14.3 Apportionment Methods 933 We have just discussed Alexander Hamilton’s method for apportionment. In this section, we will also discuss Thomas Jefferson’s method for apportionment, Daniel Webster’s method for apportionment, and John Quincy Adams’ method for apportionment. Here is a brief history of the use of these different methods to apportion members of the U.S. House of Representatives. Although Hamilton’s method for apportionment was the first method approved by Congress after the 1790 census, it was not used until 1852. Shortly after Congress approved Hamilton’s method, President George Washington vetoed it. Congress then adopted a method developed by Thomas Jefferson. As there were apportionment problems with Jefferson’s method in 1822 and 1832, Webster’s method replaced Jefferson’s method in 1842. Webster’s method had similar flaws as Jefferson’s method and was replaced in 1852 by Hamilton’s method, the first apportionment method proposed. When it was discovered in the late 1800s that there were also flaws with Hamilton’s method, Webster’s method was once again used to apportion the number of representatives for each state. Webster’s method was used from 1900 until it was replaced by the current method in 1941, the Huntington-Hill method (see Exercise 49). Adams’ method, although considered, was never actually used by Congress to apportion members. Did You Know? Mathematical Contributions from Political Leaders m Thomas Jefferson (1743–1826) Thomas Jefferson, third president of the United States, authored the Declaration of Independence. He excelled in many careers, including philosopher, educator, politician, inventor, scientist, and writer. Under his administration, the United States doubled in area with the Louisiana Purchase in 1803. Jefferson’s Method To determine the standard quota with Hamilton’s method, the population of each group was divided by the standard divisor. Then each lower quota was used. Any leftover items were distributed to the groups with the highest fractional part of the standard quota. As we saw with Examples 2 and 3, each group received either its lower quota or its upper quota. A drawback that can occur with Hamilton’s method is that the groups that received their upper quota gained an advantage over the groups that received their lower quota. Is there a way to modify the quotas to overcome this drawback in Hamilton’s method? Jefferson’s method uses a divisor other than the standard divisor, called the modified divisor, to obtain quotas. A modified divisor is a divisor that approximates the standard divisor. We will explain how to determine modified divisors shortly. A modified divisor is used in Jefferson’s method, Webster’s method, and Adams’ method, each of which will be discussed shortly. In each case, a modified divisor is used to obtain modified quotas . Modified quotas are obtained by dividing a group’s population by a modified divisor. The differences in Jefferson’s method, Webster’s method, and Adams’ method are in the modified divisor used to obtain the modified quotas and in the rounding process used to apportion the items. Hamilton’s method is the only method that uses the standard quota rather than a modified quota in determining apportionments. Now we will discuss Jefferson’s method. JEFFERSON’S METHOD To use Jefferson’s method for apportionment, do the following. 1. Determine a modified divisor, d, such that when each group’s modified quota is rounded down to the nearest integer , the total of the integers is the exact number of items to be apportioned. We will refer to the modified quotas that are rounded down as modified lower quotas. 2. Apportion to each group its modified lower quota. PROCEDURE With Jefferson’s method, we use a modified divisor that is less than the standard divisor. This results in the modified quotas being slightly greater than the standard quotas. When using Hamilton’s method, the standard quotas are rounded down and then leftover items are distributed. With Jefferson’s method, the modified quotas are also rounded down . The modified divisor used with Jefferson’s method must result in all items being distributed, with no items left over, when the modified quotas are rounded down. We will explain Jefferson’s method in Example 4. W.H. Gallagher Co., N.Y/Library of Congress Prints and Photographs Division [LC-USZ62-53985]
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