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.
