Congressional Apportionment Using General Divisor Methods
The United States Constitution states that representatives in the House of Representatives "shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed" (14th amendment, section 2). The Constitution further places a ceiling on the number of total representatives at 1 per 30,000 of the total population and states that every state shall have at least one member in the House of Representatives (Article I, section 2). The Constitution does not specify, however, how to deal with the issue of rounding in the resulting computations.
Numerous methodologies have been suggested and employed over American history. Many are examples of "general divisor methods", which solve the problem , where is the population of state , is the total number of seats in the House of Representatives, and is some rounding function that maps a real number to an integer. This Demonstration shows how different divisor methods affect the representation of the American states for all of the decennial censuses taken since 1790. You select the census of interest. You choose how many seats your House of Representatives will hold. You then choose what to display: the absolute number of seats held by each state for each method, the number of seats held by each state for each method relative to the mean number of seats over all methods, the difference between the number of seats that each state would have if fractional seats were permitted ("quota") and the integer number of seats yielded by each method, or the difference between the number of seats resulting from each method relative to the number of seats resulting from a method you choose. You can also decide whether to label the columns of the resulting array with the names of the associated states. The rows of the displayed array are labeled according to the divisor method employed and the particular divisor chosen by that method.
The data for this article was compiled by the author using publicly available census data, publicly available data on the times at which the various states were admitted to the United States, and using Mathematica to manipulate that data as needed. As specified by the United States Constitution until the ratification of the 14th amendment in 1868, slaves count as only 3/5 of a person for purposes of state representation in the House of Representatives.
An article on general divisor methods may be found in E. Park, "The Mathematics of Apportionment", The University of Chicago Law School Roundtable, 7, 2000 pp. 227–235. An article on alternative rounding methods and the relationship of the divisor problem to constrained optimization may be found in R. Agnew, "Optimal Congressional Apportionment", The American Mathematical Monthly, 115, 2008 pp. 297–303. This Demonstration draws significantly on both of these works.
The "floor" method is essentially that of Thomas Jefferson and was used from 1790 through 1830. The "round" method is essentially that proposed by Daniel Webster and was used in 1840 and again with minor variation in 1910 and 1930. The geometric method is roughly what is used today. The identric mean and logarithmic methods have newly been suggested but have never been adopted.