Benjamin Peirce

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. He derived several of its properties and gave "Peirce's reduction" for the elements.

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
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.

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.

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. 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 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  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.

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 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.

In 1870 he introduced a major  contribution  to  the development  of  modern  abstract  algebra,  his  Linear  Associative  Algebra. 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 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 posses. 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&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.