# Hamilton method

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The Hamilton method (or Largest Remainder-Hare/LR-Hare) is a version of the largest remainder method for allocating seats proportionally for representative assemblies with party list voting systems. It uses the Hare quota as the initial quota.

## Example

In 1790, the U.S. had 15 states. For the purpose of allocating seats in the House of Representatives, the state populations were as follows:

StatePopulation
Virginia630 560
Massachusetts475 327
Pennsylvania432 879
North Carolina353 523
New York331 589
Maryland278 514
Connecticut236 841
South Carolina206 236
New Jersey179 570
New Hampshire141 822
Vermont85 533
Georgia70 835
Kentucky68 705
Rhode Island68 446
Delaware55 540
Total3 615 920

Suppose that there were to be 60 seats in the House.

The quota is 3 615 920 ÷ 60 = 60 265.333. Dividing the state populations by the quota gives

State Quotient Remainder
Virginia 10 27 906.7
Massachusetts 7 53 469.7
Pennsylvania 7 11 021.7
North Carolina 5 52 196.3
New York 5 30 262.3
Maryland 4 37 452.7
Connecticut 3 56 045.0
South Carolina 3 25 440.0
New Jersey 2 59 039.3
New Hampshire 2 21 291.3
Vermont 1 25 267.7
Georgia 1 10 569.7
Kentucky 1 8 439.7
Rhode Island 1 8 180.7
Delaware 0 55 540.0
Total 52

Each state receives a number of seats equal to the integer part of the quotient. The remaining 8 seats are given to the states with the largest remainders: New Jersey, Connecticut, Delaware, Massachusetts, North Carolina, Maryland, New York, and Virginia. The final apportionment is:

State Seats District size Rel. rep.
Virginia 11 57 324 1.0513
Massachusetts 8 59 416 1.0143
Pennsylvania 7 61 840 0.9745
North Carolina 6 58 920 1.0228
New York 6 55 265 1.0905
Maryland 5 55 703 1.0819
Connecticut 4 59 210 1.0178
South Carolina 3 68 745 0.8766
New Jersey 3 59 857 1.0068
New Hampshire 2 70 911 0.8499
Vermont 1 85 533 0.7046
Georgia 1 70 835 0.8562
Kentucky 1 68 705 0.8772
Rhode Island 1 68 446 0.8805
Delaware 1 55 540 1.0851
Total 60

## Notes

Hamilton doesn't guarantee that a majority of voters will always win at least half of the seats (though LR-Hagenbach-Bischoff does, since a majority always has more votes than a majority of Hagenbach-Bischoff quotas).

35-seat example
1. "ELECTION INVERSIONS UNDER PROPORTIONAL REPRESENTATION" (PDF). p. 16. line feed character in `|title=` at position 20 (help)