# Peirce, Benjamin

# PEIRCE, BENJAMIN

(*b*. Salem, Massachusetts, 4 April 1809; *d*. Cambridge, Massachusetts, 6 October 1880)

*mathematics, astronomy*.

In an address before the American Mathematical Society during the semicentennial celebration of its founding in 1888 as the New York Mathematical Society, G. D. Birkhoff spoke of Benjamin Peirce as having been “by far the most influential scientific personage in America” and “a kind of father of pure mathematics in our country.”

Peirce’s background and training were completely American. The family was established in America by John Peirce (Pers), a weaver from Norwich, England, who settled in Watertown, Massachusetts, in 1637. His father, Benjamin Peirce, graduated from Harvard College in 1801, and served for several years as representative from Salem in the Massachusetts legislature; he was Harvard librarian from 1826 until 1831, prepared a printed catalog of the Harvard library (1830–1831), and left a manuscript history of the university from its founding to the period of the American Revolution (published 1833). Peirce’s mother, Lydia Ropes Nichols of Salem, was a first cousin of her husband. On 23 July 1833 Peirce married Sarch Hunt Mills, daughter of Harriette Blake and Elijah Hunt Mills of Northampton, Massachusetts. They had a daughter, Helen, and four sons: James Mills Peirce, professor of mathematics and an administrator at Harvard for fifty years; Charles Sanders Peirce, geodesist, mathematician, logician, and philosopher; Benjamin Mills Peirce, a mining engineer who wrote the U.S. government report on mineral resources and conditions in Iceland and Greenland; and Herbert Henry Davis Peirce, a diplomat who served on the staff of the legation in St Petersburg and who later arranged for the negotiations between Russia and Japan that led to the Treaty of Portsmouth on 5 September 1905.

Peirce attended the Salem Private Grammar School, where Henry Ingersoll Bowditch was a classmate. This relationship influenced the entire course of Peirce’s life, since Ingersoll Bowditch’s father, Nathaniel Bowditch, discovered Peirce’s unusual talent for mathematics. During Peirce’s undergraduate career at Harvard College (1825–1829), the elder Bowditch enlisted Peric’s aid in reading the proof-sheets of his translation of Laplace’s *Traité de mécanique céleste* Peirce gave evidence of his own mathematical powers in his revision and correction of Bowditch’s translation and commentary on the first four volumes (1829–1839), and also with his proof (in 1832) that there is no odd perfect number that has fewer than four prime factros.

Peirce taught at Bancroft’s Round Hill School at Northampton, Massachusetts, from 1829 until 1831, when he was appointed tutor in mathematics at Harvard College; he recevied his M.A. from that insitution in 1833. At Harvard he became University professor of mathematics and natural philosophy (1833–1842), then Perkins professor of astronomy and mathematics (1842–1880). During the early days of his teaching at Harvard, Peirce published a popular series of textbooks on elementary branches of mathematics.

Peirce’s continued interest in the theory of astronomy was apparent in his study of comets. Around 1840 he made observations in the old Harvard College observatory; his 1843 Boston lectures on the great comet of that year stimulated the support that led to the installation of the new telescope at the Harvard Observatory in June 1847. Since 1842 Peirce had also supervised the perparation of mathematics section of the ten-volume *American Almanac and Repository of Useful Knowledge*, and in 1847 he published therein a list of known orbits of comets. In 1849 Charles Henry Davis a brother-in-law of Peirce’s wife, was apponited superintendent of the newly created *American Ephemeris and Nautical Almanac*, and Peirce was appointed consulting astronomer (1849–1867).

Peirce was not only helpful to Davis in planning the general form of the *Ephemeris*, but he also began a revision of the theory of planets. He had become deeply interested in the work of Le Verrier and John Couch Adams that had permitted Galle’s discovery of the planet Neptune on 23 September 1846. In cooperation with Sears Walker, Peirce determined the orbit of Neptune and its perturbation of Uranus. Simon Newcomb wrote in his *Popular Astrononmy* (1878) that the investigation of the motion of the new planet was left in the hands of Walker and Peirce for several years, and that Peirce was “the first one to compute the perturbations of Neptune produced by the action of the other planets.” Peirce was led to believe that Galle’s “happily” discovered Neptune and Le Verrier’s calculated theoretical planet were not the same body and that the latter did not exist—an opinion that led to considerable controversy.

In conjunction with his work on the solar system, Peirce became interested in the mathematical theory of the rings of Saturn. In 1850 George Phillips Bond, assistant in the Harvard College observatory, discovered Saturn’s dusky ring and on 15 April 1851 announced to a meeting of the American Academy of Arts and Sciences his belief that the rings were fluid, multiple, and variable in number. Peirce published several mathematical papers on the constitution of the rings in which he reached the same conclsion concerning their fluidity. His review of the problem at that time led to a most unfortunate priority dispute.

Peirce also enjoyed a distinguished career in the U. S. Coast Survey. In 1852 he accepted a commission —at the request of Alexander Dallas Bache, who was then superintendent—to work on the determination of longitude for the Survey. This project involved Peirce in a thorough investigation of the question of errors of observation; his article “Criterion for the Rection of Doubtful Observations” appeared in B. A. Gould’s *Astronomical Journal* in July 1852. The criterion was designed to determine the most probable hypothesis whereby a set of observations might be divided into normal and abnormal, when “the greater part is to be regarded as normal and subject to the ordinary law of error adopted in the method of least squares, while a smaller unknown portion is abnormal and subject to some obscure soucre of error.” Some authorities regarded “Peirce’s criterion”—which gave good discrimination and acceptable practical results—as one of his most important contributions, although it has since been demonstrated to be invalid.

After Bache’s death Peirce became superintendent of the Coast Survey (1867–1874), while maintaining his association with Harvard. He arranged to carry forward Bache’s plans for a geodetic system that would extend from the Atlantic to the Gulf. This project laid the foundation for a general map of the country independent of detached local surveys. Peirce’s principal contribution to the development of the Survey is thought to have been the initiation of a geodetic connection between the surveys of the Atlantic and Pacific coasts. He superintended the measurement of the arc of the thirty-ninth parallel in order to join the Atlantic and Pacific systems of triangulation.

Peirce also took personal charge of the U.S. expedition that went to Sicily to observe the solar eclipse of 22 December 1870, and, as a member of the transit of Venus commission, sent out two Survey parties—one to Nagasaki and the other to Chatham Island—in 1874. Peirce also played a role in the acquisition of Alaska by the United States in 1867, since in that year he sent out a reconaissance party, whose reports were important aids to proponents of the purchase of that region. In 1869 he sent parties to observe the eclipse of the sun in Alaska and in the central United States.

Peirce’s eminence made him influential in the founding of scientific institutions in the United States. In 1847 the American Academy of Arts and Sciences appointed him to a committee of five in order to draw up a program for the organization of the Smithsonian Institution. From 1855 to 1858 he served with Bache and Joseph Hery on a council to organize the Dudley observatory at Albany, New York, under the direction of B. A. Gould. In 1863 he became one of the fifty incorporators of the National Academy of Sciences.

Despite his many administrative obligations, Peirce continued to do mathematics in the 1860’s. He read before the National Academy of Sciences a number of papers on algebra, which had resulted from his interest in Hamilton’s calculus of quaternions and finally led to Peirce’s study of possible systems of multiple algebras. In 1870 his *Linear Associative Algebra* appeared as a memoir for the National Academy and was lithographed in one hundred copies for private circulation. The opening sentence states that “Mathematics is the science which draws necessary conclusions.” George Bancroft received the fifty-second copy of the work, and in an accompanying letter (preserved in the manuscript division of the New York Public Library) Peirce explained that.

This work undertakes the investigation of all possible single, double, triple, quadruple, and quintuple Algebras which are subject to certain simple and almost indispensable conditions. The conditions are those wellknown to algebraists by the terms of

distributiveandassociativewhich are defined on p. 21. It also contains the investigation of all sextuple algebras of a certain class, i.e., of those which contain what is called in this treatise anidempotentelement.

D. E. Smith and J. Ginsburg, in their *History of Mathematics Before 1900,* speak of Peirce’s memoir as “one of the few noteworthy achievements in the field of mathematics in American before the last quarter of the century.” It was published posthumously in 1881 under the editorship of his son Charles Sanders Peirce *American Journal of Mathematics*, **4** no. 2, 97–229).

In *A System of Analytic Mechanics* (1855) Peirce again set forth the principles and methods of the science as a branch of mathematical thethod, a subject he development from the idea of the “potential.” The book has been described as the most important mathematical treatise that had been produced in the United States up to that time. Peirce’s treatment of mechanics has also been said, by Victor Lenzen, to be “on the highest level of any work in the field in English until the appearance of Whittaker’s *Analytical Dynamics”* in 1904. Peirce was widely honored by both American and foreign scholarly and scientific soieties.

## BIBLIOGRAPHY

I. Original Works. Peirce’s works include *An Elementary Treatise on Sound* (Boston, 1836); *An Elementary Treatise on Algebra* (Boston, 1837), to which are added exponential equations and logarithms; *An Elementary Treatise on Plane and Solid Geomerty (Boston, 1837); An Elementary Treatise on Plane and Spherical Trigonometry. . . Particularly Adapted to Explaining the Construction of Bowditch’s Navigator and the Natuical Almanac* (Boston, 1840); *An Elementary Treatise on Curves,Functions, and Forces, 2 vols. (Boston, 1841, 1846); and Tables of the Moon* (Washington, D. C. 1853) for the *American Ephemeries and Naturical Almanac. Tables of the**Moon* was used in taking the *Ephemeris* up to the volume for 1883 and was constructed from Plana’s theory, with Airy’s and Longstreth’s corrections, Hansen’s two inequalities of long period arising from the action of Venus, and Hansen’s values of the secular variations of the mean motion and of the motion of the perigee.

Later works are *A System of Analytic Mechanics* (Boston, 1855); *Linear Associative Algebra* (1870), edited by C. S. Peirce, which appeared in *American Journal of Mathematics*, **4** (1881), 97–229, and in a separate vol. (New York, 1882); and James Mills Peirce, ed., *Ideality in the Physical Sciences,* Lowell Institute Lectures of 1879 (Boston, 1881).

Peirce’s unpublished letters are in the National Archives, Washington, D. C. and in the Benjamin Peirce and Charles S. Peirce collections of Harvard University.

II. Secondary Literature. On Peirce and his work, see reminiscences by Charles W. Eliot, A. Lawrence Lowell, W. E. Byerly, Arnold B. Chace, and a biographical sketch by R. C. Archibald, in *American Mathematical Monthly*, **32** (1925), repr. as a monograph, with four new portraits and addenda (Oberlin, 1925), which contains in sec. 6 a listing with occasional commentary of Peirce’s writings and massive references to writings about him. See also Bessie Zaban Jones and Lyle Gifford Boyd, *The Harvard College Observatory* (Cambridge, Mass., 1971), esp. the chap. entitled “The Two Bonds,” which gives a detailed description of the unhappy relationship that developed between Peirce and George and William Bond.

See further R. C. Archibald, in *Dictionary of American Biography* (New York, 1934); A. Hunter Dupree, “The Founding of the National Academy of Sciences℄A Reinterpretation,” in *Proceedings of the American Philosophical Society*, **101** , no. 5 (1957), 434–441; M. King, ed., *Benjamin Peirce. . . A Memorial Collection* (Cambridge, Mass., 1881); Victor Lenzen, *Benjamin Peirce and the United States Coast Survey* (San Franciso, 1968); Simpon Newcomb, *Popular Astronomy* (New York, 1878), esp. pp. 350 (on the rings of Saturn), 363 (on the perturbation of Neptune), and 403 (on comets); H. A. Newton, “Benjamin Peirce,” in *Proceedings of the American Academy of Arts and Sciences*, 167–178; James Mills Peirce, in *Lamb’s Biographical Dictionary of the United States*, VI (Boston, 1903), 198; and Poggendorff, II (1863), 387–388; and III (1858–1883), 1012–1013. See also F. C. Pierce, *Peirce Genealogy* (Worcester, Mass., 1880).

Carolyn Eisele

# Benjamin Peirce

# Benjamin Peirce

Mathematician, physicist, and astronomer, Benjamin Peirce (1809-1880) has been called the "fatherof American mathematics." He distinguished himself as superintendent of the U.S. Coast Survey. Peircewas a professor at Harvard College from 1833 until his death, and contributed to the founding of its observatory.

Peirce is remembered for calculating the exact location and path of the planet Neptune, using a series of lengthy and complex mathematical equations. He also tracked and calculated the paths of many comets and charted the phases of the moon for the United States government. Soon after the discovery of Saturn's rings in 1850, he calculated and published his assessment of the possible composition of the rings. Peirce's work and his writings on practical mathematics extended to the fields of astronomy, mechanics, and geology. His contributions to the field of theoretical mathematics included commentary on hypercomplex number theory.

## A Prominent Puritan

Benjamin Peirce was born in Salem, Massachusetts, on April 4, 1809. He was the third child of Benjamin Peirce and the former Lydia Ropes Nichols and was descended from a long line of Puritans. The Peirce family came from Norfolk County, England, to Watertown, Massachusetts, around 1634. The Peirces were craftsmen, weavers, shopkeepers, farmers, and East India traders. The elder Benjamin Peirce served as a state legislator and later joined the staff at Harvard as a distinguished librarian.

Peirce attended Salem Private Grammar School. There he came under the influence of the mathematician and scientist Nathaniel Bowditch, whose son attended school with Peirce. Bowditch let Peirce read proofs of his English translation of P.S. Laplace's astronomy treatise, *Traite de Mecanique Celeste* ( *Celestial Mechanics* ). The translation was published as Peirce was graduating from Harvard in 1829. Until 1839, Peirce contributed to revising and refining the Bowditch translation. Peirce had a lifelong professional relationship with Bowditch and later paid tribute to Bowditch in the introduction to a treatise Peirce wrote on analytical mechanics, calling him "the father of American geometry."

## Harvard Professor

After graduating from Harvard, Peirce joined the teaching staff at Round Hill School in Northampton, Massachusetts. After two years at Round Hill, Peirce joined the faculty at Harvard, initially as a mathematics tutor. In 1833 he became professor of mathematics and astronomy at Harvard, and in 1842 he was named Perkins professor at the college.

So complex was Peirce's mind that it was difficult for all but the most proficient mathematicians to understand much of his work. The speed of his mental processes made it difficult for him to describe his work coherently with pen and paper. His lectures were far too advanced for many students to follow. Those students who heard him speak often said they felt awed by his commitment to the discipline of mathematics.

During his tenure at Harvard, Peirce published a variety of mathematics texts. These included *Plane Trigonometry* in 1835, *Spherical Trigonometry* in 1836, and *Plane and Solid Geometry* in 1837. He published his *Treatise on Sound* in 1836. In 1840 he published another text on plane geometry, followed in 1841 by the first volume of a text on mechanics; a second volume followed in 1846.

Peirce was instrumental in the founding of the Harvard College Observatory in 1839. When in March 1843 he observed a comet, he spearheaded an intensive effort to get a 15-inch telescope at the observatory. The telescope was installed in 1847, allowing Peirce better opportunity to study celestial phenomena. The telescope was instrumental in the discovery of the planet Neptune that year and in calculating its path. As the mathematics editor of the *American Almanac,* Peirce published his calculations and observations of the known orbits of comets in the almanac's 1847 edition.

## Lazzaroni

Around 1840, a small but prominent cadre of American scientists and mathematicians assembled to seek support and funding for national development in science. The group was known sarcastically as Lazzaroni, or beggars. It had close ties to a formidable European scientific community. Peirce aligned himself with the Lazzaroni, who eventually became the nucleus of the American Association for the Advancement of Science. The association solicited leading industrialists for political backing and support for scientific advancement.

In 1843 Alexander Dallas Bache, a self-ordained leader of the Lazzaroni, became the second superintendent of the U.S. Coast Survey. The Coast Survey, authorized in 1807, was the first scientific agency established by the U.S. Government. Under the influence of the Lazzaroni, the Coast Survey helped spur the scientific movement in the United States. In 1852, at superintendent Bache's request, Peirce calculated the longitudes of various places in the United States.

Peirce and his colleagues in the educational and scientific communities persuaded Congress to authorize funding for the publication of the *American Ephemeris and Nautical Almanac.* A bureau for the publication was established in Washington, D.C., in 1849, and the project was delegated to Harvard College, with Peirce as consulting mathematician. The moon tables, as calculated by Peirce, were published in 1853 and described the phases of the moon through 1883.

In the 1840s Peirce joined with one of his former Harvard students, Benjamin Apthorp Gould, and a Swiss naturalist, Louis Agassiz, to lobby for expansion of the science facility at Harvard College. They were instrumental in the opening of Harvard's Lawrence Scientific School in 1847. Peirce, with his high-visibility projects during those years, contributed significantly to the development of a Harvard-based sphere of scientific influence.

In 1863, Peirce and the American Association for the Advancement of Science helped get Congress to establish the National Academy of Sciences. Peirce was among its 50 charter members. He was a member of the organizing committee and served as chairman of the mathematics and physics groups of the academy.

In 1867 Peirce relinquished his post as consultant for the American Ephemeris bureau to become superintendent of the U.S. Coast Survey. His three-year association with the Coast Survey is considered one of his most significant contributions. Among the far-reaching projects undertaken by the Survey under Peirce was surveying Alaska, which had just been purchased from Russia. Peirce also sent representatives worldwide to observe solar eclipses.

## Astronomer of Note

Because of his exceptional competence in performing lengthy calculations and comprehending complex geometric concepts, Peirce made many calculations about objects in space. In 1847 he published a work containing the known orbits of comets. That same year he performed a series of extended calculations to determine the precise location, orbit, and weight of Neptune, which had been discovered in 1846.

In 1850 rings were discovered around Saturn, leading to much speculation about their composition. George Phillips Bond of the Harvard College Observatory, who discovered the rings, suggested that they might be fluid. Peirce contested that theory and put forth a comprehensive set of formulas to disprove it. He published an initial analysis of the nature of the rings in 1851. In 1866 he published his formulas for the rings in *Memoirs of the National Academy of Sciences.*

In 1852 Peirce described a hypotheses concerning probability, which came to be known as Peirce's criterion. Although his premise was ultimately disproved, he left an extraordinary amount of detailed calculations about probability. In the *Astronomical Journal* of December 31, 1858, he published a theory detailing the hyperbolic orbit of comets. His *System of Analytical Mechanics,* which he dedicated to Bowditch, was regarded well into the 20th century as definitive.

Peirce lectured widely, and during his lectures he routinely introduced new concepts. In an 1870 presentation called "Linear Associative Algebra," which he presented to the National Academy of Sciences, he introduced the terms *idempotent* and *nilpotent.* The former describes a number that approaches itself when calculated to a power of two or greater, and the latter refers to a number that approaches zero. He later self-published the speech, which contained his widely quoted statement: "Mathematics is the science which draws necessary conclusions." He also coined the phrase "indeterminate form" to describe the incalculable equation of zero divided by zero.

Peirce died on June 10, 1880, in Cambridge, Massachusetts. His final significant work, *Ideality in the Physical Sciences,* was published posthumously in 1881. It was the text of a lecture that he delivered early in 1879.

Peirce's use of mathematical calculations to predict accurately such phenomena as the recurrence of comets and the phases of the moon demonstrated the practicality of his knowledge. He performed his extensive calculations without the convenience of an adding machine, much less a calculator or computer.

Peirce married Sarah Hunt Mills on July 23, 1833. Together they had five children: James Mills, born in 1834; Charles Sanders, born in 1839; Benjamin Mills, born in 1844; Helen Huntington, born in 1845; and Herbert Henry Davis, born in 1849. The children, like their father, grew to be prominent citizens: a Harvard professor, a noted Harvard-based scientist and philosopher, a mining engineer, the wife of a realtor, and a foreign diplomat respectively.

## Books

*Dictionary of American Biography,* American Council of Learned Societies, 1928-1936.

Lenzen, V.F., *Benjamin Peirce and the U.S. Coast Survey,* San Francisco Press, 1968.

## Periodicals

*New Solidarity,* August 22, 1986.

## Online

"Earliest Known Uses of Some of the Words of Mathematics," http://members.aol.com/jeff570/i.html (February 5, 2001). □

# Benjamin Peirce

# Benjamin Peirce

**1809-1880**

Benjamin Peirce is generally regarded as the first American research mathematician. Peirce was a professor of mathematics and astronomy at Harvard from 1833 to 1880. He also served in the important position of superintendent of the Coast Survey from 1867 to 1874. Peirce's most important work in mathematics was *Linear Associative Algebra,* in which he investigated various possible systems of algebra. Written and circulated privately in 1870, *Linear Associative Algebra *was published posthumously in 1881.

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