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. Benjamin Peirce (1809–1880) was born at Salem; Mass., and graduated at Harvard College, having as undergraduate carried the study of mathematics far beyond the limits of the college course.[2] When Bowditch was preparing his translation and commentary of the Mécanique Céleste, young Peirce helped in reading the proof-sheets. He was made professor at Harvard in 1833, a position which he retained until his death. For some years he was in charge of the Nautical Almanac and superintendent of the United States Coast Survey. He published a series of college text-books on mathematics, an Analytical Mechanics, 1855, and calculated, together with Sears C. Walker of Washington, the orbit of Neptune. Profound are his researches on Linear Associative Algebra. The first of several papers thereon was read at the first meeting of the American Association for the Advancement of Science in 1864. Lithographed copies of a memoir were distributed among friends in 1870, but so small seemed to be the interest taken in this subject that the memoir was not printed until 1881 (Am. Jour. Math., Vol. IV., No. 2). Peirce works out the multiplication tables, first of single algebras, then of double algebras, and so on up to sextuple, making in all 162 algebras, which he shows to be possible on the consideration of symbols $\scriptstyle{A}$, $\scriptstyle{B}$, etc., which are linear functions of a determinate number of letters or units $\scriptstyle{i}$, $\scriptstyle{j}$, $\scriptstyle{k}$, $\scriptstyle{l}$, etc., with coefficients which are ordinary analytical magnitudes, real or imaginary,—the letters $\scriptstyle{i}$, $\scriptstyle{j}$, etc., being such that every binary combination $\scriptstyle{i^2}$, $\scriptstyle{ij}$, $\scriptstyle{ji}$, etc., is equal to a linear function of the letters, but under the restriction of satisfying the associative law.[56] Charles S. Peirce, a son of Benjamin Peirce, and one of the foremost writers on mathematical logic, showed that these algebras were all defective forms of quadrate algebras which he had previously discovered by logical analysis, and for which he had devised a simple notation. Of these quadrate algebras quaternions is a simple