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The sum of the __ was only 111, and so the states with the nine largest remainders were given one more seat. Thus, Virginia increased to ________ . Congress approved this apportionment, but George Washington used the first presidential veto to re ject it for two reasons: 1) He applied the 30,000 people - per - representative minimum to each state instead of the national total, and eight states violat ed this criterion. 2) He felt that using the largest remainders did not award seats proportionally— New York ’ s population was more than five times Delaware ’ s, but since Delaware had a larger re mainder, it received an additional representative instead of New York. Therefore, Congress had to start over with both a smaller total and a different method. Luckily, Sec retary of State Thomas Jefferson had already pro posed a different method.

Virginia received 19 out of 105 representatives. Congress did not codify the method so future Con gresses would have to debate this again. The meth od of Jefferson was used again following the 1800, 1810, and 1820 Censuses, with the size of the House increasing so that no state lost a seat. However, as more data went into this method, a bias toward larger states began to emerge. From 1790 to 1830, Delaware ’ s quota was always be tween 1.52 and 1.95, but only once did it round up to two seats. However, New York always rounded up, and worse, in 1820, its quota was 32.50 but Jefferson ’ s method “ rounded ” up past the next in teger to 34, and in 1830, its 38.59 rounded up to 40. Congressmen from smaller states demanded an alternative method. Jefferson ’ s method is one example of a class of methods called Divisor Methods . All Divisor Methods have the same basic structure of adjusting the value of D until the correct sum is awarded, with the differences being how the methods round. In 1830, Rep. John Quincy Adams of Massachu setts and former president proposed a different di visor method that rounded all the quotas up instead of down. Predictably, this was biased toward smaller states and was rejected. Sen. Daniel Web ster of Massachusetts who introduced the natural compromise, Quotas, which are rounded in the standard manner, round down if the quota is below ______ and up otherwise. That is, the rounding point is the arithmetic mean of the integers above and below the quota. 1. Start with the divisor D=P/N. Compute each state ’ s quota 2. If __________, then round down so that _______ , but if __________, round up so that _________. 3. If the sum of the __ is less than N , decrease the divisor D until the number of seats awarded increases to the desired N . If the sum of the __ is greater than N, increase D until the number of seats awarded decreases to the desired N . The Method of Webster

The Method of Jefferson

1. Choose a divisor D to be approximately the number of people per representative for the na tion. Start by setting D = P/N. Compute each state ’ s quota in terms of D as

2. Round each quota down so that

3. If the sum of the __ is less than N , decrease the divisor D and repeat until the number of seats awarded increases to the desired N . They found N = 105 to be the largest value that made each state have at least 30,000 people per representative. For this N , the “ true ” divisor was P/N = 34,437, but that awarded only 97 seats after the quotas were rounded down. The correct num ber of seats would be awarded for any divisor in the interval _______________ and so D = 33,000 was chosen. Under this method, Virginia received

representatives. Congress passed this in the Appor tionment Act of 1792, and President Washington signed it into law.

1800 - 1910: Exploration of Methods

The Apportionment Act of 1792 only stated that

Virginia Mathematics Teacher vol. 48, no. 1

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