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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
, that is, 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
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criticism directed towards the work of rival astronomers, and in his Brahmasphutasiddhanta is found one of the
earliest attested
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.
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
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
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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
or of a positive is positive; [the square] of zero is zero. That of which [the square] is the
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,
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
]
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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
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]
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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 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]
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.
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
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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
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:
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"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]
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]
1. 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
theAryabhatiya, 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 worldin
this case, the choices of astronomical parametersand 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
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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
fractionsin short, diverse rules for computing
with known quantities. The second, on the otherhand, 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 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
2. only figures inscribed in circles, but it is implied by
these rules for computing their circumradius.3. ^(Stillwell 2004, p. 77)
4. ^(Plofker 2007, p. 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
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straight-forward rules for the volumes of a
rectangular prism and pyramid are followed by a
5. 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.
6. ^Joseph(2000, pp.28586).
7. ^Brahmagupta, and the influence on Arabia.
Retrieved 23 December 2007.
8. ^Al Biruni, India translated by Edward sachau.
9. ^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.10.^(Plofker 2007, p. 420)
11.^Teresi, Dick (2002). Lost Discoveries: The
Ancient Roots of Modern Science. Simon and
Schuster. p. 135.ISBN0-7432-4379-X.
12.^Al-Biruni(1030), Ta'rikh al-Hind(Indica)
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13.^Brahmagupta, Brahmasphutasiddhanta (628)
(cf.al-Biruni(1030), Indica)
14.^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]
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
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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
mathematical constant
Uses
Area of disk
Circumference
Use in other formulae
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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
Ludolph van Ceulen
Aryabhata
History
Chronology
Book
In culture
Legislation
Holiday
http://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/Ludolph_van_Ceulenhttp://en.wikipedia.org/wiki/Ludolph_van_Ceulenhttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80http://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80http://en.wikipedia.org/wiki/A_History_of_Pihttp://en.wikipedia.org/wiki/A_History_of_Pihttp://en.wikipedia.org/wiki/Indiana_Pi_Billhttp://en.wikipedia.org/wiki/Indiana_Pi_Billhttp://en.wikipedia.org/wiki/Pi_Dayhttp://en.wikipedia.org/wiki/Pi_Dayhttp://en.wikipedia.org/wiki/Pi_Dayhttp://en.wikipedia.org/wiki/Indiana_Pi_Billhttp://en.wikipedia.org/wiki/A_History_of_Pihttp://en.wikipedia.org/wiki/Chronology_of_computation_of_%CF%80http://en.wikipedia.org/wiki/Aryabhatahttp://en.wikipedia.org/wiki/Ludolph_van_Ceulenhttp://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_irrational -
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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 of
Macclesfield.
Jones initially served at sea, teaching mathematics on board Navy ships
between 1695 and 1702 where he became very interested in navigation andpublishedA 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
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.
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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
Born 27 August 1858
Spinetta,Piedmont,Kingdom of
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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
Known for Peano axioms
Peano existence theoremFormulario 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 Italy
Commendatore of the Crown of
Italy
Correspondent of theAccademia
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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]
Peano was born and raised on a farm at Spinetta, a hamlet now belonging toCuneo,Piedmont,Italy. He
attended theLiceo classico CavourinTurin, and enrolled at theUniversity of Turinin 1876, graduating in 1880
with high
In 1890 Peano founded the journal Rivista di Matematica, which published its first issue in January 1891.[3]In
1891 Peano started theFormulario Project. It was to be an "Encyclopedia of Mathematics", containing all
known formulae and theorems of mathematical science using a standard notation invented by Peano. In 1897,
the firstInternational Congress of Mathematicianswas held inZrich. Peano was a key participant, presenting
a paper on mathematical logic. He also started to become increasingly occupied with Formulario to the
detriment of his other work.
In 1898 he presented a note to the Academy aboutbinary numerationand its ability to be used to represent the
sounds of languages. He also became so frustrated with publishing delays (due to his demand that formulae be
printed on one line) that he purchased a printing press.
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He [Peano] was a man I greatly
admired from the moment I met
him for the first time in 1900 at a
Congress of Philosophy, which he
dominated by the exactness of hismind.
Bertrand Russell, 1932,[4]
Pariswas the venue for the SecondInternational Congress of Mathematiciansin 1900. The conference was
preceded by the FirstInternational Conference of Philosophywhere Peano was a member of the patronage
committee. He presented a paper which posed the question of correctly formed definitions in
mathematics, i.e. "how do you define a definition?". This became one of Peano's main philosophical interests
for the rest of his life. At the conference Peano metBertrand Russelland gave him a copy ofFormulario.
Russell was so struck by Peano's innovative logical symbols that he left the conference and returned home to
study Peano's text.
Peano's studentsMario PieriandAlessandro Padoahad papers presented at the philosophy congress also.
For the mathematical congress, Peano did not speak, but Padoa's memorable presentation has been
frequently recalled. A resolution calling for the formation of an "international auxiliary language" to facilitate the
spread of mathematical (and commercial) ideas, was proposed; Peano fully supported it.
By 1901, Peano was at the peak of his mathematical career. He had made advances in the areas ofanalysis,
foundations and logic, made many contributions to the teaching of calculus and also contributed to the fields
ofdifferential equationsandvectoranalysis. Peano played a key role in theaxiomatizationof mathematics and
was a leading pioneer in the development of mathematical logic. Peano had by this stage become heavily
involved with the Formulario project and his teaching began to suffer. In fact, he became so determined to
teach his new mathematical symbols that the calculus in his course was neglected. As a result he was
dismissed from the Royal Military Academy but retained his post at Turin University.
In 1903 Peano announced his work on an international auxiliary language calledLatino sine
flexione("Latinwithout inflexion," later called Interlingua, but which should not be confused with the
laterInterlinguaof theIALA). This was an important project for him (along with finding contributors for
'Formulario'). The idea was to use Latin vocabulary, since this was widely known, but simplify the grammar as
much as possible and remove all irregular and anomalous forms to make it easier to learn. In one speech, he
started speaking in Latin and, as he described each simplification, introduced it into his speech so that by the
end he was talking in his new language.
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The year 1908 was big for Peano. The fifth and final edition of the Formulario project, titled Formulario
Mathematico, was published. It contained 4200 formulae and theorems, all completely stated and most of them
proved. The book received little attention since much of the content was dated by this time. However, it remains
a significant contribution to mathematical literature. The comments and examples were written in Latino sine
flexione.
Also in 1908, Peano took over the chair of higher analysis at Turin (this appointment was to last for only two
years). He was elected the director ofAcademia pro Interlingua. Having previously createdIdiom Neutral, the
Academy effectively chose to abandon it in favor of Peano'sLatino sine flexione.
After his mother died in 1910, Peano divided his time between teaching, working on texts aimed for secondary
schooling including a dictionary of mathematics, and developing and promoting his and otherauxiliary
languages, becoming a revered member of the international auxiliary language movement. He used his
membership of theAccademia dei Linceito present papers written by friends and colleagues who were not
members (the Accademia recorded and published all presented papers given in sessions).
In 1925 Peano switched Chairs unofficially from Infinitesimal Calculus to Complementary Mathematics, a field
which better suited his current style of mathematics. This move became official in 1931. Giuseppe Peano
continued teaching at Turin University until the day before he died, when he suffered a fatal heart attack.
Milestones and honors received [edit]
1881: Published first paper.
1884: Calcolo Differenziale e Principii di Calcolo Integrale.
1887:Applicazioni Geometriche del Calcolo Infinitesimale.
1889: Appointed Professor First Class at the Royal Military Academy.
1890: Appointed Extraordinary Professor ofinfinitesimal calculusat theUniversity of Turin.
1891: Made a member of the Academy of Science, Torino.
1893: Lezioni di Analisi Infinitesimale, 2 vols.
1895: Promoted to Ordinary Professor.
1901: Made Knight of theOrder of Saints Maurizio and Lazzaro.
1903: AnnouncesLatino sine flexione.
1905: Made Knight of theOrder of the Crown of Italy. Elected a corresponding member of theAccademia
dei LinceiinRome, the highest Italian honour for scientists.
1908: Fifth and final edition of theFormulario mathematico.
1917: Made an Officer of the Crown of Italy.
1921: Promoted to Commendatore of the Crown of Italy.
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Bibliography [edit]
Peano's writings in English translation
1889. "The principles of arithmetic, presented by a new method" inJean van Heijenoort, 1967.A SourceBook in Mathematical Logic, 18791931. Harvard Univ. Press: 8397.
1973. Selected works of Giuseppe Peano. Kennedy, Hubert C., ed. and transl. With a biographical sketch
and bibliography. London: Allen & Unwin.
Secondary literature
Gillies, Douglas A., 1982. Frege, Dedekind, and Peano on the foundations of arithmetic. Assen,
Netherlands: Van Gorcum.
Ivor Grattan-Guinness, 2000. The Search for Mathematical Roots 18701940. Princeton University Press.
Kennedy, Hubert C., 1980.Peano: Life and Works of Giuseppe Peano. Reidel. Biography with complete
bibliography (p. 195209).
References [edit]
1. ^Richard N. Aufmann; Joanne Lockwood (29 January 2010).Intermediate
Algebra: An Applied Approach. Cengage Learning. pp. 10.ISBN978-1-
4390-4690-6. Retrieved 14 August 2011.
2. ^The man who painted the MTA. Luigi Crosio 1835 - 1916. Schoenstatt
webpage
3. ^Ziwet, Alexander (1891)."A New Italian Mathematical Journal".Bull. Amer.
Math. Soc.1 (2): 4243.
4. ^Hubert C. Kennedy (30 May 1980).Peano, life and works of Giuseppe
Peano. D. Reidel Pub. Co. p. 169.ISBN978-90-277-1067-3. Retrieved 14
August 2011.
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