Brahmagupta - Citizendia

Brahmagupta (listen (help·info)) (598–668) was an Indian mathematician and astronomer. Events Battle of Catraeth at Catterick North Yorkshire: The Celtic British (Brythonic people defeat the Anglo-Saxon Bernicians (approximate Events By Place Europe Childeric II succeeds Clotaire III as King of the Franks. India, officially the Republic of India (भारत गणराज्य inc-Latn Bhārat Gaṇarājya; see also other Indian languages) is a country Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. Indian astronomy —the earliest textual mention of which is given in the religious literature of India (2nd millennium BCE—became an established tradition by the 1st millennium BCE

Life and work

Brahmagupta was born in 598 CE in Bhinmal city in the state of Rajasthan of northwest India. WikipediaWikiProject Indian cities for details --> Bhinmal (भीनमाल also known as Shrimal, is a town in the Jalore District of Rajasthan Rājasthān ( Devanāgarī: राजस्थान raːdʒəst̪ʰaːn is the largest state of the Republic of India in terms of area He likely lived most of his life in Bhillamala (modern Bhinmal in Rajasthan) in the empire of Harsha during the reign (and possibly under the patronage) of King Vyaghramukha. WikipediaWikiProject Indian cities for details --> Bhinmal (भीनमाल also known as Shrimal, is a town in the Jalore District of Rajasthan Rājasthān ( Devanāgarī: राजस्थान raːdʒəst̪ʰaːn is the largest state of the Republic of India in terms of area Harsha or Harshavardhana (हर्षवर्धन or "Harsha vardhan" ( 590 &ndash 647) was an Indian emperor who ruled Northern India ^[1] As a result, Brahmagupta is often referred to as Bhillamalacarya, that is, the teacher from Bhillamala Bhinmal. WikipediaWikiProject Indian cities for details --> Bhinmal (भीनमाल also known as Shrimal, is a town in the Jalore District of Rajasthan 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. WikipediaWikiProject Indian cities for details --> Ujjain ( Hindi:उज्जैन (also known as Ujain, Ujjayini, Avanti The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including

Undoubtedly, the Brahmasphutasiddhanta (Corrected Treatise of Brahma) is 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. The Caliph is the Head of state in a Caliphate, and the title for the leader of the Islamic Ummah, an Islamic community ruled by the Shari'ah Abu Jafar al-Ma'mun ibn Harun (also spelled Almamon and el-Mâmoûn) ( September 14, 786 &ndash August 9, 833) (المأمون It is generally presumed that Sindhind is none other than Brahmagupta's Brahmasphuta-siddhanta. The main work of Brahmagupta, Brahmasphuta-siddhanta (The Opening of the Universe, written in the year 628, contains some remarkably advanced ideas including ^[2]

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. Āryabhatīya, an astronomical treatise is the Magnum opus and only extant work of the 5th century Indian mathematician Aryabhata. Bhāskara (commonly called Bhāskara I to avoid confusion with the 12th century mathematician Bhāskara II) (c ^[1] Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, and in his Brahmasphutasiddhanta 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. ^[1] 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. ^[1]

Mathematics

Brahmagupta's most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common practice in Indian mathematics, and consequently has a poetic ring to it. Indian mathematics &mdashwhich here is the mathematics that emerged in South Asia zero, Negative numbers, Arithmetic, and Algebra. As no proofs are given, it is not known how Brahmagupta's mathematics was derived. ^[3]

Algebra

Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta,

18. A linear equation is an Algebraic equation in which each term is either a Constant or the product of a constant and (the first power of a single Variable 43 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. ^[4]

Which is a solution equivalent to $x = \tfrac{e-c}{b-d}$ , where rupas represents constants. He further gave two equivalent solutions to the general quadratic equation,

18. In Mathematics, a quadratic equation is a Polynomial Equation of the second degree. 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. ^[4]

Which are, respectively, solutions equivalent to,

$x = \tfrac{\sqrt{4ac+b^2}-b}{2a}$

and

$x = \tfrac{\sqrt{ac+\tfrac{b^2}{4}}-\tfrac{b}{2}}{a}$

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. An indeterminate equation, in Mathematics, is an equation for which there is an infinite set of solutions for example 2x = y is a simple indeterminate equation In Mathematics, a coefficient is a Constant multiplicative factor of a certain object 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]. ^[4]

Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should 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. ^[5] 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. Elementary algebra is the branch of Mathematics that deals with solving for the Operands of Arithmetic Equations. ^[5]

Arithmetic

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, $\tfrac{a}{c} + \tfrac{b}{c}$ , $\tfrac{a}{c} \cdot \tfrac{b}{d}$ , $\tfrac{a}{1} + \tfrac{b}{d}$ , $\tfrac{a}{c} + \tfrac{b}{d} \cdot \tfrac{a}{c} = \tfrac{a(d+b)}{cd}$ , and $\tfrac{a}{c} - \tfrac{b}{d} \cdot \tfrac{a}{c} = \tfrac{a(d-b)}{cd}$ . ^[6]

Series

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]. ^[7]

It is important to note 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. ^[8]

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.

Zero

Brahmagupta made use of an important concept in mathematics, the number zero. 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. Babylonia was an Amorite state in lower Mesopotamia (modern southern Iraq) with Babylon as its capital Claudius Ptolemaeus ( Greek: Klaúdios Ptolemaîos; after 83 &ndash ca Ancient Rome was a Civilization that grew out of a small agricultural community founded on the Italian Peninsula as early as the 10th century BC 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. ^[4]

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. ^[4]

But then he spoils the matter some what when he describes division,

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. ^[4]

Here Brahmagupta states that $\tfrac{0}{0} = 0$ and as for the question of $\tfrac{a}{0}$ where $a \neq 0$ he did not commit himself. ^[9] His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone A negative number is a Number that is less than zero, such as −2 In Mathematics, defined and undefined are used to explain whether or not expressions have meaningful sensible and unambiguous values

Diophantine analysis

Pythagorean triples

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta finds Pythagorean triples,

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. ^[7]

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. ^[7]

Pell's equation

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such as $N x 2 + 1 = y 2$ (called Pell's equation) by using the Euclidean algorithm. Pell's equation is any Diophantine equation of the form x^2-ny^2=1\ where n is a nonsquare integer and x In Number theory, the Euclidean algorithm (also called Euclid's algorithm) is an Algorithm to determine the Greatest common divisor (GCD The Euclidean algorithm was known to him as the "pulverizer" since it breaks numbers down into ever smaller pieces. ^[10]

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 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. ^[4]

The key to his solution was the identity,^[11]

$(x^2_1 - Ny^2_1)(x^2_2 - Ny^2_2) = (x_1 x_2 + Ny_1 y_2)^2 - N(x_1 y_2 + x_2 y_1)^2$

which is a generalization of an identity that was discovered by Diophantus,

$(x^2_1 - y^2_1)(x^2_2 - y^2_2) = (x_1 x_2 + y_1 y_2)^2 - (x_1 y_2 + x_2 y_1)^2.$

Using his identity and the fact that if $(x 1,$ $y 1)$ and $(x 2,$ $y 2)$ are solutions to the equations $x 2 - N y 2 = k 1$ and $x 2 - N y 2 = k 2$ , respectively, then $(x 1 x 2 + N y 1 y 2,$ $x 1 y 2 + x 2 y 1)$ is a solution to $x 2 - N y 2 = k 1 k 2$ , he was able to find integral solutions to the Pell's equation through a series of equations of the form $x 2 - N y 2 = k i$ . Diophantus of Alexandria ( Greek: b between 200 and 214 d between 284 and 298 AD sometimes called "the father of Algebra " a title some claim should 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 $x 2 - N y 2 = k$ has an integral solution for k = $\pm 1, \pm 2, \pm 4$ then $x 2 - N y 2 = 1$ has a solution. The solution of the general Pell's equation would have to wait for Bhaskara II in c. Bhaskara (1114 &ndash 1185 also known as Bhaskara II and Bhaskara Achārya ("Bhaskara the teacher" was an Indian mathematician 1150 CE. ^[11]

Geometry

Brahmagupta's formula

Diagram for reference

Brahmagupta's most famous result in geometry is his formula for cyclic quadrilaterals. In Geometry, Brahmagupta 's formula finds the Area of any Quadrilateral given the lengths of the sides and some of their angles In Geometry, a cyclic quadrilateral is a Quadrilateral whose vertices all lie on a single Circle. 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. ^[7]

So given the lengths a, b, c and d of a cyclic quadrilateral, the approximate area is $(\tfrac{a + c}{2}) (\tfrac{b + d}{2})$ while, letting $s = \tfrac{a + b + c + d}{2}$ , the exact area is $\sqrt{(s - a)(s - b)(s - c)(s - d)}$ . Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. ^[12] Heron's formula is a special case of this formula and it can be derived by setting one of the sides equal to zero. In Geometry, Heron's (or Hero's formula states that the Area (A of a Triangle whose sides have lengths a, b, and

Triangles

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. ^[7]

Thus the lengths of the two segments are $b \pm (c^2 - a^2)/b$ .

He further gives a theorem on rational triangles. In Geometry, a Heronian triangle is a Triangle whose sidelengths and area are all Rational numbers It is named after Hero of Alexandria A triangle with rational sides $a, b, c$ and rational area is of the form:

$a = \frac{1}{2}(\frac{u^2}{v}+v), \ \ b = \frac{1}{2}(\frac{u^2}{w}+w), \ \ c = \frac{1}{2}(\frac{u^2}{v} - v + \frac{u^2}{w} - w)$

for some rational numbers $u, v,$ and $w$ . ^[13]

Brahmagupta's theorem

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]. ^[7]

So, in a "non-unequal" cyclic quadrilateral (that is, an isosceles trapezoid), the length of each diagonal is $\sqrt{ac + bd}$ . A trapezoid (in North America or a trapezium (in Britain and elsewhere is a Quadrilateral (a closed plane shape with four linear sides that has at least one

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. Brahmagupta's theorem is a result in Geometry. It states that if a Cyclic quadrilateral has Perpendicular diagonals then the perpendicular to a side from 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]. ^[7]

Pi

In verse 40, he gives values of [pi],

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. ^[7]

So Brahmagupta uses 3 as a "practical" value of pi, and $\sqrt{10}$ as an "accurate" value of pi.

Measurements and constructions

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 frustrum 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. Elements special cases and related concepts Each plane section is a base of the frustum ^[14]

Trigonometry

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 moonl the moon, arrows, suns [. . . ]^[15]

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 4, 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. ^[16]

In his Paitamahasiddhanta, Brahmagupta uses the initial sine value of 225 with a radius of approximately 3438, although the rest of the sine table is lost. The value of 3438 for the radius is a traditional value that was also used by Aryabhata, although it is not known why Brahmagupta used 3270 instead of the 3438 in his Brahmasphutasiddhanta. ^[16]

Astronomy

It was through the Brahmasphutasiddhanta that the Arabs learned of Indian astronomy. ^[17] 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. Al-Mansur Almanzor or Abu Ja'far Abdallah ibn Muhammad al-Mansur (712&ndash775 Arabic: ابو جعفر عبدالله ابن محمد المنصور was the second Baghdad (بغداد) is the Capital of Iraq and of Baghdad Governorate, with which it is also coterminous The Tigris is the eastern member of the two great Rivers that define Mesopotamia, along with the Euphrates, which flows from the mountains of southeastern The caliph invited a scholar of Ujjain by the name of Kankah in 770 A. WikipediaWikiProject Indian cities for details --> Ujjain ( Hindi:उज्जैन (also known as Ujain, Ujjayini, Avanti Events By Place Asia Emperor Kōnin ascends to the throne of Japan, succeeding Empress Shōtoku. D. Kankah used the Brahmasphutasiddhanta to explain the Hindu system of arithmetic astronomy. Muhammad al-Fazari translated Brahmugupta's work into Arabic upon the request of the caliph. Abu abdallah Muhammad ibn Ibrahim al-Fazari (d 796 or 806 was a Muslim Philosopher, Mathematician and Astronomer.

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. ^[18]

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. [. . . ]^[19]

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. ^[18]

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. An ephemeris (plural ephemerides; from the Greek word ἐφήμερος ephemeros "daily" is a table of values that gives the positions of Conjunction is a term used in Positional astronomy and Astrology. An eclipse is an astronomical event that occurs when one Celestial object moves into the shadow of another ^[20] Brahmagupta criticized the Puranic view that the Earth was flat or hollow. For other meanings see Purana (disambiguation. The Puranas ( Sanskrit: sa पुराण purāṇa, "of ancient times" 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:

"If such were the case, stones would and trees would fall from the earth. Latin ( lingua Latīna, laˈtiːna is an Italic language, historically spoken in Latium and Ancient Rome. "^[21]

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. Gravitation is a natural Phenomenon by which objects with Mass attract one another Prana (प्राण) is the Sanskrit for " Breath " (from the root prā "to fill" cognate to Latin plenus "full" [. . . ] 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. "^[22]

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. "^[23]

Citations and footnotes

^ ^a ^b ^c ^d Plofker, Kim (2007). , 418-419. “The Paitamahasiddhanta also directly inspired another major siddhanta, written by a contemporary of Bhaskara: The Brahmasphutasiddhanta (Corrected Treatise of Brahma) completed by Brahmagupta in 628. This astronomer was born in 598 and apparently worked in Bhillamal (identified with modern Bhinmal in Rajasthan), during the reign (and possibly under the patronage) of King Vyaghramukha.
Although we do not know whether Brahmagupta encountered the work of his contemporary Bhaskara, he was certainly aware of the writings of other members of the tradition of the Aryabhatiya, about which he has nothing good to say. This is almost the first trace we possess of the division of Indian astronomer-mathematicians into rival, sometimes antagonistic "schools. " [. . . ] it was in the application of mathematical models to the physical world - in this case, the choices of astronomical parameters and theories - that disagreements arose. [. . . ]
Such critiques of rival works appear occasionally throughout the first ten astronomical chapters of the Brahmasphutasiddhanta, and its eleventh chapter is entirely devoted to them. But they do not enter into the mathematical chapters that Brahmagupta devotes respectively to ganita (chapter 12) and the pulverizer (chapter 18). This division of mathematical subjects reflects a different twofold classification from Bhaskara's "mathematics of fields" and "mathematics of quantities. " Instead, the first is concerned with arithmetic operations beginning with addition, proportion, interest, series, formulas for finding lengths, areas, and volumes in geometrical figures, and various procedures with fractions - in short, diverse rules for computing with known quantities. The second, on the other hand, deals with what Brahmagupta calls "the pulverizer, zero, negatives, positives, unknowns, elimination of the middle term, reduction to one [variable], bhavita [the product of two unknowns], and the nature of squares [second-degree indeterminate equations]" - that is, techniques for operating with unknown quantities. This distinction is more explicitely presented in later works as mathematics of the "manifest" and "unmanifest," respectively: i. e. , what we will henceforth call "arithmetic" manipulations of known quantities and "algebraic" manipulation of so-called "seeds" or unknown quantities. The former, of course, may include geometric problems and other topics not covered by the modern definition of "arithmetic. " (Like Aryabhata, Brahmagupta relegates his sine-table to an astronomical chapter where the computations require it, instead of lumping it in with other "mathematical" topics. ”
^ Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "The Arabic Hegemony", , 226. “By 766 we learn that an astronomical-mathematical work, known to the Arabs as the Sindhind, was brought to Baghdad from India. It is generally thought that this was the Brahmasphuta Siddhanta, although it may have been the Surya Siddhanata. A few years later, perhaps about 775, this Siddhanata was translated into Arabic, and it was not long afterwards (ca. 780) that Ptolemy's astrological Tetrabiblos was translated into Arabic from the Greek. ”
^ Brahmagupta biography
^ ^a ^b ^c ^d ^e ^f ^g Plofker, Kim (2007). , 428 - 434.
^ ^a ^b (Boyer 1991, "China and India" p. 221) "he was the first one to give a general solution of the linear Diophantine equation ax + by = c, where a, b, and c are integers. [. . . ] It is greatly to the credit of Brahmagupta that he gave all integral solutions of the linear Diophantine equation, whereas Diophantus himself had been satisfied to give one particular solution of an indeterminate equation. Inasmuch as Brahmagupta used some of the same examples as Diophantus, we see again the likelihood of Greek influence in India - or the possibility that they both made use of a common source, possibly from Babylonia. It is interesting to note also that the algebra of Brahmagupta, like that of Diophantus, was syncopated. Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The operations of multiplication and evolution (the taking of roots), as well as unknown quantities, were represented by abbreviations of appropriate words. "
^ Plofker, Kim (2007). , 422. “The reader is apparently expected to be familiar with basic arithmetic operations as far as the square-root; Brahmagupta merely notes some points about applying them to fractions. The procedures for finding the cube and cube-root of an integer, however, are described (compared the latter to Aryabhata's very similar formulation). They are followed by rules for five types of combinations: [. . . ]”
^ ^a ^b ^c ^d ^e ^f ^g ^h Plofker, Kim (2007). , 421 - 427.
^ Plofker, Kim (2007). , 423. “Here the sums of the squares and cubes of the first n integers are defined in terms of the sum of the n integers itself;”
^ Boyer (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he "China and India", , 220. “However, here again Brahmagupta spoiled matters somewhat by asserting that $0 \div 0 = 0$ , and on the touchy matter of $a \div 0$ , he did not commit himself:”
^ Stillwell, John (2004). , 44 - 46. “In the seventh century CE the Indian mathematician Brahmagupta gave a recurrence relation for generating solutions of $x 2 - D y 2 = 1$ , as we shall see in Chapter 5. The Indians called the Euclidean algorithm the "pulverizer" because it breaks numbers down to smaller and smaller pieces. To obtain a recurrence one has to know that a rectangle proportional to the original eventually recurs, a fact that was rigorously proved only in 1768 by Lagrange. ”
^ ^a ^b Stillwell, John (2004). , 72 - 74.
^ Plofker, Kim (2007). , 424. “Brahmagupta does not explicitly state that he is discussing only figures inscribed in circles, but it is implied by these rules for computing their circumradius. ”
^ (Stillwell 2004, p. 77)
^ "Plofker, Kim (2007). , 427. “After the geometry of plane figures, Brahmagupta discusses the computation of volumes and surface areas of solids (or empty spaces dug out of solids). His straight-forward rules for the volumes of a rectangular prism and pyramid are followed by a more ambiguous one, which may refer to finding the average depth of a sequence of puts with different depths. The next formula apparently deals with the volume of a frustum of a square pyramid, where the "pragmatic" volume is the depth times the square of the mean of the edges of the top and bottom faces, while the "superficial" volume is the depth times their mean area. ”
^ Plofker, Kim (2007). , 419.
^ ^a ^b Plofker, Kim (2007). , 419-420. “Brahmagupta's sine table, like much other numerical data in Sanskrit treatises, is encoded mostly in concrete-number notation that uses names of objects to represent the digits of place-value numerals, starting with the least significant. [. . . ]
There are fourteen Progenitors ("Manu") in Indian cosmology; "twins" of course stands for 2; the seven stars of Ursa Major (the "Sages") for 7, the four Vedas, and the four sides of the traditional dice used in gambling, for 4, and so on. Thus Brahmagupta enumerates his first six sine-values as 214, 427, 638, 846, 1051, 1251. (His remaining eighteen sines are 1446, 1635, 1817, 1991, 2156, 2312, 1459, 2594, 2719, 2832, 2933, 3021, 3096, 3159, 3207, 3242, 3263, 3270. The Paitamahasiddhanta, however, specifies an initial sine-value of 225 (although the rest of its sine-table is lost), implying a trigonometric radius of R= 3438 aprox= C(')/2π: a tradition followed, as we have seen, by Aryabhata. Nobody knows why Brahmagupta chose instead to normalize these values to R = 3270. ”
^ Brahmagupta, and the influence on Arabia. Retrieved 23 December 2007.
^ ^a ^b Plofker, Kim (2007). , 419-420. “Brahmagupta discusses the illumination of the moon by the sun, rebutting an idea maintained in scriptures: namely, that the moon is farther from the earth than the sun is. In fact, as he explains, because the moon is closer the extent of the illuminated portion of the moon depends on the relative positions of the moon and the sun, and can be computed from the size of the angular separation α between them. ”
^ Plofker, Kim (2007). , 420.
^ Dick Teresi, Lost Discoveries: The Ancient Roots of Modern Science, Simon and Schuster, 2002. p. 135. ISBN 074324379X.
^ Al-Biruni (1030), Ta'rikh al-Hind (Indica)
^ Brahmagupta, Brahmasphutasiddhanta (628) (cf. al-Biruni (1030), Indica)
^ Thomas Khoshy, Elementary Number Theory with Applications, Academic Press, 2002, p. cf is an abbreviation for the Latin -derived (but also modern English) word confer, meaning "compare" or "consult" 567. ISBN 0124211712.

References

Plofker, Kim (2007). "Mathematics in India", The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 9780691114859.
Boyer, Carl B. (1991). Carl Benjamin Boyer ( November 3, 1906 – April 26, 1976) has been called the " Gibbon of math history"he A History of Mathematics, Second Edition, John Wiley & Sons, Inc. ISBN 0471543977.
Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823.
Stillwell, John (2004). Mathematics and its History, Second Edition, Springer Science + Buisiness Media Inc. . ISBN 0387953361.

External links

Brahmagupta's Brahma Sphuta Siddhanta. PDF format. Four volumes. (English introduction, Sanskrit text with Sanskrit and Hindi commentaries. )

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