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In the United States Congress, there are two legislative bodies: the House of Representatives and the Senate. These two bodies are structured differently from the very beginning in Article 1 of the U.S. Constitution in that the Senate represents each U.S. state equally (two senators for each state) and House represents populations roughly equally: "Representatives ... shall be apportioned among the several States which may be included within this Union, according to their respective Numbers... The actual Enumeration shall be made ... within every subsequent Term of ten Years, in such Manner as they shall by Law direct."

So big states (w.r.t. population) get more representatives than do small states. The Constitution does not spell out how many total representatives there are in the House nor exactly the manner that the apportionment shall determine how many each state gets. It's just that every 10 years, we must have a Census for which the primary Constitutional purpose is to determine how the U.S. House of Representatives is apportioned among the now 50 U.S. states. The data from the Census gets used in a variety of ways and purposes, but this is the original primary purpose of the U.S. Decennial Census. It's a big deal.

(I would presume, in the European Union, that they have the same issue, but maybe settle the issue differently.)

So then, the specific number of U.S. representatives and the specific manner of proportionately distributing their seats to the various states is a matter of federal law (one notch lower than the Constitution) and is spelled out in U.S.C. Title 2. Well, not really completely spelled out, hence this question.

Okay, I know and understand what "the method known as the method of equal proportions" is. It's also known as the Huntington-Hill Method and is referred to as "the method known as the method of equal proportions" in §2a and §2b of U.S.C. Title 2.

There is also some useful history to read but I want to see the actual document in law that spells it out. Even if it was a report transmitted to Congress circa 1941, I would like to see what a federal judge would be saying that "this is what the law is" and I cannot find that.

Where is this "method of equal proportions" actually spelled out in the actual law? I cannot find it. I understand how the H-H method works and I have read a little history about it.

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There is nothing legally authoritative (e.g. codified regulations) beyond 2 U.S.C. §§ 2a and 2b, except for the legislative history of those acts.

In the case of Section 2a, this is the legislative history of 46 Stat. 26, c. 28, § 22 (June 18, 1929), 54 Stat. 162, c. 152 (Apr. 25, 1940), 55 Stat. 761, c. 470, § 1 (Nov. 15, 1941), and Pub.L. 104-186, Title II, § 201, 110 Stat. 1724 (Aug. 20, 1996).

In the case of Section 2b, this is the legislative history of 55 Stat. 762, c. 470, § 2(a) (Nov. 15, 1941).

But primarily, the relevant legislative history is the legislative history of the 1941 act mentioned above that introduced the concept.

Only one case has litigated § 2b and it does not address the mechanics of the equal proportions method. While many cases have been litigated under § 2a (59 annotations in some cases with more than one annotation from a single case), none of them addressed the meaning of the words "method of equal proportions". The census bureau, however, notes two cases that it looks to: Department of Commerce v. Montana, 112 S.Ct. 1415 (1992) and Franklin v. Massachusetts 112 S.Ct. 2767 (1992). The court in the first case found that the method adopted by Congress was constitutional and that its meaning was clarified by the National Academy of Science Committee report referenced by Congress in the legislative history involved in adopting the 1941 Act. The second case only touches on the issue tangentially because that issue was mooted by the resolution of the first case.

The meaning of the words "method of equal proportions" has never been reduced to a federal regulation.

The Congressionally intended meaning of the term method of equal proportions has always been sufficiently clear to avoid all disputes over it, even though it is not, for example, defined in Black's Law Dictionary.

The U.S. Census Bureau has, however, explained how the "method of equal proportions" works on multiple occasions in non-authoritative documents like press releases, and those together with secondary sources explaining it, could be offered up to prove it if necessary.

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    Well, @Barmar , some people might say that the Court was unlawfully legislating from the bench. Not that it's never happened before.
    – robert bristow-johnson
    Commented Feb 21, 2023 at 21:52
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    @robertbristow-johnson True, but it's the lack of definition that makes that possible. The point is that lawmakers aren't like mathematicians or computer programmers, they don't define everything rigidly. Much language is just taken from accepted usage.
    – Barmar
    Commented Feb 21, 2023 at 21:54
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    In the case of marriage, it was statutorily defined in the Defense of Marriage Act, and many courts and ultimately the U.S. Supreme Court, held that the 14th Amendment to the U.S. Constitution rendered that statutory definition unconstitutional.
    – ohwilleke ♦
    Commented Feb 21, 2023 at 21:57
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    @robertbristow-johnson one thing about law that people sometimes don't appreciate is that when everyone understands what is meant by a word or phrase, the legislature may not even think of the possibility that it should be further defined, let alone actually enact an explicit definition. It's only later, when people who weren't around at the time and therefore do not take the context for granted, start saying things like "hey, wait a minute, how do we know that this really means?" that serious questions arise that may need to be decided by a court or clarified by the legislature.
    – phoog
    Commented Feb 22, 2023 at 17:00
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    @robertbristow-johnson if you find it unacceptable then your best course of action is probably to start a letter writing campaign to members of congress to call for the enactment of a specific definition. I suspect, however, that members of congress won't give it much priority because (as this answer notes) the meaning is clear. Everyone knows what it means or can find out with minimal effort. The procedure is in fact well defined. The benefits of enacting the definition into law are therefore small, while the costs are relatively large.
    – phoog
    Commented Feb 22, 2023 at 23:43

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