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The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of science.
Aspects of Science, 1925.
Apportioning Representatives in the United States Congress
Hamilton's Method of Apportionment
The first census was to be taken in 1790, less than three years after the ratification of the Constitution. Once the numbers were in, the Congress had to decide how to use the data to apportion the Representatives. They also had to decide how many Representatives the House should have. In the spring of 1792 they passed a bill to apportion the House, using a method proposed by Alexander Hamilton and now known as Hamilton’s method.
Here is the procedure:
The divisor “D” in the method is the ratio of all U.S. residents to Representatives; it came out to a bit over 30,000 in this case. (Recall the Constitutional threshold above.) The State Quota is the number of seats each state is due, according to the divisor D. But while the apportionment must be in terms of whole numbers, these State Quotas are not. Therefore the method rounds the Quotas down, since rounding up could cause us to have more than the specified number of Representatives. But rounding down means we have apportioned fewer than the requisite number of seats. Hamilton proposes to allot the surplus seats, in order, to the states with the highest remainders after dividing by D. This process is followed until the given number of seats is assigned.
To see how this method apportioned the proposed 120 seats in the House, see the spreadsheet 1792 Hamilton.
Caulfield, Michael J., "Apportioning Representatives in the United States Congress," Loci (November 2008), DOI: 10.4169/loci003163