Mathematical 123

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Mathematical notation comprises thesymbolsused to write

mathematicalequationsandformulas. It includesHindu-Arabic numerals,

letters from theRoman,Greek,Hebrew, andGermanalphabets, and a host of

symbols invented by mathematicians over the past several centuries.

The development of mathematical notation for algebra can be divided in three

stages. The first is "rhetorical", where all calculations are performed by words

and no symbols are used. Most medieval Islamic mathematicians belonged to

this stage. The second is "syncopated", where frequently used operations and

quantities are represented by symbolic abbreviations. To this

stageDiophantusbelonged. The third is "symbolic", which is a more

comprehensive system of notation that replaces much of rhetoric. This system

was in use by medieval Indian mathematicians and in Europe since the middle

of the 17th century,[1]

and has continued to develop down to the present day.

BrahmaguptaFrom Wikipedia, the free encyclopedia

Brahmagupta(Sanskrit:; listen(helpinfo)) (597668 AD) was

aIndianmathematicianandastronomerwho wrote many important works on mathematics and astronomy. His

best known work is theBrhmasphuasiddhnta(Correctly Established Doctrine of Brahma), written in 628

inBhinmal. Its 25 chapters contain several unprecedented mathematical results.

Brahmagupta was the first to use zero as a number. He gave rules to compute withzero. 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 inIndian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is

not known how Brahmagupta's mathematics was derived.[1]

Life and work [edit]

Brahmagupta is believed to have been born in 598 AD inBhinmalcity in the state ofRajasthanof Northwest

India. In ancient times Bhillamala was the seat of power of theGurjars. His father was Jisnugupta.[2]

He likely

lived most of his life in Bhillamala (modern Bhinmal inRajasthan) during the reign (and possibly under the

patronage) of King Vyaghramukha.[3]

As a result, Brahmagupta is often referred to as Bhillamalacharya, that is,
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the teacher from Bhillamala. He was the head of the astronomical observatory atUjjain, and during his tenure

there wrote four texts on mathematics and astronomy: the Cadamekela in 624, theBrahmasphutasiddhantain

628, the Khandakhadyaka in 665, and the Durkeamynarda in 672. The Brahmasphutasiddhanta (Corrected

Treatise of Brahma) is arguably his most famous work. The historianal-Biruni(c. 1050) in his book Tariq al-

Hindstates that theAbbasidcaliphal-Ma'munhad 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 Sindhindis none other

than Brahmagupta'sBrahmasphuta-siddhanta.[4]

Although Brahmagupta was familiar with the works of astronomers following the tradition ofAryabhatiya, it is

not known if he was familiar with the work ofBhaskara I, a contemporary.[3]

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.[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]

rce.[6]

Arithmetic [edit]

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

hisBrahmasphutasiddhanta, 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]

Series [edit]

Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers.
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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

ofn 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).

Zero [edit]

Brahmagupta's Brahmasphuasiddhanta is the first book that mentions zero as a number, hence

Brahmagupta is considered the first to formulate the concept ofzero. 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 theBabyloniansor as a symbol for a lack of quantity as was done

byPtolemyand theRomans. In chapter eighteen of hisBrahmasphutasiddhanta, 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]

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]

But his description ofdivision by zerodiffers from our modern understanding,

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
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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 forarithmeticonnegative numbersand zero are quite close to the modern

understanding, except that in modern mathematics division by zero is leftundefined.

Diophantine analysis [edit]

Pythagorean triples [edit]

In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta findsPythagorean 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.[9]

or in other words, for a given length m and an arbitrary multiplierx, let a = mxand b = m + mx/(x + 2).

Then m, a, and b form a Pythagorean triple.[9]

Pell's equation [edit]

Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of

Diophantine equations of the second degree such as (calledPell's equation) by

using theEuclidean 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]

The key to his solution was the identity,[13]

which is a generalization of an identity that was discovered byDiophantus,

Using his identity and the fact that if and are solutions to the

equations and , respectively,
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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 ofN, 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 forBhaskara IIin c. 1150 CE.[13]

Geometry [edit]

Brahmagupta's formula [edit]

Diagram for reference

Main article:Brahmagupta's formula

Brahmagupta's most famous result in geometry is hisformulaforcyclic 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 lengthsp, q, rand s of a cyclic quadrilateral, the approximate area

is while, letting , the exact area is
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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 formulais a special case of this

formula and it can be derived by setting one of the sides equal to zero.

Triangles [edit]

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 onrational triangles. A triangle with rational sides a, b, cand

rational area is of the form:

for some rational numbers u, v, and w.[15]

Brahmagupta's theorem [edit]

Main article:Brahmagupta theorem

Brahmagupta's theorem states thatAF=FD.

Brahmagupta continues,
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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 isoscelestrapezoid), 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

toBrahmagupta'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 areseparately 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]

Pi [edit]

In verse 40, he gives values of,

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, and as an "accurate"

value of.

Measurements and constructions [edit]

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.
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21, 3096, 3159, 3207, 3242, 3263, and 3270, with the radius being 3270.[18]

Interpolation formula [edit]

See main article:Brahmagupta's interpolation formula

In 665 Brahmagupta devised and used a special case of the NewtonStirling

interpolation formula of the second-order tointerpolatenew values of

thesinefunction 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 1x 1) when its value is already known at ah, a and a + h.

The formula for the estimate is:

where is the first-order forward-difference operator, i.e.

Astronomy [edit]

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 famousAbbasidcaliphAl-

Mansur(712775) foundedBaghdad, which is situated on the banks of

theTigris, and made it a center of learning. The caliph invited a scholar

ofUjjainby the name of Kankah in 770 A.D. Kankah used

the Brahmasphutasiddhanta to explain the Hindu system of arithmetic

astronomy.Muhammad al-Fazaritranslated 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
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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 lunareclipses.[24]

Brahmagupta criticized thePuranicview 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, theMuslim

astronomerAbu al-Rayhan al-Biruni, in his Ta'rikh al-Hind, later translated

intoLatinas 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."[25]

According to al-Biruni, Brahmagupta responded to these criticisms with the

following argument ongravitation:

"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,

thepranasof 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]
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10/23

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]

See also [edit] BrahmaguptaFibonacci identity

Brahmagupta's formula

Brahmagupta theorem

Chakravala method

Citations and footnotes [edit]

1. ^Brahmagupta biography2. ^Shashi S. Sharma.Mathematics & Astronomers

of Ancient India. Pitambar Publishing. "He was

born in bhillamala. In ancient times it was the seat

of power of theGurjars...Jisnu Gupta.."

3. ^abcd(Plofker 2007, pp. 418419)

The Paitamahasiddhanta also directly inspired

another majorsiddhanta, 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 contemporaryBhaskara, he was certainly aware of the writings

of other members of the tradition of

theAryabhatiya, about which he has nothing

good to say. This is almost the first trace we

possess of the division of Indian astronomer-
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11/23

mathematicians into rival, sometimes antagonistic

"schools." [...] it was in the application of

mathematical models to the physical worldin

this case, the choices of astronomical parameters

and theoriesthat 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 subjectsreflects 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

fractionsin 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 explicitly presented in laterworks 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


12/23

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

4. empty |title=(help)

5. ^(Plofker 2007, p. 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.

6. ^(Stillwell 2004, p. 77)

7. ^(Plofker 2007, p. 427) After the geometry of

plane figures, Brahmagupta discusses the

computation of volumes and surface areas ofsolids (or empty spaces dug out of solids). His

straight-forward rules for the volumes of a

rectangular prism and pyramid are followed by a

8. 7, the four Vedas, and the four sides of the

traditional dice used in gambling, for 6, 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

ofR= 3438 aprox= C(')/2: a tradition followed,

as we have seen, by Aryabhata. Nobody knows

why Brahmagupta chose instead to normalize

these values to R = 3270.9. ^Joseph(2000, pp.28586).

10.^Brahmagupta, and the influence on Arabia.

Retrieved 23 December 2007.

11.^Al Biruni, India translated by Edward sachau.
http://en.wikipedia.org/wiki/Help:CS1_errors#citation_missing_titlehttp://en.wikipedia.org/wiki/Help:CS1_errors#citation_missing_titlehttp://en.wikipedia.org/wiki/Help:CS1_errors#citation_missing_titlehttp://en.wikipedia.org/wiki/Brahmagupta#cite_ref-14http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-14http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-15http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-15http://en.wikipedia.org/wiki/Brahmagupta#CITEREFStillwell2004http://en.wikipedia.org/wiki/Brahmagupta#CITEREFStillwell2004http://en.wikipedia.org/wiki/Brahmagupta#CITEREFStillwell2004http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-16http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-16http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-19http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-19http://en.wikipedia.org/wiki/Brahmagupta#joseph-2000http://en.wikipedia.org/wiki/Brahmagupta#joseph-2000http://en.wikipedia.org/wiki/Brahmagupta#joseph-2000http://en.wikipedia.org/wiki/Brahmagupta#joseph-2000http://en.wikipedia.org/wiki/Brahmagupta#joseph-2000http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-20http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-20http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_3.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_3.htmlhttp://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_3.htmlhttp://en.wikipedia.org/wiki/Brahmagupta#cite_ref-21http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-21http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-21http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-21http://www-groups.dcs.st-and.ac.uk/~history/Projects/Pearce/Chapters/Ch8_3.htmlhttp://en.wikipedia.org/wiki/Brahmagupta#cite_ref-20http://en.wikipedia.org/wiki/Brahmagupta#joseph-2000http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-19http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-16http://en.wikipedia.org/wiki/Brahmagupta#CITEREFStillwell2004http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-15http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-14http://en.wikipedia.org/wiki/Help:CS1_errors#citation_missing_title


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12. ab(Plofker 2007, pp. 419420) 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

between them.

13.^(Plofker 2007, p. 420)

14.^Teresi, Dick (2002). Lost Discoveries: The

Ancient Roots of Modern Science. Simon andSchuster. p. 135.ISBN0-7432-4379-X.

15.^Al-Biruni(1030), Ta'rikh al-Hind(Indica)

16.^Brahmagupta, Brahmasphutasiddhanta (628)

(cf.al-Biruni(1030), Indica)

17.^Khoshy, Thomas (2002). Elementary Number

Theory with Applications. Academic Press.

p. 567.ISBN0-12-421171-2.

References [edit]

Plofker, Kim (2007). "Mathematics in India". The Mathematics of

Egypt, Mesopotamia, China, India, and Islam: A Sourcebook.

Princeton University Press.ISBN978-0-691-11485-9.

Boyer, Carl B.(1991).A History of Mathematics (Second Edition ed.).

John Wiley & Sons, Inc.ISBN0-471-54397-7.

Cooke, Roger (1997). The History of Mathematics: A Brief Course.

Wiley-Interscience.ISBN0-471-18082-3.

Joseph, George G. (2000). The Crest of the Peacock. Princeton, NJ:

Princeton University Press.ISBN0-691-00659-8.

Stillwell, John (2004). Mathematics and its History(Second Edition

ed.). Springer Science + Business Media Inc.ISBN0-387-95336-1.

External links [edit]
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ational_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Carl_Benjamin_Boyerhttp://en.wikipedia.org/wiki/Special:BookSources/978-0-691-11485-9http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/w/index.php?title=Brahmagupta&action=edit&section=22http://en.wikipedia.org/wiki/Special:BookSources/0-12-421171-2http://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Brahmagupta#cite_ref-27http://en.wikipedia.org/wiki/Al-Birunihttp://en.wikipedia.org/wiki/Cf.http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-26http://en.wikipedia.org/wiki/Al-Birunihttp://en.wikipedia.org/wiki/Brahmagupta#cite_ref-25http://en.wikipedia.org/wiki/Special:BookSources/0-7432-4379-Xhttp://en.wikipedia.org/wiki/International_Standard_Book_Numberhttp://en.wikipedia.org/wiki/Brahmagupta#cite_ref-24http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-23http://en.wikipedia.org/wiki/Brahmagupta#CITEREFPlofker2007http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-Plofker_420_22-1http://en.wikipedia.org/wiki/Brahmagupta#cite_ref-Plofker_420_22-0


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Brahmagupta's Biography

Brahmagupta's Brahma-sphuta-siddhantaEnglish introduction,

Sanskrit text, Sanskrit and Hindi commentaries (PDF

For other people named William Jones, seeWilliam Jones (disambiguation).

William Jones

Portrait of William Jones byWilliam Hogarth,

1740 (National Portrait Gallery)

Born 1675

Llanfihangel Tre'r Beirdd,

Isle of Anglesey

Died 3 July 1749

Part ofa series of articleson the
http://www.brahmagupta.net/http://www.brahmagupta.net/http://www.wilbourhall.org/index.html#BSShttp://www.wilbourhall.org/index.html#BSShttp://en.wikipedia.org/wiki/William_Jones_(disambiguation)http://en.wikipedia.org/wiki/William_Jones_(disambiguation)http://en.wikipedia.org/wiki/William_Jones_(disambiguation)http://en.wikipedia.org/wiki/William_Hogarthhttp://en.wikipedia.org/wiki/William_Hogarthhttp://en.wikipedia.org/wiki/William_Hogarthhttp://en.wikipedia.org/w/index.php?title=Llanfihangel_Tre%27r_Beirdd&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Llanfihangel_Tre%27r_Beirdd&action=edit&redlink=1http://en.wikipedia.org/wiki/Isle_of_Angleseyhttp://en.wikipedia.org/wiki/Isle_of_Angleseyhttp://en.wikipedia.org/wiki/Category:Pihttp://en.wikipedia.org/wiki/Category:Pihttp://en.wikipedia.org/wiki/Category:Pihttp://en.wikipedia.org/wiki/File:William_Jones,_the_Mathematician.jpghttp://en.wikipedia.org/wiki/Category:Pihttp://en.wikipedia.org/wiki/Isle_of_Angleseyhttp://en.wikipedia.org/w/index.php?title=Llanfihangel_Tre%27r_Beirdd&action=edit&redlink=1http://en.wikipedia.org/wiki/William_Hogarthhttp://en.wikipedia.org/wiki/William_Jones_(disambiguation)http://www.wilbourhall.org/index.html#BSShttp://www.brahmagupta.net/


15/23

mathematical constant

Uses

Area of disk

Circumference

Use in other formulae

Properties

Irrationality

Transcendence

Value

Less than 22/7

Approximations

Memorization

People

Archimedes

Liu Hui

Zu Chongzhi

Madhava of Sangamagrama

William Jones

John Machin

John Wrench
http://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Pihttp://en.wikipedia.org/wiki/Area_of_a_diskhttp://en.wikipedia.org/wiki/Area_of_a_diskhttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80http://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrationalhttp://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrationalhttp://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theoremhttp://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theoremhttp://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80http://en.wikipedia.org/wiki/Approximations_of_%CF%80http://en.wikipedia.org/wiki/Approximations_of_%CF%80http://en.wikipedia.org/wiki/Piphilologyhttp://en.wikipedia.org/wiki/Piphilologyhttp://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedeshttp://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedeshttp://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithmhttp://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithmhttp://en.wikipedia.org/wiki/Zu_Chongzhihttp://en.wikipedia.org/wiki/Zu_Chongzhihttp://en.wikipedia.org/wiki/Madhava_of_Sangamagramahttp://en.wikipedia.org/wiki/Madhava_of_Sangamagramahttp://en.wikipedia.org/wiki/John_Machinhttp://en.wikipedia.org/wiki/John_Machinhttp://en.wikipedia.org/wiki/John_Wrenchhttp://en.wikipedia.org/wiki/John_Wrenchhttp://en.wikipedia.org/wiki/John_Wrenchhttp://en.wikipedia.org/wiki/John_Machinhttp://en.wikipedia.org/wiki/Madhava_of_Sangamagramahttp://en.wikipedia.org/wiki/Zu_Chongzhihttp://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithmhttp://en.wikipedia.org/wiki/Method_of_exhaustion#Use_by_Archimedeshttp://en.wikipedia.org/wiki/Piphilologyhttp://en.wikipedia.org/wiki/Approximations_of_%CF%80http://en.wikipedia.org/wiki/Proof_that_22/7_exceeds_%CF%80http://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theoremhttp://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrationalhttp://en.wikipedia.org/wiki/List_of_formulae_involving_%CF%80http://en.wikipedia.org/wiki/Circumferencehttp://en.wikipedia.org/wiki/Area_of_a_diskhttp://en.wikipedia.org/wiki/Pi


16/23

Ludolph van Ceulen

Aryabhata

History

Chronology

Book

In culture

Legislation

Holiday

Related topics

William Jones was born the son of Sin Sir (John George Jones) and

Elizabeth Rowland in the village ofLlanfihangel Tre'r Beirdd, on theIsle of

Anglesey. He attended a local charity school at Llanfechell, where his

mathematical talents were spotted by the local landowner who arranged for

him to be given a job in London working in a merchant's counting-house. He

owed his successful career partly to the patronage of the

distinguishedBulkeleyfamily of northWales, and later to theEarl ofMacclesfield.

Jones initially served at sea, teaching mathematics on board Navy ships

between 1695 and 1702 where he became very interested in navigation and

publishedA New Compendium of the Whole Art of Navigation in

1702[2]

dedicated to a benefactorJohn Harris.[3]

In this work he applied

mathematics to navigation, studying methods to calculate position at sea.

After his voyages were over he became a mathematics teacher inLondon,

both in coffee houses and as a private tutor to the son of the future Earl of

Macclesfield and also the futureBaron Hardwicke. He also held a number of

undemanding posts in government offices with the help of his former pupils.

Jones published Synopsis Palmariorum Matheseos in 1706, a work which was

intended for beginners and which included theorems ondifferential
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calculusandinfinite series. This used as an abbreviation forperimeter. His

1711 workAnalysis per quantitatum series, fluxiones ac

differentias introduced the dot notation fordifferentiationin calculus.[4]

In 1731

he publishedDiscourses of the Natural Philosophy of the Elements.

He married twice, firstly the widow of his counting-house employer, whose

property he inherited on her death, and secondly, in 1731, Mary, the 22 year

old daughter of cabinet-maker George Nix with whom he had two surviving

children. His son, also namedWilliam Jonesborn in 1746, was a

renownedphilologistwho first recognised the existence of theIndo-European

languagegroup.

References[edit]

1. ^"Library and Archive catalogue". Royal Society. Retrieved 1 November

2010.

2. ^"Jones biography". University of St. Andrews. Retrieved 12 December

2010.

3. ^William Jones (1702).A New Compendium of the Whole Art of Navigation.

Retrieved 2011-02-03.

4. ^Garland Hampton Cannon (1990).The life and mind Oriental Jones.

Retrieved 2011-02-03.

External links [edit]

William Jonesand other important Welsh mathematicians

William Jones and his Circle: The Man who invented Pi

Giuseppe PeanoFrom Wikipedia, the free encyclopedia

Giuseppe Peano
http://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Differential_calculushttp://en.wikipedia.org/wiki/Infinite_serieshttp://en.wikipedia.org/wiki/Infinite_serieshttp://en.wikipedia.org/wiki/Infinite_serieshttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_note-4http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_note-4http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_note-4http://en.wikipedia.org/wiki/William_Jones_(philologist)http://en.wikipedia.org/wiki/William_Jones_(philologist)http://en.wikipedia.org/wiki/William_Jones_(philologist)http://en.wikipedia.org/wiki/Philologisthttp://en.wikipedia.org/wiki/Philologisthttp://en.wikipedia.org/wiki/Philologisthttp://en.wikipedia.org/wiki/Indo-European_languagehttp://en.wikipedia.org/wiki/Indo-European_languagehttp://en.wikipedia.org/wiki/Indo-European_languagehttp://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=2http://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=2http://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=2http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-1http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=21&dsqSearch=%28Surname%3D%27jones%27%29http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=21&dsqSearch=%28Surname%3D%27jones%27%29http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=21&dsqSearch=%28Surname%3D%27jones%27%29http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-2http://www-history.mcs.st-and.ac.uk/Biographies/Jones.htmlhttp://www-history.mcs.st-and.ac.uk/Biographies/Jones.htmlhttp://www-history.mcs.st-and.ac.uk/Biographies/Jones.htmlhttp://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-3http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-3http://books.google.com/books?id=Gm9bAAAAQAAJhttp://books.google.com/books?id=Gm9bAAAAQAAJhttp://books.google.com/books?id=Gm9bAAAAQAAJhttp://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-4http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-4http://books.google.com/books?id=V7vgzG1BtYgChttp://books.google.com/books?id=V7vgzG1BtYgChttp://books.google.com/books?id=V7vgzG1BtYgChttp://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=3http://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=3http://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=3http://www.famouswelsh.com/scientists/1053-william-jones--scientist-with-a-welsh-connection!http://www.famouswelsh.com/scientists/1053-william-jones--scientist-with-a-welsh-connection!http://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pihttp://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pihttp://www.historytoday.com/patricia-rothman/william-jones-and-his-circle-man-who-invented-pihttp://www.famouswelsh.com/scientists/1053-william-jones--scientist-with-a-welsh-connection!http://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=3http://books.google.com/books?id=V7vgzG1BtYgChttp://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-4http://books.google.com/books?id=Gm9bAAAAQAAJhttp://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-3http://www-history.mcs.st-and.ac.uk/Biographies/Jones.htmlhttp://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-2http://www2.royalsociety.org/DServe/dserve.exe?dsqIni=Dserve.ini&dsqApp=Archive&dsqCmd=Show.tcl&dsqDb=Persons&dsqPos=21&dsqSearch=%28Surname%3D%27jones%27%29http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_ref-1http://en.wikipedia.org/w/index.php?title=William_Jones_(mathematician)&action=edit&section=2http://en.wikipedia.org/wiki/Indo-European_languagehttp://en.wikipedia.org/wiki/Indo-European_languagehttp://en.wikipedia.org/wiki/Philologisthttp://en.wikipedia.org/wiki/William_Jones_(philologist)http://en.wikipedia.org/wiki/William_Jones_(mathematician)#cite_note-4http://en.wikipedia.org/wiki/Derivativehttp://en.wikipedia.org/wiki/Infinite_serieshttp://en.wikipedia.org/wiki/Differential_calculus


18/23

Born 27 August 1858

Spinetta,Piedmont,Kingdom of

Sardinia

Died 20 April 1932 (aged 73)

Turin,Italy

Residence Italy

Citizenship Italian

Fields Mathematics

Institutions University of Turin,Accademia

dei Lincei

Alma mater University of Turin

Doctoral advisor Enrico D'Ovidio

Other

academic advisorsFrancesco Fa di Bruno
http://en.wikipedia.org/wiki/File:Giuseppe_Peano.jpghttp://en.wikipedia.org/wiki/Cuneohttp://en.wikipedia.org/wiki/Cuneohttp://en.wikipedia.org/wiki/Piedmonthttp://en.wikipedia.org/wiki/Piedmonthttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Turinhttp://en.wikipedia.org/wiki/Turinhttp://en.wikipedia.org/wiki/Kingdom_of_Italyhttp://en.wikipedia.org/wiki/Kingdom_of_Italyhttp://en.wikipedia.org/wiki/Kingdom_of_Italyhttp://en.wikipedia.org/wiki/Italyhttp://en.wikipedia.org/wiki/Italyhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/University_of_Turinhttp://en.wikipedia.org/wiki/Accademia_dei_Linceihttp://en.wikipedia.org/wiki/Accademia_dei_Linceihttp://en.wikipedia.org/wiki/Accademia_dei_Linceihttp://en.wikipedia.org/wiki/Accademia_dei_Linceihttp://en.wikipedia.org/wiki/Alma_materhttp://en.wikipedia.org/wiki/Alma_materhttp://en.wikipedia.org/wiki/University_of_Turinhttp://en.wikipedia.org/wiki/University_of_Turinhttp://en.wikipedia.org/wiki/Doctoral_advisorhttp://en.wikipedia.org/w/index.php?title=Enrico_D%27Ovidio&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Enrico_D%27Ovidio&action=edit&redlink=1http://en.wikipedia.org/wiki/Francesco_Fa%C3%A0_di_Brunohttp://en.wikipedia.org/wiki/File:Giuseppe_Peano.jpghttp://en.wikipedia.org/wiki/Francesco_Fa%C3%A0_di_Brunohttp://en.wikipedia.org/w/index.php?title=Enrico_D%27Ovidio&action=edit&redlink=1http://en.wikipedia.org/wiki/Doctoral_advisorhttp://en.wikipedia.org/wiki/University_of_Turinhttp://en.wikipedia.org/wiki/Alma_materhttp://en.wikipedia.org/wiki/Accademia_dei_Linceihttp://en.wikipedia.org/wiki/Accademia_dei_Linceihttp://en.wikipedia.org/wiki/University_of_Turinhttp://en.wikipedia.org/wiki/Mathematicshttp://en.wikipedia.org/wiki/Italyhttp://en.wikipedia.org/wiki/Italyhttp://en.wikipedia.org/wiki/Kingdom_of_Italyhttp://en.wikipedia.org/wiki/Turinhttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Kingdom_of_Sardiniahttp://en.wikipedia.org/wiki/Piedmonthttp://en.wikipedia.org/wiki/Cuneo


19/23

Known for Peano axioms

Peano existence theorem

Formulario mathematico

Latino Sine Flexione

Influences Euclid,Angelo

Genocchi,Gottlob Frege

Influenced Bertrand Russell,Giovanni

Vailati

Notable awards Knight of the Order of Saints

Maurizio and Lazzaro

Knight of the Crown of ItalyCommendatore of the Crown of

Italy

Correspondent of theAccademia

dei Lincei

Giuseppe Peano (Italian:[duzppe peano]; 27 August 1858 20 April 1932) was anItalianmathematician,

whose work was ofphilosophicalvalue. The author of over 200 books and papers, he was a founder

ofmathematical logicandset theory, to which he contributed much notation. The standardaxiomatizationof

thenatural numbersis named thePeano axiomsin his honor. As part of this effort, he made key contributions

to the modern rigorous and systematic treatment of the method ofmathematical induction. He spent most of his

career teaching mathematics at theUniversity of Turin.

Contents

[hide]

1 Biography

2 Milestones and honors received

3 See also

4 Bibliography

5 References

6 External links

Biography [edit]
http://en.wikipedia.org/wiki/Peano_axiomshttp://en.wikipedia.org/wiki/Peano_axiomshttp://en.wikipedia.org/wiki/Peano_existence_theoremhttp://en.wikipedia.org/wiki/Peano_existence_theoremhttp://en.wikipedia.org/wiki/Formulario_mathematicohttp://en.wikipedia.org/wiki/Formulario_mathematicohttp://en.wikipedia.org/wiki/Latino_Sine_Flexi

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