 1150 CE.[13]
Diagram for reference
Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. Given the lengths of the sides of any cyclic quadrilateral,
Brahmagupta gave an approximate and an exact formula for the figure's area,
12.21. The approximate area is the product of
the halves of the sums of the sides and opposite sides of a triangle and a
quadrilateral. The accurate [area] is the square root from the product of the
halves of the sums of the sides diminished by [each] side of the quadrilateral.[9]
So given the lengths p, q, r and s of a cyclic
quadrilateral, the approximate area is 
while, letting 
, the exact area is
while, letting , the exact area is
Although Brahmagupta does not explicitly state that these
quadrilaterals are cyclic, it is apparent from his rules that this is the case.[14]Heron's formula is a special case of this formula and it can be
derived by setting one of the sides equal to zero.
Brahmagupta dedicated a substantial portion of his work to
geometry. One theorem states that the two lengths of a triangle's base when
divided by its altitude then follows,
12.22. The base decreased and increased by the
difference between the squares of the sides divided by the base; when divided
by two they are the true segments. The perpendicular [altitude] is the
square-root from the square of a side diminished by the square of its segment.[9]
Thus the lengths of the two segments are 
.
.
He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form:
Brahmagupta's theorem
states that AF =FD.
Brahmagupta continues,
12.23. The square-root
of the sum of the two products of the sides and opposite sides of a non-unequal
quadrilateral is the diagonal. The square
of the diagonal is
diminished by the square of half the sum of the base and the top; the
square-root is the perpendicular [altitudes].[9]
So, in a "non-unequal" cyclic quadrilateral (that is, an
isosceles trapezoid),
the length of each diagonal is 
.
.
He continues to give formulas for the lengths and areas of
geometric figures, such as the circumradius of an isosceles trapezoid and a
scalene quadrilateral, and the lengths of diagonals in a scalene cyclic
quadrilateral. This leads up to Brahmagupta's famous theorem,
12.30-31. Imaging two
triangles within [a cyclic quadrilateral] with unequal sides, the two diagonals
are the two bases. Their two segments are separately the upper and lower
segments [formed] at the intersection of the diagonals. The two [lower
segments] of the two diagonals are two sides in a triangle; the base [of the
quadrilateral is the base of the triangle]. Its perpendicular is the lower
portion of the [central] perpendicular; the upper portion of the [central]
perpendicular is half of the sum of the [sides] perpendiculars diminished by
the lower [portion of the central perpendicular].[9]
12.40. The diameter and
the square of the radius [each] multiplied by 3 are [respectively] the
practical circumference and the area [of a circle]. The accurate [values] are
the square-roots from the squares of those two multiplied by ten.[9]
So Brahmagupta uses 3 as a "practical" value of π, and 
as an "accurate" value of π.
as an "accurate" value of π.
In some of the verses before verse 40, Brahmagupta gives
constructions of various figures with arbitrary sides. He essentially
manipulated right triangles to produce isosceles triangles, scalene triangles,
rectangles, isosceles trapezoids, isosceles trapezoids with three equal sides,
and a scalene cyclic quadrilateral.
After giving the value of pi, he deals with the geometry of plane
figures and solids, such as finding volumes and surface areas (or empty spaces
dug out of solids). He finds the volume of rectangular prisms, pyramids, and
the frustum of a square pyramid. He further finds the average depth of a series
of pits. For the volume of a frustum of a pyramid, he gives
the "pragmatic" value as the depth times the square of the mean of
the edges of the top and bottom faces, and he gives the "superficial"
volume as the depth times their mean area.[16]
In Chapter 2 of his Brahmasphutasiddhanta, entitled Planetary True Longitudes, Brahmagupta presents a sine table:
2.2-5. The sines: The
Progenitors, twins; Ursa Major, twins, the Vedas; the gods, fires, six;
flavors, dice, the gods; the moon, five, the sky, the moon; the moon, arrows,
suns [...][17]
Here Brahmagupta uses names of objects to represent the digits of
place-value numerals, as was common with numerical data in Sanskrit treatises.
Progenitors represents the 14 Progenitors ("Manu") in Indian
cosmology or 14, "twins" means 2, "Ursa Major" represents
the seven stars of Ursa Major or 7, "Vedas" refers to the 4 Vedas or
4, dice represents the number of sides of the tradition die or 6, and so on.
This information can be translated into the list of sines, 214, 427, 638, 846,
1051, 1251, 1446, 1635, 1817, 1991, 2156, 2312, 1459, 2594, 2719, 2832, 2933,
3021, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270.[18]
In 665 Brahmagupta devised and used a special case of the
Newton–Stirling interpolation formula of the second-order to interpolate new values of the sine function from other
values already tabulated.[19] The formula gives an estimate for the value of a
function 
at a valuea + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when
its value is already known at a − h, a and a + h.
at a valuea + xh of its argument (with h > 0 and −1 ≤ x ≤ 1) when
its value is already known at a − h, a and a + h.
The formula for the estimate is:
It was through the Brahmasphutasiddhanta that the Arabs learned of Indian astronomy.[20] Edward Saxhau stated that "Brahmagupta, it
was he who taught Arabs astronomy",[21] The famous Abbasid caliph Al-Mansur (712–775) founded Baghdad, which is situated on the banks of the Tigris, and made it a center of learning. The caliph invited a scholar
of Ujjain by the name of Kankah in 770 A.D. Kankah used theBrahmasphutasiddhanta to explain the Hindu system of arithmetic
astronomy. Muhammad al-Fazari translated Brahmugupta's work into Arabic upon the request of the
caliph.
In chapter seven of his Brahmasphutasiddhanta, entitled Lunar Crescent, Brahmagupta rebuts the idea that the Moon is
farther from the Earth than the Sun, an idea which is maintained in scriptures.
He does this by explaining the illumination of the Moon by the Sun.[22]
7.1. If the moon were
above the sun, how would the power of waxing and waning, etc., be produced from
calculation of the [longitude of the] moon? the near half [would be] always
bright.
7.2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.
7.3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...][23]
7.2. In the same way that the half seen by the sun of a pot standing in sunlight is bright, and the unseen half dark, so is [the illumination] of the moon [if it is] beneath the sun.
7.3. The brightness is increased in the direction of the sun. At the end of a bright [i.e. waxing] half-month, the near half is bright and the far half dark. Hence, the elevation of the horns [of the crescent can be derived] from calculation. [...][23]
He explains that since the Moon is closer to the Earth than the
Sun, the degree of the illuminated part of the Moon depends on the relative
positions of the Sun and the Moon, and this can be computed from the size of the
angle between the two bodies.[22]
Some of the important contributions made by Brahmagupta in
astronomy are: methods for calculating the position of heavenly bodies over
time (ephemerides), their rising and setting, conjunctions,
and the calculation of solar and lunar eclipses.[24] Brahmagupta criticized thePuranic view that the Earth was flat or hollow. Instead,
he observed that the Earth and heaven were spherical and that the Earth is
moving. In 1030, the Muslim astronomer Abu al-Rayhan al-Biruni, in his Ta'rikh al-Hind, later translated into Latin as Indica, commented on Brahmagupta's work and wrote that
critics argued:
According to al-Biruni, Brahmagupta responded to these criticisms
with the following argument on gravitation:
"On the contrary,
if that were the case, the earth would not vie in keeping an even and uniform
pace with the minutes of heaven, the pranas of the times. [...] All heavy things are attracted towards the
center of the earth. [...] The earth on all its sides is the same; all people
on earth stand upright, and all heavy things fall down to the earth by a law of
nature, for it is the nature of the earth to attract and to keep things, as it
is the nature of water to flow, that of fire to burn, and that of wind to set
in motion... The earth is the only low thing, and seeds always return to it, in
whatever direction you may throw them away, and never rise upwards from the
earth."[26]
About the Earth's gravity he said: "Bodies fall towards the
earth as it is in the nature of the earth to attract bodies, just as it is in
the nature of water to flow."[27]
Euclid
Euclid (/ˈjuːklɪd/ ewk-lid; Ancient
Greek: Εὐκλείδης Eukleidēs), fl. 300 BC, also known asEuclid of Alexandria, was a Greek
mathematician, often referred to as the "Father of Geometry". He was
active in Alexandria during the reign of Ptolemy I (323–283 BC). HisElements is one of
the most influential works in the history of mathematics, serving as the main textbook
for teaching mathematics (especially geometry) from the
time of its publication until the late 19th or early 20th century.[1][2][3] In the Elements,
Euclid deduced the principles of what is now called Euclidean geometry from a
small set of axioms. Euclid
also wrote works onperspective, conic
sections, spherical geometry, number
theory and rigor.
"Euclid" is the
anglicized version of the Greek name Εὐκλείδης, meaning "Good Glory"
Life
Little is known about Euclid's
life, as there are only a handful of references to him. The date and place of
Euclid's birth and the date and circumstances of his death are unknown, and
only roughly estimated in proximity to contemporary figures mentioned in
references. The few historical references to Euclid were written centuries
after he lived, by Proclus and Pappus of Alexandria.[5] Proclus introduces Euclid only briefly in his fifth-century Commentary on the Elements, as
the author of Elements,
that he was mentioned by Archimedes, and that
when King
Ptolemy asked if
there was a shorter path to learning geometry than Euclid's Elements, "Euclid replied
there is no royal road to geometry."[6] Although the purported citation of Euclid by Archimedes has been
judged to be an interpolation by later editors of his works, it is still believed
that Euclid wrote his works before those of Archimedes.[7][8][9] In addition, the "royal road" anecdote is questionable
since it is similar to a story told about Menaechmus and Alexander the Great.[10] In the only other key reference to Euclid, Pappus briefly
mentioned in the fourth century that Apollonius "spent
a very long time with the pupils of Euclid at Alexandria, and it was thus that
he acquired such a scientific habit of thought. Elements
One of the oldest surviving fragments of Euclid's Elements, found at Oxyrhynchus and dated to circa AD 100 (P. Oxy. 29). The diagram accompanies Book II, Proposition
5.[12]
Although many of the results in Elements originated with earlier
mathematicians, one of Euclid's accomplishments was to present them in a
single, logically coherent framework, making it easy to use and easy to
reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23
centuries later.[13]
There is no mention of Euclid in the earliest remaining copies of
the Elements, and most of the copies say they are "from the edition of Theon" or the "lectures of Theon",[14] while the text considered to be primary, held by
the Vatican, mentions no author. The only reference that historians rely on of
Euclid having written the Elements was from Proclus, who briefly in his Commentary on the Elements ascribes Euclid as its author.
Although best known for its geometric results, the Elements also includes number theory. It considers the connection between perfect numbers and Mersenne primes, the infinitude of prime numbers, Euclid's lemma on factorization (which leads to the fundamental
theorem of arithmetic on uniqueness of prime factorizations), and the Euclidean algorithm for finding the greatest common divisor of two numbers.
The geometrical system described in the Elements was long known simply as geometry, and was considered to be the only geometry possible. Today,
however, that system is often referred to as Euclidean geometry to distinguish it from other so-called non-Euclidean
geometries that mathematicians
discovered in the 19th century.
Other works
Construction of a dodecahedron basing on a cube
In addition to the Elements, at least five works of Euclid have survived to
the present day. They follow the same logical structure as Elements, with definitions and proved propositions.
·
Data deals with the nature and implications of "given"
information in geometrical problems; the subject matter is closely related to
the first four books of the Elements.
·
On
Divisions of Figures, which survives only
partially in Arabic translation, concerns the division of
geometrical figures into two or more equal parts or into parts in given ratios. It is similar to a third century AD work by Heron of Alexandria.
·
Catoptrics, which concerns the mathematical theory of
mirrors, particularly the images formed in plane and spherical concave mirrors.
The attribution is held to be anachronistic however by J J O'Connor and E F
Robertson who name Theon of Alexandria as a more likely author.[15]
·
Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.
·
Optics is the earliest surviving Greek treatise on perspective. In its
definitions Euclid follows the Platonic tradition that vision is caused by discrete rays which
emanate from the eye. One important
definition is the fourth: "Things seen under a greater angle appear
greater, and those under a lesser angle less, while those under equal angles
appear equal." In the 36 propositions that follow, Euclid relates the
apparent size of an object to its distance from the eye and investigates the
apparent shapes of cylinders and cones when viewed from different angles.
Proposition 45 is interesting, proving that for any two unequal magnitudes,
there is a point from which the two appear equal. Pappus believed these results to be important in
astronomy and included Euclid's Optics, along with his Phaenomena,
in the Little Astronomy, a compendium of smaller works to be studied
before the Syntaxis (Almagest) of Claudius Ptolemy.
Other works are credibly attributed to Euclid, but have been lost.
·
Conics was a work on conic sections that was later extended by Apollonius of Perga into his famous work on the subject. It is
likely that the first four books of Apollonius's work come directly from
Euclid. According to Pappus, "Apollonius, having completed Euclid's four
books of conics and added four others, handed down eight volumes of conics."
The Conics of Apollonius quickly supplanted the former work, and by the time of
Pappus, Euclid's work was already lost.
·
Porisms might have been an outgrowth of Euclid's work
with conic sections, but the exact meaning of the title is controversial.
·
Surface
Loci concerned either loci (sets of points) on surfaces or loci which were
themselves surfaces; under the latter interpretation, it has been hypothesized
that the work might have dealt with quadric surfaces.
·
Several works on mechanics are attributed to Euclid by Arabic sources. On the Heavy and the Light contains, in nine definitions and five
propositions, Aristotelian notions of moving bodies and the concept of specific
gravity. On the Balance treats the theory of the lever in a similarly
Euclidean manner, containing one definition, two axioms, and four propositions.
A third fragment, on the circles described by the ends of a moving lever,
contains four propositions. These three works complement each other in such a
way that it has been suggested that they are remnants of a single treatise on
mechanics written by Euclid.
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