1

Over the centuries there have been several methods used for calculating the number of representatives each state gets in Congress. From what I have read, though, the results of the apportionment are the only thing that Congress ratifies, not the formulas themselves.

Is there anywhere that the actual apportionment formulas have been written into law, instead of just the results?

The closest I could find was the 1850 reapportionment.

Those I found that only mention the resulting numbers are: 1802, 1811, 1822, 1832, 1842, 1872, 1882, 1901, and 1911.

2

Here is a little bit on different methods, and Wikipedia says that this is the Huntington-Hill method being the currently employed one. Here is the statute, 2 USC 2a, which requires using "the method known as the method of equal proportions", which is not statutorily defined. Here is the original act of Congress. This document from the Census bureau, see also this, states what it believes that the method requires. The mathematicians, or at least some of them, offer some history of methods, and refer the matter to Huntington's Equal Proportions Method. In that context, "method of equal proportions" is probably unambiguous, but that's properly a "history of math" question.

2

Under 2 U.S. Code § 2a,

(a) On the first day, or within one week thereafter, of the first regular session of the Eighty-second Congress and of each fifth Congress thereafter, the President shall transmit to the Congress a statement showing the whole number of persons in each State, excluding Indians not taxed, as ascertained under the seventeenth and each subsequent decennial census of the population, and the number of Representatives to which each State would be entitled under an apportionment of the then existing number of Representatives by the method known as the method of equal proportions, no State to receive less than one Member.

The law doesn't spell out what "the method of equal proportions" means, but it's the name of a specific algorithm (also known as the Huntington-Hill method). In any case, the law does clearly show that Congress has chosen a specific algorithm that must be used, so the President has no discretion to choose his own method.

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