Return the House of Representatives to the People
return to home page
A Brief History of Apportionment
  Click here to buy the shirt.
 A.  Apportionment Overview

Article I, Section 2 of the Constitution requires that:

  • the House Representatives be apportioned among the several States according to their respective populations;
  • the number of Representatives shall not exceed one for every 30,000 persons;
  • each State shall have at least one Representative; and,
  • the reapportionment shall occur once every 10 years as a result of the decennial census.

Over the years, Congress adopted various mathematical formulas to determine how the number of Representatives should be apportioned to each state and, more specifically, how to deal with this dilemma: when the population of a state is divided by the desired district size (e.g., 30,000), the resulting number will inevitably include a fractional amount. However, because a state cannot have 3-and-¾ Representatives, there has always been the problem of how to fairly minimize, across all the states, this fractional remainder of representation. Moreover, as is explained elsewhere in this pamphlet, the larger the district size, the greater the inequity created by the fractional remainders.

From 1790 to 1910, the size of the House of Representatives was augmented (increased) every ten years to keep up with our nation’s growing population. In 1911, Congress passed legislation that changed the apportionment method (again), and also ultimately fixed — by default — the number of Representatives at 435.

After the 1920 census, Congress failed to pass a reapportionment act as they could not agree on a reallocation. There were disputes over whether the rural population had been undercounted, thereby favoring districts with urban centers. In addition, debates continued about whether or not the problem of fractional remainders was properly resolved by the apportionment formula. This stalemate left the size of the House unchanged at 435.


Section, act Aug. 8, 1911, ch. 5, Sec. 1, 2, 37 Stat. 13, 14, fixed composition of House of Representatives at 435 Members, to be apportioned to the States therein enumerated. For provisions dealing with reapportionment of Representatives and manner of election, etc., see sections 2a and 2b of this title.

Citation: 2 USC Sec. 2 (from the “Office of the Law Revision Counsel”) (Also: Title 2, Chapter 1 (Election of Senators and Representatives)

In the beginning, the number of Representatives was determined through a trial-and-error process which began with dividing the states’ populations with a target district size. If the resulting number of Representatives were thought to be too large or too small (relative to a desired total), then different target districts sizes were tried until a politically palatable result was achieved. This simple approach to apportionment had the following advantages:

  • It could be readily understood by anyone.
  • If the target district size is small enough, it can achieve uniformly consistent House district sizes across the country (in compliance with the one-person-one-vote rule).
Unfortunately, as is shown in Section III of this pamphlet, the inevitable result of this process was to increase district sizes over time as Representatives naturally wished to maintain & grow their power base.

In contrast to the simple apportionment method described above, the apportionment imposed by the 435-seat limitation requires a complex mathematical contrivance that defies easy understanding and, in fact, is virtually its own arcane branch of mathematics. (Anyone who needs to be convinced on this last point need only read this 23-page PDF: “The House Apportionment Formula in Theory and Practice”.)
 B.  Apportionment Milestones

Apportionment Milestones:

 — Constitution drafted by the Constitutional Convention
 — First Census
 — After much debate, Congress approved a bill for a 120 member House and Hamilton’s method to apportion seats among the states. Hamilton’s method won out over Jefferson’s method. Hamilton’s method was supported by the Federalists while Jefferson’s method was supported by the Republicans.
 — President Washington vetoes the above bill (the first veto in US history).
 — The House, unable to override the veto, passed a new bill for a 105 member House and Jefferson’s method to apportion seats among the states. (This method was used until 1840.)
 — Rep. William Lowndes (SC) proposed an apportionment method now known as the Lowndes method. It never passed.
 — John Quincy Adams (former President and, at this time, a representative from Massachusetts) proposes the Adam’s method for apportionment. It fails.
 —  Senator Daniel Webster (Mass) proposes Webster’s method. It fails.
 —  Congress passes a bill that retains Jefferson’s method but changes the size of the House to 240.

 — Webster’s method is adopted and the size of the House is reduced to 223.
 —  Rep. Samuel Vinton (Ohio) proposed a bill adopting Hamilton’s method with a House size of 233. Congress passes this bill with a change to a House size of 234, a size for which Hamilton’s and Webster’s methods give the same apportionment.
For additional information, see Section III Chart A2
 — A very confusing year: First the House size was chosen to be 283 so that Hamilton’s and Webster’s methods would again agree. After much political infighting, 9 more seats were added and the final apportionment did not agree with either method.
 — Rutherford B. Hayes became President based on the botched apportionment of 1872. The electoral college vote was 185 for Hayes and 184 for Tilden. Tilden would have won if the correct apportionment as required by law had been used.
 — The Alabama Paradox surfaced as a major flaw of Hamilton’s method.
 — Concerns continued over the flaws in Hamilton’s method. Congress passed a bill that kept Hamilton’s method but changed the House size to 325 so that Hamilton’s method gave the same apportionment as Webster’s.
 — The Census Bureau gave Congress tables showing apportionments based on Hamilton’s method for all House sizes between 350 and 400.
 — For all House sizes in this range (except for 357) Colorado would get 3 seats. For 357, Colorado would get 2 seats. Rep. Albert Hopkins (IL), chm of the House Committee on Apportionment, submitted a bill using a House size of 357 — causing an uproar.
 —  Congress defeated Hopkin’s bill and instead adopted Webster’s method with a House size of 386.
 — Oklahoma joined the union and the New States Paradox was discovered as a result.
 — Webster’s method was readopted with a House size of 433. A provision was made to give Arizona and New Mexico each 1 seat if they were admitted to the union.
 — Joseph Hill (chief statistician of the Census Bureau) proposed the Huntington-Hill method.
 — No reapportionment was done after the 1920 census in direct violation of the Constitution.
 —  Webster’s method was adopted with a House size of 435.
 —  The Huntington-Hill method was adopted with a House size of 435.
 — The U.S. Census Bureau, for only the second time since 1900, allocated Defense Department overseas employees for apportionment purposes. This resulted in Massachusetts losing a seat to Washington. Massachusetts filed suit.
 —  Overruling a U. S. district court decision, the U. S. Supreme Court ruled against Massachusetts on technical grounds involving “the separation of powers and the unique constitutional position of the President.” (The President is charged with calculating and transmitting the apportionment to Congress.)
 — Montana challenged the constitutionality of the Huntington-Hill method (Montana v. US Dept. of Commerce). The Supreme Court upheld the method. Montana was upset because it lost a seat to Washington based on the results of the 1990 census.
 C.  History of Apportionment Legislation
The text below was excerpted from from U. S. Department of Commerce v. Montana:

The general admonition in Article I, § 2, that Representatives shall be apportioned among the several States “according to their respective Numbers” is constrained by three requirements. The number of Representatives shall not exceed one for every 30,000 persons; each State shall have at least one Representative; and district boundaries may not cross state lines. [n.14] Although the text of Article I determined the original apportionment that the Framers had agreed upon, [n.15] it did not explain how that specific allocation had been made.

When Congress first confronted the task of apportionment after the census of 1790 (and after Vermont and Kentucky had been admitted to the Union), it considered using the constitutional minimum of 30,000 persons as the size of each district. Dividing that number into the total population of 3,615,920 indicated that the House of Representatives should contain 120 members. When that number was divided into the population of individual States, each quotient was a whole number with a fractional remainder. Thus, the use of the 30,000 divisor for Connecticut’s population of 236,841 indicated that it should have 7.89 Representatives, while Rhode Island, with a population of 68,446, should have 2.28 Representatives. Because each State must be represented by a whole number of legislators, it was necessary either to disregard fractional remainders entirely or to treat some or all of them as equal to a whole Representative. [n.16]

In the first apportionment bill passed by Congress, an additional Representative was assigned to the nine States whose quotas had the highest fractional remainders. Thus, Connecticut’s quota of 7.89 gave it 8 and Rhode Island’s smaller remainder was disregarded, giving it only 2. Although that method was supported by Alexander Hamilton, Thomas Jefferson persuaded President Washington to veto the bill, in part because its allocation of eight Representatives to Connecticut exceeded the constitutional limit of one for every 30,000 persons. [n.17]

In response to that veto, Congress adopted a proposal sponsored by Thomas Jefferson that disregarded fractional remainders entirely (thus giving Connecticut only 7 Representatives). To overcome the basis for the veto, the size of the House was reduced from 120 to 105 members, giving each Representative an approximate constituency of 33,000 instead of 30,000 persons. Although both the total number of Representatives and the size of their districts increased, [n.18] Jefferson’s method of disregarding fractional remainders was used after each of the next four censuses. Today mathematicians sometimes refer to that method as the “method of greatest divisors,” and suggest that it tends to favor large States over smaller States. [n.19]

In 1832, Congress considered, but did not adopt, a proposal sponsored by John Quincy Adams that was the exact opposite of the Jefferson method. Instead of disregarding fractional remainders, Adams would have treated every fraction as a unit. Thus, using the former example as a hypothetical, both Connecticut and Rhode Island would have received one more Representative under the Adams method than they actually received under the Jefferson method. The Adams method is sometimes described as the “method of smallest divisors” and is said to favor the smaller States. [n.20] It has never been endorsed by Congress.

In 1842, Congress abandoned the Jefferson method in favor of an approach supported by Senator Daniel Webster. The Webster method took account of fractional remainders that were greater than one half by allocating “one additional representative for each State having a fraction greater than one moiety.” [n.21] Thus, if that method had been used in 1790, Connecticut’s quota of 7.89 would have entitled it to 8 Representatives, whereas Rhode Island, with a quota of 2.28, would have received only 2. The Webster method is also described as the “method of major fractions.”

In 1850, Congress enacted legislation sponsored by Representative Vinton endorsing the approach that had been sponsored by Alexander Hamilton after the first census. [n.22] Although this method was used during the balance of the 19th century, it occasionally seemed to produce paradoxical results. [n.23] Congress rejected it in 1911, reverting to the Webster method. In that year Congress also passed legislation that ultimately fixed the number of Representatives at 435. [n.24]

After the 1920 census Congress failed to pass a reapportionment Act, but debates over the proper method of apportionment ultimately led to a request to the National Academy of Sciences to appoint a committee of experts to review the subject. That committee, composed of respected mathematicians, recommended the adoption of the “method of equal proportions.” Congress used that method in its apportionment after the 1930 census, and formally adopted it in the 1941 statute at issue in this case. [n.25]

The report of the National Academy of Sciences committee noted that Congress had properly rejected the Hamilton/Vinton method, and concluded that the use of only five methods could lead to a workable solution of the fractional remainder problem. [n.26] In the opinion of the committee members, given the fact that it is impossible for all States to have districts of the same size, the best method was the one that minimized the discrepancy between the size of the districts in any pair of States. Under their test of fairness, a method was satisfactory if, for any pair of States, the transfer of one Representative would not decrease the discrepancy between those States’ districts. [n.27] The choice of a method depended on how one decided to measure the discrepancy between district sizes. Each of the five methods could be described as the “best” in the sense of minimizing the discrepancy between districts, depending on the discrepancy measure selected. The method of the harmonic mean, for example, yielded the fairest apportionment if the discrepancy was measured by the absolute difference between the number of persons per Representative. The method of major fractions was the best method if the discrepancy was measured by the absolute difference between the number of Representatives per person (also known as each person’s “share” of a Representative [n.28] ). The method of equal proportions produced the fairest apportionment if the discrepancy was measured by the “relative difference” [n.29] in either the size of the district or the share of a Representative. [n.30]

The report concluded by endorsing the method of equal proportions. The committee apparently preferred this method for two reasons. First, the method of equal proportions minimized the relative difference both between the size of congressional districts and between the number of Representatives per person. Second, in comparison with the other four methods considered, this method occupied an intermediate position in terms of favoring small States over large States: it favored small States more than major fractions and greatest divisors, but not as much as smallest divisors or the harmonic mean. [n.31]


17 See id., at 16 22. President Washington’s veto message read as follows:

Gentlemen of the House of Representatives:

“I have maturely considered the act passed by the two Houses entitled ‘An act for an Apportionment of Representatives among the several States, according to the first Enumeration;’ and I return it to your House, wherein it originated, with the following objections:

“First. The Constitution has prescribed that Representatives shall be apportioned among the several States according to their respective numbers; and there is no one proportion or divisor which, applied to the respective numbers of the States, will yield the number and allotment of Representatives proposed by the bill.

“Second. The Constitution has also provided that the number of Representatives shall not exceed one for every thirty thousand; which restriction is, by the context, and by fair and obvious construction, to be applied to the separate and respective numbers of the States; and the bill has allotted to eight of the States more than one for every thirty thousand,

“G. Washington”

3 Annals of Cong. 539 (1792).

18 The 1802 apportionment Act continued the ratio of 33,000, which then corresponded to a House of 141 Members. Act of Jan. 14, 1802, 2Stat. 128. The third apportionment established a ratio of 35,000, which provided a House of 181 Members. Act of Dec. 21, 1811, 2 Stat. 669. The 1822 apportionment Act increased the ratio to 40,000 and the size of the House to 213. Act of Mar. 7, 1822, 3 Stat. 651. The 1832 apportionment Act provided for 240 districts representing an average of 47,700 persons each. Act of May 22, 1932, 4 Stat. 516. See generally L. Schmeckebier, Congressional Apportionment 111 113 (1941).

22 Act of May 23, 1850, § § 24 26, 9 Stat. 432 433. Under the Hamilton/Vinton method, the Nation’s population was divided by the size of the House (set at 233 in 1850) to determine the ratio of persons per Representative. This ratio was then divided into the population of a State to establish its quota. Each State would receive the number of Representatives corresponding to the whole number of the quota (ignoring the fractional remainders). The remaining seats necessary to bring the nationwide total to the proper size (233 in 1850) would then be distributed to the States with the largest fractional remainders. In practice, the method was not strictly followed. See Balinski & Young 37; Chafee, Congressional Reapportionment, 42 Harv. L. Rev. 1015, 1025 (1929).

23 The Hamilton/Vinton method was subject to the “ Alabama paradox,” a mathematical phenomenon in which a State’s number of Representatives may decrease when the size of the House is increased. See Balinski & Young 38 40; Chafee, Congressional Reapportionment, 42 Harv. L. Rev., at 1026.

24 The 1911 statute actually specified 433 Representatives but authorized an additional Representative for Arizona and New Mexico when they were admitted to the Union. See 37 Stat. 13. Additional Representatives were also authorized when Alaska and Hawaii were admitted to the Union in 1959, but the number thereafter reverted to 435, where it has remained ever since. See 72 Stat. 345; 73 Stat. 8.

25 Act of Nov. 15, 1941, § 1, 55 Stat. 761-762, 2 U.S.C. § 2a. That Act also made the reapportionment process self executing, eliminating the need for Congress to enact an apportionment Act after each decennial census:

“(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.

“(b). . . It shall be the duty of the Clerk of the House of Representatives, within fifteen calendar days after the receipt of such statement, to send to the executive of each State a certificate of the number of Representatives to which such State is entitled under this section.”

Source: No. 91-860
 D. Apportionment Judicial History

The summary below was excerpted from FindLaw®

Clause 3. Apportionment of Seats in the House

  The Census Requirement

While Sec. 2 expressly provides for an enumeration of persons, Congress has repeatedly directed an enumeration not only of the free persons in the States, but also of those in the territories, and has required all persons over eighteen years of age to answer an ever-lengthening list of inquiries concerning their personal and economic affairs. This extended scope of the census has received the implied approval of the Supreme Court; 314 it is one of the methods whereby the national legislature exercises its inherent power to obtain the information necessary for intelligent legislative action. Although taking an enlarged view of its power in making the enumeration of persons called for by this section, Congress has not always complied with its positive mandate to reapportion representatives among the States after the census is taken. 315 It failed to make such a reapportionment after the census of 1920, being unable to reach agreement for allotting representation without further increasing the size of the House. Ultimately, by the act of June 18, 1929, 316 it provided that the membership of the House of Representatives should henceforth be restricted to 435 members, to be distributed among the States by the so-called “method of major fractions,” which had been earlier employed in the apportionment of 1911 and which has now been replaced with the “method of equal proportions.” Following the 1990 census, a State that had lost a House seat as a result of the use of this formula sued, alleging a violation of the “one person, one vote” rule. Exhibiting considerable deference to Congress and a stated appreciation of the difficulties in achieving interstate equalities, the Supreme Court upheld the formula and the resultant apportionment. 317 


[Footnote 314] Knox v. Lee (Legal Tender Cases). 79 U.S. (12 Wall.) 457, 536 (1871).

[Footnote 315] For an extensive history of the subject, see L. Schmeckebier, Congressional Apportionment (Washington: 1941).

[Footnote 316] 46 Stat. 26, 22, as amended by 55 Stat. 761 (1941), 2 U.S.C. Sec. 2a.

[Footnote 317] U.S. Department of Commerce v. Montana, 112 S.Ct. 1415 (1992). The practice of the Secretary of Commerce in allocating overseas federal employees and military personnel to the States of last residence was attacked but upheld in Franklin v. Massachusetts, 112 S.Ct. 2767 (1992). The mandate of the clause of an enumeration of “their respective numbers” was complied with, it having been the practice since the first enumeration to allocate persons to the place of their “usual residence,” and to construe both this term and the word “inhabitant” broadly to include people temporarily absent. Another census controversy was resolved in Wisconsin v. City of New York, 116 S. Ct. 1091 (1996), in which the Court held that the decision of the Secretary of Commerce not to conduct a post-enumeration survey and statistical adjustment for an undercount in the 1990 Census was reasonable and within the bounds of discretion conferred by the Constitution and statute.

“If a nation expects to be ignorant and free ... 
     it expects what never was and never will be.”

– Thomas Jefferson

Last updated: 13July2004
Web Pamphlet