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An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - References

Nicole M. Wessman-Enzinger (Illinois State University)
Arithmetic Books (1658 – 1882)

Adams, D. (1808). The scholar’s arithmetic or federal accountant (5th ed.). Troy, NY: John Prentiss.

Adams, D. (1830). Adam’s new arithmetic. Keene, NH: J. & J.W. Prentiss.

Adams, D. (1848). Adam’s new arithmetic: Revised edition. Keene, NH: J. & J.W. Prentiss.

Ayres, J. (1711). Arithmetick: A treatise defined for the use and benefit of trades-men (11th ed.). London, UK: Tho. Norris at Looking Glass on London Bridge.

Barreme, N. (1747). L’Arithmetique du Sr Barreme ou le livre facile pour apprendre l’arithmetique. Paris, France: Gandouin.

Botham, P.E. (1835). Botham’s common school arithmetic: The common school arithmetic. Hartford, CT: Henry Benton.

Braune, A. (1882). Vollständige Kaufmännische Arithmetik für Handels-, Real- und Gewerbescheulen sowie zum Selbstunterrichte für Angehende Geschäftsleute. Leipzig: Verlag von Ferdinand Hirt & Sohn.

Brookes, J. (1776). A treatise on arithmetic. New Castle, UK: J. Smith.

Cocker, E. (1702). Cocker’s arithmetick. London, UK:  W. Richardson.

Colburn, W. (1824). First lessons in arithmetic (2nd ed.). Boston, MA: Hilliard & Metcalf.

Colburn, D. P. (1855). Arithmetic and its applications: Designed as a text book for common schools, high schools, and academies. Philadelphia, PA: H. Cowperthwait & Co. 

Colburn, D .P. (1858). The common-school arithmetic: A practical treatise on the science of numbers. Philadelphia, PA: H. Cowperthwait & Co. 

Daboll, N. (1829). Daboll’s schoolmaster’s assistant, a plain practical system of arithmetic. Utica, NY: Hastings & Tracy.

Dilworth, T. (1802). The schoolmaster’s assistant: Being a compendium of arithmetic, both practical and theoretical. New York, NY: Lewis Nichols.

Dilworth, T. (1810). The schoolmaster’s assistant: Being a compendium of arithmetic, both practical and theoretical. New York, NY: E. Duyckinck & R. McDermott. 

Farrar, J. (1818). Arithmetic taken primarily from the arithmetic of S.F. Lacroix. Cambridge, MA: Hilliard & Metcalf.

Fish, D.W. (1874). Robinson’s shorter course: The complete arithmetic oral and written. New York, NY: Ivison, Blakeman, & Co.

Fisher, G. (1785). The instructor. Edinburgh, Scotland: J. Dickson.

Gough, J. (1803). Practical arithmetick. Dublin, Ireland: Author. 

Greenleaf, B. (1866). The national arithmetic, on the inductive system, combining the analytic and synthetic methods; forming a complete course of higher arithmetic. Boston, MA: Robert S. Davis & Co.

Greenwood, I. (1729). Arithmetick vulgar and decimal with the application thereof to a variety of cases in trade and commerce. Boston, MA: Kneeland & Green. 

Lee, C. (1797). The American accomptant: Being a plain, practical and systematic compendium of federal arithmetic. Lansingburgh, NY: William W. Wands.

Pike, N. (1788). The new complete system of arithmetic, composed for the use of the citizens of the United States. Newburyport, MA: John Mycall.

Pinnock, William (1823). Pinnock’s catechism of arithmetic containing the most useful rules with exercises upon each (6th ed.). London, UK: G. & W.B. Whittaker.

Ray, J. (1856). Ray’s higher arithmetic: The principles of arithmetic. Cincinnati, OH: Winthrop B. Smith & Co.

Ray, J. (1877). Eclectic educational series: Ray’s new practical arithmetic. Cincinnati, OH: Van Antwerp, Bragg & Co.

Record[e], R. (1658). Record’s arithmetick or the ground of arts. London, England: James Mether.

[Spence, David] (1721). Arithmetick compendiz’d or, a short treatise of arithmetick. Edinburgh, Scotland: William Monro.

Walsh, M. (1828). The mercantile arithmetic. Boston, MA: Richardson & Lord.

Wentworth, G.A. (1897). A practical arithmetic. Boston, MA: Ginn & Company.  

Weston, T. (1729). A treatise of arithmetic in whole numbers and fractions. London, UK: J. Hooke.   

Wingate, W. H. (1865). The parallel arithmetic:  A collection of examples composed and arranged on a new plan. London, UK: Longman, Green, Longman, Roberts, & Green.

Some Cyphering Books from the Clements-Ellerton Collection (1702 – 1834)

Bigelow, G. (c. 1820).  NY: Unpublished manuscript.

Brayton, O. (1792). Unpublished manuscript.

Burlingame, T. (1779 –1784). Burlingame, RI: Unpublished manuscript.

Butlar, J. (1785). Bucks County, PA: Unpublished manuscript. 

Cuming, M. (c. 1820). RI: Unpublished manuscript.

Green, W. (1795-1800). Connecticut: Unpublished manuscript.

Hall, E. (1780). Unpublished manuscript.

Johnson, L. (1798-1801). MA: Unpublished manuscript. 

Prust, T. (1702). Unpublished manuscript.

Steiner, M. (1801). Germany: Unpublished manuscript.

Senter, W. (1795-1801). Portsmouth, NH:  Unpublished manuscript.

Thompson, J. (1820-1834). ME:  Unpublished manuscript.

Townsend, W.  (1824-1830).  NY:  Unpublished manuscript.  

Unknown author (c. 1820). RI: Unpublished manuscript.

Wilson, U.  (1823-1827).  PA: Unpublished manuscript.

References (1905 – 2012)

Ellerton, N. F. & Clements, M. A. (2012). Rewriting the history of school mathematics in North America 1607-1861. New York, NY: Springer. 

Johnson, J. T. (1938). The relative merits of three methods of subtraction: An experimental comparison of the decomposition method of subtraction with the equal additions and the Austrian method. New York, NY: Teachers College, Columbia University.

National Governors Association Center for Best Practices. Council of Chief State School Officers. (2010). Common core state standards for mathematics. Washington DC: Author. Retrieved from

Osburn, W. J. (1927). How shall we subtract? Journal of Educational Research, 16(4): 237 -246.  

Ross, S., & Pratt-Cotter, M. (2000). Subtraction in the United States: An historical perspective. Mathematics Educator, 8(1), 49-56.

Smith, D. E. (1905). Handbook to Smith’s arithmetic. Boston, MA: Ginn & Co.

Smith, D. E. (1925). History of mathematics: Special topics of elementary mathematics (Vol. 2). Boston, MA: Ginn & Co.

Smith, D. E. (1909). The teaching of arithmetic. Boston, MA: Ginn & Co.

Swetz, F. J. (1987). Capitalism and arithmetic. La Salle, IL: Open Court.

Nicole M. Wessman-Enzinger (Illinois State University) , "An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - References," Loci (January 2014)