Hamilton's Apportionment Method

The apportionment method suggested by Alexander Hamilton was approved by Congress in 1791, but was subsequently vetoed by president Washington - in the very first exercise of the veto power by President of the United States. Hamilton's method was adopted by the US Congress in 1852 and was in use through 1911 when it was replaced by Webster's method.

(Bold numbers could be clicked upon. To increase the number, click to the right of its vertical center line. To decrease it click to the left of the line. Dragging the mouse near the center line will accomplish the same task, but faster.)

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The given total number of seats (23 in the applet) is to be apportioned between several (3 at the ouset) states proportionally to their populations. To accomplish that task according to Hamilton,

  1. Compute the divisor D = (Total population)/(Number of seats)
  2. Find and round down state quotas {(State population)/D}. The leftover fractional parts add up to a whole number of seats.
  3. Distribute the surplus seats, one per state, starting with the largest leftover fractional part, then proceeding to the next largest, and so on, until all the surplus seats have been dealt with.

This is probably the most controversial method of apportionment that leads to the so called paradoxes: Alabama paradox, population paradox, new-states paradox.

(One of the applets at this site combines Hamilton's and four additional methods of apportionment under a single umbrella.)


  1. For All Practical Purposes by COMAP, 5th edition, W. H. Freeman & Company, 2008 (8th edition)
  2. G. Szpiro, Numbers Rule: The Vexing Mathematics of Democracy, from Plato to the Present, Princeton University Press, 2010.
  3. P. Tannebaum & R. Arnold, Excursions In Modern Mathematics, 7th edition, Prentice Hall, 2009

Related material

  • The Constitution and Paradoxes
  • Student's Social Choice
  • Adams' Apportionment Method
  • Banzhaf Power Index Calculator
  • Fair Division: Method of Lone Divider
  • Fair Division: Method of Markers
  • Fair Division: Method of Sealed Bids
  • Fair Division: Method of Sealed Bids II
  • Fast Power Indices
  • Five Methods of Apportionment
  • Four Voting Methods
  • Huntington-Hill Apportionment Method
  • Jefferson's Apportionment Method
  • Method of Markers II
  • Shapley-Shubik Power Index Calculator
  • Voting Methods and Social Choice
  • Webster's Apportionment Method
  • Weighted Voting and Power Indices
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