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 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 the 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 these different rounding methodologies affect the representation of 13 imaginary states for 10 different scenarios and for between 50 and 500 possible seats. It also checks to make sure that the total number of votes held by the states under each of the methodologies in fact equals the desired number of total votes. A bar chart shows the relative population of each of the 13 states for each dataset.
An article on general divisor methods may be found in: E. Park, "The Mathematics of Apportionment," University of Chicago Law School Roundtable, 7, 2000 pp. 227-235. This Demonstration draws significantly on his work.
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.