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{{Image|Benjamin Peirce.jpeg|right|250px|Portrait of Benjamin Peirce (1857) by [[Daniel Huntington]]. }}
{{Image|Benjamin Peirce.jpeg|right|250px|Portrait of Benjamin Peirce (1857) by [[Daniel Huntington]]. }}
'''Benjamin Peirce''' (April 4, 1809, [[Salem, Massachusetts]] &ndash; October 6, 1880, [[Cambridge, Massachusetts]]) was the first internationally known American-born mathematician and is sometimes called "the father of American mathematics". He was the first to recognize as an important mathematical structure  the [[associative algebra|linear associative algebra]].<ref>In 1870 B. Peirce's ''Linear associative algebra'' existed in handwritten form only. It was copied by a lady with fine handwriting on lithograph stone and about one hundred  copies were printed and distributed over the world. In 1881 the text was published posthumously in the American Journal of  Mathematics, [vol '''4''', pp. 97-215 (1881)] at  the insistence  of his son Charles  Peirce, who thought  it  represented his father's  best work. </ref> He derived several of its properties and gave "[[Peirce's reduction]]" for the elements.<ref>H. Weyl, ''The Theory of Groups and Quantum Mechanics'', Dover (1950)</ref>
'''Benjamin Peirce''' (April 4, 1809, [[Salem, Massachusetts]] &ndash; October 6, 1880, [[Cambridge, Massachusetts]]) was the first renowned American-born mathematician and is sometimes called "the father of American mathematics". He was the first to recognize as an important mathematical structure  the [[associative algebra|linear associative algebra]].<ref>In 1870 B. Peirce's ''Linear associative algebra'' existed in handwritten form only. It was copied by a lady with fine handwriting on lithograph stone and about one hundred  copies were printed and distributed over the world. In 1881 the text was published posthumously in the American Journal of  Mathematics, [vol '''4''', pp. 97-215 (1881)] at  the insistence  of his son Charles  Peirce, who thought  it  represented his father's  best work. </ref> He derived several of its properties and gave "[[Peirce's reduction]]" for the elements.<ref>H. Weyl, ''The Theory of Groups and Quantum Mechanics'', Dover (1950)</ref>


Peirce  was also a highly respected theoretical astronomer who performed some significant work on the orbit of the newly discovered (1846) planet [[Neptune]].   
Peirce  was also a highly respected theoretical astronomer who performed some significant work on the orbit of the newly discovered (1846) planet [[Neptune]].   
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==Life==
==Life==
Benjamin  Peirce  graduated from [[Harvard]]  in 1829 and accepted a teaching position with [[George Bancroft]] at his Round Hill School in [[Northampton, Massachusetts]]. Two years later, at  the  age  of  twenty-two, Peirce was asked to join the faculty at Harvard as a tutor in mathematics. In 1833 Peirce received his M.A. from Harvard and was promoted to professor of mathematics. In 1842 he became Harvard's Perkins Professor of Mathematics and Astronomy, a position he held until his death in  1880.
 
, at  the  age  of  twenty-two,  


In the  same year that he received his M.A. (1833),  Peirce married Sarah Hunt  Mills;  four sons were born to the couple at intervals of five years.  The  eldest,  James Mills (1834–1906),  was for forty-five years a  prominent  mathematician  at  Harvard;  Charles  Sanders (1839–1914),  was known  for  his  work  in  mathematics  and  physics, but also recognized for  his  discoveries  in  logic  and  philosophy;  Benjamin  Mills (1844–1870) a mining engineer, brilliant  but  undisciplined,  died  in  early  manhood; and  Herbert  Henry  Davis  (1849–1916) was  a  Cambridge businessman and diplomat.  
In the  same year that he received his M.A. (1833),  Peirce married Sarah Hunt  Mills;  four sons were born to the couple at intervals of five years.  The  eldest,  James Mills (1834–1906),  was for forty-five years a  prominent  mathematician  at  Harvard;  Charles  Sanders (1839–1914),  was known  for  his  work  in  mathematics  and  physics, but also recognized for  his  discoveries  in  logic  and  philosophy;  Benjamin  Mills (1844–1870) a mining engineer, brilliant  but  undisciplined,  died  in  early  manhood; and  Herbert  Henry  Davis  (1849–1916) was  a  Cambridge businessman and diplomat.  


In 1847 Benjamin Peirce was appointed to a five-man committee by the American Academy of Arts and Sciences to plan and organize what was to become the [[Smithsonian Institution]]. From 1849 to 1867 Peirce served as consulting astronomer to the newly created [[American Ephemeris and Nautical Almanac]].  Peirce was also one of the 50 founders of the [[National Academy of Sciences]] (1863). He stimulated the forming of the Harvard Observatory by  lecturing  on  [[Encke's  Comet]]  in 1843 and was an organizer of the [[Dudley Observatory]], [[Albany]], N.Y.  
Peirce was also one of the 50 founders of the [[National Academy of Sciences]] (1863). He stimulated the forming of the Harvard Observatory by  lecturing  on  [[Encke's  Comet]]  in 1843 and was an organizer of the [[Dudley Observatory]], [[Albany, New York]].  
 
a US government agency that was renamed to U.S. Coast and Geodetic Survey in 1878; it maintained  this name until 1970.


In 1852 he began a long association with the U.S. Coast  Survey, a US government agency that was renamed to U.S. Coast and Geodetic Survey in 1878; it maintained  this name until 1970.
. In 1871  Peirce convinced Congress to mandate a transcontinental geodetic survey<ref>R. P. Crease, ''Charles Sanders Peirce and the First Absolute Measurement Standard'', ''Physics Today'', December 2009, pp. 39-44.</ref> along the 39th parallel (the transcontinental arc that passes approximately through Baltimore-Denver-San Francisco). In addition, he oversaw the first geodetic map of the US.   
Starting as director of longitude determinations, he eventually became superintendent (from 1867 until 1874). In 1871  Peirce convinced Congress to mandate a transcontinental geodetic survey<ref>R. P. Crease, ''Charles Sanders Peirce and the first absolute measurement standard'', Physics Today, pp. 39-44, December 2009</ref> along the 39th parallel (the transcontinental arc that passes approximately through Baltimore-Denver-San Francisco). In addition, he oversaw the first geodetic map of the US.   


Before the [[American  Civil  War]], Peirce  was  a  pro-slavery  [[Democratic Party|Democrat]] with many  good  friends  in  the  South.  When the war started in 1861 with  the taking  of  [[Fort  Sumter]] (near [[Charleston]] SC) by the [[Confederates]],  Peirce changed his mind and  became  a  strong  Union  supporter.  Peirce was a deeply  religious  man, he  clung to  the  fundamental doctrine  of a personal,  loving God,  to whom  he made frequent  reference  in even his most technical  books and papers.
Before the [[American  Civil  War]], Peirce  was  a  pro-slavery  [[Democratic Party|Democrat]] with many  good  friends  in  the  South.  When the war started in 1861 with  the taking  of  [[Fort  Sumter]] (near [[Charleston]] SC) by the [[Confederates]],  Peirce changed his mind and  became  a  strong  Union  supporter.  Peirce was a deeply  religious  man, he  clung to  the  fundamental doctrine  of a personal,  loving God,  to whom  he made frequent  reference  in even his most technical  books and papers.


==Peirce's science==
==Peirce's science==
Peirce is mainly remembered for his work on the linear associative algebra of 1870. But before that
Peirce is mainly remembered for his work on the linear associative algebra of 1870. But before that he did other important work.  When he was not yet twenty he found an error in the proof of his countryman [[Nathaniel Bowditch]]'s translation of [[Pierre-Simon Laplace]]'s ''Traité de mécanique céleste'' [Treatise on Celestial Mechanics]. From then on he assisted regularly in the proof-reading of the translation.
he did other important work.  When he was not yet twenty he found an error in the proof of his countryman [[Nathaniel Bowditch]]'s translation of [[Pierre-Simon Laplace]]'s ''Traité de mécanique céleste'' [Treatise on Celestial Mechanics]. From then on he assisted regularly in the proof-reading of the translation.  


Noticeable work (1832) was his solution to a mathematical problem published in the journal Mathematical Diary, in which he proved that there is no odd [[perfect number]] (a positive integer that is equal to the sum of its proper divisors, such as 6=1+2+3) with fewer than four distinct prime factors.
Noticeable work (1832) was his solution to a mathematical problem published in the journal ''Mathematical Diary'', in which he proved that there is no odd [[perfect number]] (a positive integer that is equal to the sum of its proper divisors, such as 6=1+2+3) with fewer than four distinct prime factors.


In his early years of teaching, Peirce wrote a series of elementary textbooks in the fields of Trigonometry, Sound, Geometry, Algebra, and Mechanics.  All these texts were used in his own courses at Harvard as soon as they came out, but only the Trigonometry  became widely popular. These textbooks, although considered terse and difficult, had a lasting influence on the teaching of mathematics in America.<ref> S. R. Peterson, ''Benjamin Peirce: Mathematician and Philosopher'', Journal of the History of Ideas, Vol. '''16''', pp. 89-112 (1955)</ref>
In his early years of teaching, Peirce wrote a series of elementary textbooks in the fields of trigonometry, sound, geometry, algebra, and mechanics.  All these texts were used in his own courses at [[Harvard University]] as soon as they came out, but only the trigonometry became widely popular. These textbooks, although considered terse and difficult, had a lasting influence on the teaching of mathematics in America.<ref> S. R. Peterson, ''Benjamin Peirce: Mathematician and Philosopher'', Journal of the History of Ideas, Vol. '''16''', pp. 89-112 (1955)</ref>


In addition Peirce wrote on a wide range of topics, mostly astronomical or physical. Some of the problems he discussed were: the motion of two adjacent pendulums, the motion  
In addition, Peirce wrote on a wide range of topics, mostly astronomical or physical. Some of the problems he discussed were: the motion of two adjacent pendulums, the motion of a top, the fluidity and tides of Saturn's rings, and Encke's comet.   
of a top, the fluidity and tides of Saturn's rings,   and Encke's comet.   


Peirce's work on the orbits for [[Uranus]] and [[Neptune]] was triggered by the discovery of Neptune. In 1846 [[Le Verrier]] concluded from certain irregularities in the orbit of [[Uranus]] that there must exist another, yet unobserved, planet. He predicted its orbit and position.  His prediction was quickly verified by the observation of a new planet that was baptized [[Neptune]].  Peirce, however, pointed out that two solutions of the problem were possible and that Neptune would not have been discovered at all, except that by chance both possible locations lay at that particular time in the same direction from the earth.  Later, however, it was found that both men were wrong: Le Verrier because he had simply made an error in his calculations which resulted in a wrong orbit; Peirce because he accepted this wrong orbit as mathematically valid, and from it derived a second solution. Le Verrier had indicated the correct direction in which to look, but had predicted the wrong distance.  Nevertheless, the net result of the controversy was that Peirce gained international recognition as a mathematician and astronomer.
Peirce's work on the orbits for [[Uranus]] and [[Neptune]] was triggered by the discovery of Neptune. In 1846 [[Le Verrier]] concluded from certain irregularities in the orbit of [[Uranus]] that there must exist another, yet unobserved, planet. He predicted its orbit and position.  His prediction was quickly verified by the observation of a new planet that was baptized [[Neptune]].  Peirce, however, pointed out that two solutions of the problem were possible and that Neptune would not have been discovered at all, except that by chance both possible locations lay at that particular time in the same direction from the earth.  Later, however, it was found that both men were wrong: Le Verrier because he had simply made an error in his calculations which resulted in a wrong orbit; Peirce because he accepted this wrong orbit as mathematically valid, and from it derived a second solution. Le Verrier had indicated the correct direction in which to look, but had predicted the wrong distance.  Nevertheless, the net result of the controversy was that Peirce gained international recognition as a mathematician and astronomer.


Peirce's advanced treatise ''A System of Analytical Mechanics'' of 1855 was considered one of the most important mathematical books produced in the United States up to that date and was praised as being the best book on the subject at the time.
Peirce's 1855 advanced treatise ''[[A System of Analytical Mechanics]]'' was considered one of the most important mathematical books produced in the United States up to that date and was praised as being the best book on the subject at the time.


{{Image|Peirce symbols.JPG|right|475px|<small>Paragraph from Peirce's posthumously published (1882) book. Note the symbols for &pi; (<nowiki>=</nowiki>3.14&hellip;) and e (<nowiki>=</nowiki>2.71&hellip;). Further ''J''<sup>&nbsp;&minus;''J''</sup><nowiki>=</nowiki> &radic;e <sup>&pi;</sup></small> (<nowiki>=</nowiki>4.81&hellip;)}}
{{Image|Peirce symbols.JPG|right|475px|<small>Paragraph from Peirce's posthumously published (1882) book. Note the symbols for &pi; (<nowiki>=</nowiki>3.14&hellip;) and e (<nowiki>=</nowiki>2.71&hellip;). Further ''J''<sup>&nbsp;&minus;''J''</sup><nowiki>=</nowiki> &radic;e <sup>&pi;</sup></small> (<nowiki>=</nowiki>4.81&hellip;)}}
In 1870 he introduced a major contribution to the development of modern abstract algebra, his ''Linear Associative Algebra''. <ref>Helena M. Pycior, ''Benjamin Peirce's Linear Associative Algebra'' Isis, vol. '''70''',  pp. 537-551 (1979) </ref> He established the foundation for a general theory, classified all complex associative algebras of dimension less than seven, and presented multiplication tables for over 150 new algebras. This work originated in Peirce's interest in [[William Rowan Hamilton]]'s theory of [[quaternions]] (1843). The quaternions are a generalization of [[complex number]]s. They can be added and multiplied and thus form a structure that is now called an associative algebra.  Peirce recognized their essential properties and generalized it to an abstract concept. He found that algebra elements may have peculiar properties that ordinary numbers do not posses. For instance, it is possible that ''a·b'' = 0, while both algebra elements ''a'' and ''b'' are not equal to zero ([[null divisor]]s). He introduced [[nilpotent]] elements that have the property that ''a''<sup>''n''</sup> = 0 for some natural number ''n''. Most of these properties are now very well-known for matrices, which is not surprising since the set of ''n''&times;''n'' matrices is but an example of a linear associative algebra.  One of the great algebraist of the time, [[Arthur Cayley]] emphasized the significance of Peirce's approach and described it as the "analytical basis, and the true basis" of the complex numbers, the quaternions, and other algebras.
In 1870, he introduced a major contribution to the development of modern abstract algebra, his ''[[Linear Associative Algebra]]''.<ref>Helena M. Pycior, "Benjamin Peirce's ''Linear Associative Algebra''," Isis 70 (1979): 537-551.</ref> He established the foundation for a general theory, classified all complex associative algebras of dimension less than seven, and presented multiplication tables for over 150 new algebras. This work originated in Peirce's interest in [[William Rowan Hamilton]]'s 1843 theory of [[quaternions]]. The quaternions are a generalization of [[complex number]]s. They can be added and multiplied and thus form a structure that is now called an associative algebra.  Peirce recognized their essential properties and generalized it to an abstract concept. He found that algebra elements may have peculiar properties that ordinary numbers do not possess. For instance, it is possible that ''a•b'' = 0, while both algebra elements ''a'' and ''b'' are not equal to zero ([[null divisor]]s). He introduced [[nilpotent]] elements that have the property that ''a''<sup>''n''</sup> = 0 for some natural number ''n''. Most of these properties are now very well-known for matrices, which is not surprising since the set of ''n''&times;''n'' matrices is but an example of a linear associative algebra.  One of the great algebraist of the time, [[Arthur Cayley]], emphasized the significance of Peirce's approach and described it as the "analytical basis, and the true basis" of the complex numbers, the quaternions, and other algebras.


==References==
==References==
<references />
{{reflist}}[[Category:Suggestion Bot Tag]]

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Portrait of Benjamin Peirce (1857) by Daniel Huntington.

Benjamin Peirce (April 4, 1809, Salem, Massachusetts – October 6, 1880, Cambridge, Massachusetts) was the first renowned American-born mathematician and is sometimes called "the father of American mathematics". He was the first to recognize as an important mathematical structure the linear associative algebra.[1] He derived several of its properties and gave "Peirce's reduction" for the elements.[2]

Peirce was also a highly respected theoretical astronomer who performed some significant work on the orbit of the newly discovered (1846) planet Neptune.

Benjamin Peirce is the father of Charles Sanders Peirce, a well-known philosopher and mathematician.

Life

, at the age of twenty-two,

In the same year that he received his M.A. (1833), Peirce married Sarah Hunt Mills; four sons were born to the couple at intervals of five years. The eldest, James Mills (1834–1906), was for forty-five years a prominent mathematician at Harvard; Charles Sanders (1839–1914), was known for his work in mathematics and physics, but also recognized for his discoveries in logic and philosophy; Benjamin Mills (1844–1870) a mining engineer, brilliant but undisciplined, died in early manhood; and Herbert Henry Davis (1849–1916) was a Cambridge businessman and diplomat.

Peirce was also one of the 50 founders of the National Academy of Sciences (1863). He stimulated the forming of the Harvard Observatory by lecturing on Encke's Comet in 1843 and was an organizer of the Dudley Observatory, Albany, New York.

a US government agency that was renamed to U.S. Coast and Geodetic Survey in 1878; it maintained this name until 1970.

. In 1871 Peirce convinced Congress to mandate a transcontinental geodetic survey[3] along the 39th parallel (the transcontinental arc that passes approximately through Baltimore-Denver-San Francisco). In addition, he oversaw the first geodetic map of the US.

Before the American Civil War, Peirce was a pro-slavery Democrat with many good friends in the South. When the war started in 1861 with the taking of Fort Sumter (near Charleston SC) by the Confederates, Peirce changed his mind and became a strong Union supporter. Peirce was a deeply religious man, he clung to the fundamental doctrine of a personal, loving God, to whom he made frequent reference in even his most technical books and papers.

Peirce's science

Peirce is mainly remembered for his work on the linear associative algebra of 1870. But before that he did other important work. When he was not yet twenty he found an error in the proof of his countryman Nathaniel Bowditch's translation of Pierre-Simon Laplace's Traité de mécanique céleste [Treatise on Celestial Mechanics]. From then on he assisted regularly in the proof-reading of the translation.

Noticeable work (1832) was his solution to a mathematical problem published in the journal Mathematical Diary, in which he proved that there is no odd perfect number (a positive integer that is equal to the sum of its proper divisors, such as 6=1+2+3) with fewer than four distinct prime factors.

In his early years of teaching, Peirce wrote a series of elementary textbooks in the fields of trigonometry, sound, geometry, algebra, and mechanics. All these texts were used in his own courses at Harvard University as soon as they came out, but only the trigonometry became widely popular. These textbooks, although considered terse and difficult, had a lasting influence on the teaching of mathematics in America.[4]

In addition, Peirce wrote on a wide range of topics, mostly astronomical or physical. Some of the problems he discussed were: the motion of two adjacent pendulums, the motion of a top, the fluidity and tides of Saturn's rings, and Encke's comet.

Peirce's work on the orbits for Uranus and Neptune was triggered by the discovery of Neptune. In 1846 Le Verrier concluded from certain irregularities in the orbit of Uranus that there must exist another, yet unobserved, planet. He predicted its orbit and position. His prediction was quickly verified by the observation of a new planet that was baptized Neptune. Peirce, however, pointed out that two solutions of the problem were possible and that Neptune would not have been discovered at all, except that by chance both possible locations lay at that particular time in the same direction from the earth. Later, however, it was found that both men were wrong: Le Verrier because he had simply made an error in his calculations which resulted in a wrong orbit; Peirce because he accepted this wrong orbit as mathematically valid, and from it derived a second solution. Le Verrier had indicated the correct direction in which to look, but had predicted the wrong distance. Nevertheless, the net result of the controversy was that Peirce gained international recognition as a mathematician and astronomer.

Peirce's 1855 advanced treatise A System of Analytical Mechanics was considered one of the most important mathematical books produced in the United States up to that date and was praised as being the best book on the subject at the time.

PD Image
Paragraph from Peirce's posthumously published (1882) book. Note the symbols for π (=3.14…) and e (=2.71…). Further J −J= √e π (=4.81…)

In 1870, he introduced a major contribution to the development of modern abstract algebra, his Linear Associative Algebra.[5] He established the foundation for a general theory, classified all complex associative algebras of dimension less than seven, and presented multiplication tables for over 150 new algebras. This work originated in Peirce's interest in William Rowan Hamilton's 1843 theory of quaternions. The quaternions are a generalization of complex numbers. They can be added and multiplied and thus form a structure that is now called an associative algebra. Peirce recognized their essential properties and generalized it to an abstract concept. He found that algebra elements may have peculiar properties that ordinary numbers do not possess. For instance, it is possible that a•b = 0, while both algebra elements a and b are not equal to zero (null divisors). He introduced nilpotent elements that have the property that an = 0 for some natural number n. Most of these properties are now very well-known for matrices, which is not surprising since the set of n×n matrices is but an example of a linear associative algebra. One of the great algebraist of the time, Arthur Cayley, emphasized the significance of Peirce's approach and described it as the "analytical basis, and the true basis" of the complex numbers, the quaternions, and other algebras.

References

  1. In 1870 B. Peirce's Linear associative algebra existed in handwritten form only. It was copied by a lady with fine handwriting on lithograph stone and about one hundred copies were printed and distributed over the world. In 1881 the text was published posthumously in the American Journal of Mathematics, [vol 4, pp. 97-215 (1881)] at the insistence of his son Charles Peirce, who thought it represented his father's best work.
  2. H. Weyl, The Theory of Groups and Quantum Mechanics, Dover (1950)
  3. R. P. Crease, Charles Sanders Peirce and the First Absolute Measurement Standard, Physics Today, December 2009, pp. 39-44.
  4. S. R. Peterson, Benjamin Peirce: Mathematician and Philosopher, Journal of the History of Ideas, Vol. 16, pp. 89-112 (1955)
  5. Helena M. Pycior, "Benjamin Peirce's Linear Associative Algebra," Isis 70 (1979): 537-551.