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Table 2: The “ Alabama Paradox ” in 1880

Congress chose this method in 1830. The page ti tled, “1830 Census ” compares the methods of Jef ferson, Adams, and Webster for the 1830 census. All three methods agree for 9 of the 24 states and each method gives a different value for the two largest states. Webster gives New York 39, where as Jefferson gives it 40 and Adams only 37. Penn sylvania gets 27, 28, and 26, respectively. Web ster ’ s method was also chosen for the 1840 Cen sus. In 1850, Rep. Samuel F. Vinton of Ohio proposed a different approach—choosing a method and House size before the Census results became avail able to prevent arguing over each methods ’ impact on individual states. His arguments were persua sive to a point, but he was apparently unaware that his proposed “ new ” method was actually just the method of Hamilton. “ Vinton ’ s ” method became law, but his idea of capping the size of the House was quickly abandoned. In 1850, 1860, and 1870, some states were given extra seats after the method was applied for political reasons. This was to justi fy, the 1850 Census that gave California only one seat, but Congress knew California would grow very rapidly during the coming decade, so they gave it an extra seat. In 1870, Congress applied Hamilton ’ s method with N = 283, but months later gave nine states an extra seat for reasons that were not immediately obvious.

reasonable, it suffers from subtle paradoxes that were not immediately apparent. In 1880, as they usually did, Congress asked the Census Office to prepare tables using Hamilton ’ s method with val ues of N from 275 to 350. All this data revealed a problem to the Census Office. In the words of its chief clerk C.W. Seaton, “ I met with the so - called ‘ Alabama ’ paradox where Alabama was allotted 8 Representatives out of a total of 299, receiving but 7 when the total becomes 300,” which he felt was “ conclusive proof that the process employed in ob taining it is defective. ” Increasing the total number of representatives can make a state lose a seat. How can that happen? Ta ble 2 shows the details. When increasing from N = 299 to N = 300, each state ’ s quota increases by a factor of 300/299, which means a larger state ’ s re mainder increases by a larger amount than a small er state ’ s. Specifically, when N = 299, 279 seats are awarded based on the initial rounding down. Since Alabama has the 20th largest remainder, it gets the last seat followed by the larger states of Texas and Illinois. When N increases to 300, Tex as ’ s quota increases by 0.032, whereas Alabama ’ s quota increases by 0.025, which is enough for Tex as to pass Alabama in the remainder rankings. The fourth - largest state Illinois ’ s increase of 0.062 is enough to pass both Alabama and Texas in the re mainder rankings. Therefore, adding a new seat to the total shifts a seat away from Alabama to Illi nois or Texas.

While the method of Hamilton does not seem un

Table 3: The number of seats for Maine and Colorado under the method of Hamilton as a function of the House Size

Virginia Mathematics Teacher vol. 48, no. 1

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