U.S. Supreme Court


503 U.S. 442

No. 91-860

Argued March 4, 1992
Decided March 31, 1992

Article I, 2, of the Constitution requires apportionment of Representatives among the States "according to their respective Numbers." A 1941 federal statute provides that after each decennial census "the method known as the method of equal proportions" shall be used to determine the number of Representatives to which each State is entitled. Application of that method to the 1990 census caused Montana to lose one of its two seats in the House of Representatives. If it had retained both seats, each district would have been closer to the ideal size of a congressional district than the reapportioned single district. The State and several of its officials (hereinafter Montana) sued appropriate federal defendants (hereinafter the Government) in the District Court, alleging, inter alia, that the existing apportionment method violates Article I, 2. A three-judge court, convened pursuant to 28 U.S.C. 2284, granted Montana summary judgment on this claim, holding the statute unconstitutional because the variance between the single district's population and that of the ideal district could not be justified under the "one-person, one-vote" standard developed in Wesberry v. Sanders, 376 U.S. 1 , and other intrastate districting cases.


Congress exercised its apportionment authority within the limits dictated by the Constitution. Pp. 447-466.

775 F.Supp. 1358, (D.Mont. 1991), reversed.

STEVENS, J., delivered the opinion for a unanimous Court. [503 U.S. 442, 444]  

Solicitor General Starr argued the cause for appellants. With him on the briefs were Assistant Attorney General Gerson, Deputy Solicitor General Roberts, Edwin S. Kneedler, Michael Jay Singer, and Mark B. Stern.

Marc Racicot, Attorney General of Montana, argued the cause for appellees. With him on the brief were Clay R. Smith, Solicitor, and Elizabeth S. Baker, Assistant Attorney General. *  

[ Footnote * ] Kenneth O. Eikenberry, Attorney General of Washington, James M. Johnson, Senior Assistant Attorney General, and Carole A. Ressler, Assistant Attorney General, filed a brief for the State of Washington as amicus curiae urging reversal.

Briefs of amici curiae urging affirmance were filed for the Commonwealth of Massachusetts by Scott Harshbarger, Attorney General, Dwight Golann and Steve Berenson, Assistant Attorneys General, and John P. Driscoll, Jr., Edward P. Leibensperger, and Neil P. Motenko, Special Assistant Attorneys General; and for the Crow Tribe of Indians et al. by Dale T. White, Jeanne S. Whiteing, and Daniel F. Decker.

JUSTICE STEVENS delivered the opinion of the Court.

Article I, 2, of the Constitution requires apportionment of Representatives among the several States "according to their respective Numbers." 1 An Act of Congress passed in 1941 provides that, after each decennial census, "the method known as the method of equal proportions" shall be used to determine the number of Representatives to which each State is entitled. 2 In this case, a three-judge District Court [503 U.S. 442, 445]   held that statute unconstitutional because it found that the method of equal proportions resulted in an unjustified deviation from the ideal of equal representation. 3 The Government's appeal from that holding requires us to consider the standard that governs the apportionment of Representatives among the several States. In view of the importance of the issue and its significance in this year's congressional and Presidential elections, we noted probable jurisdiction and ordered expedited briefing and argument. 502 U.S. 101 (1991). We now reverse.


The 1990 census revealed that the population of certain States, particularly California, Florida, and Texas, had increased more rapidly than the national average. The application of the method of equal proportions to the 1990 census caused 8 States to gain a total of 19 additional seats in the House of Representatives 4 and 13 States to lose an equal number. 5 Montana was one of those States. Its loss of one seat cut its delegation in half and precipitated this litigation.

According to the 1990 census, the population of the 50 States that elect the members of the House of Representatives is 249,022,783. 6 The average size of the 435 congressional districts is 572,466. Montana's population of 803,655 forms a single congressional district that is 231,189 persons larger than the ideal congressional district. If it had retained its two districts, each would have been 170,638 persons smaller than the ideal district. In terms of absolute [503 U.S. 442, 446]   difference, each of the two districts would have been closer to ideal size than the single congressional district.

The State of Montana, its Governor, Attorney General, and Secretary of State, 7 and the State's two Senators and Representatives (hereinafter collectively referred to as Montana) filed suit against appropriate federal defendants (the Government) in the United States District Court for the District of Montana, asserting that Montana was entitled to retain its two seats. They alleged that the existing apportionment method violates Article I, 2, of the Constitution because it "does not achieve the greatest possible equality in the number of individuals per representative," 8 and also violates Article I, 2, and Article I, 7, because reapportionment is effected "through application of a mathematical formula by the Department of Commerce and the automatic transmittal of the results to the states, 9 " rather than by legislation on which Members of Congress vote in the normal manner. A three-judge District Court, convened pursuant to 28 U.S.C. 2284, granted Montana's motion for summary judgment on the first claim. 10  

The majority of the three-judge District Court decided that the principle of equal representation for equal numbers of people that was applied to intrastate districting in Wesberry v. Sanders, 376 U.S. 1 (1964), should also be applied to the apportionment of seats among the States. Under that standard, the only population variances that are acceptable are those that "are unavoidable despite a good faith effort to achieve absolute equality, or for which justification is shown," Kirkpatrick v. Preisler, 394 U.S. 526, 531 (1969). The District Court held that the variance between [503 U.S. 442, 447]   the population of Montana's single district and the ideal district could not be justified under that standard. The majority refused to accord deference to the congressional decision to adopt the method of equal proportions in 1941 because that decision was made without the benefit of this Court's later jurisprudence adopting the "one-person, one-vote" rule. Accordingly, the District Court entered a judgment declaring the statute void and enjoining the Government from effecting any reapportionment of the House of Representatives pursuant to the method of equal proportions. 11  

Circuit Judge O'Scannlain dissented. After noting that Congress has used four different apportionment formulas during the country's history, and that it is not possible to create 435 districts of equal size when each district must be located entirely within a single State, he concluded that the goal of any apportionment formula must be a "`practical approximation'" to a population-based allocation. 12 He analyzed the two formulae proposed by Montana and concluded that the State had failed to demonstrate that either was better than the one that had been chosen by Congress. 13  


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 [503 U.S. 442, 448]   one for every 30,000 persons; each State shall have at least one Representative; and district boundaries may not cross state lines. 14 Although the text of Article I determined the original apportionment that the Framers had agreed upon, 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 the number 30,000 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. 16   [503 U.S. 442, 449]  

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. 17  

In response to that veto, Congress adopted a proposal sponsored by Thomas Jefferson that disregarded fractional remainders entirely (thus giving Connecticut only seven 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 [503 U.S. 442, 450]   increased, 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. 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. 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." 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, [503 U.S. 442, 451]   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. 22 Although this method was used during the balance of the 19th century, it occasionally seemed to produce paradoxical results. 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. 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 [503 U.S. 442, 452]   apportionment after the 1930 census, and formally adopted it in the 1941 statute at issue in this case. 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. 26 In the opinion of the committee [503 U.S. 442, 453]   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 [503 U.S. 442, 454]   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. 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). 28 The method of equal proportions produced the fairest apportionment if the discrepancy [503 U.S. 442, 455]   was measured by the "relative difference" 29 in either the size of the district or the share of a Representative. 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. 31  

If either the method of smallest divisors or the method of the harmonic mean, also known as the "Dean Method," had been used after the 1990 census, Montana would have received a second seat. Under the method of equal proportions, which was actually used, five other States had stronger claims to an additional seat, because Montana's claim to a second seat was the 441st on the equal proportions "priority list," see n. 26, supra. 32 Montana would not have received [503 U.S. 442, 456]   a second seat under either the method of major fractions or greatest divisors.


The Government argues that Congress' selection of any of the alternative apportionment methods involved in this litigation is not subject to judicial review. Relying principally on Baker v. Carr, 369 U.S. 186 , (1962), the Government contends that the choice among these methods presents a "political question" not amenable to judicial resolution.

In Baker v. Carr, after an extensive review of our prior cases involving political questions, we concluded:

The Government insists that each of the factors identified in Baker supports the conclusion that the question presented here is committed to the "political branches," to the exclusion of the Judiciary. Significantly, however, the Government does not suggest that all congressional decisions relating to apportionment are beyond judicial review. The Government does not, for instance, dispute that a court could set aside an apportionment plan that violated the constitutional requirement that "[t]he number of Representatives shall not exceed one for every thirty Thousand." 33 Further, with respect to the provision that Representatives "shall be apportioned among the several States . . . according to their respective Numbers," 34 the Government acknowledges that Congress has a judicially enforceable obligation to select an apportionment plan that is related to population. 35 The gravamen of the Government's argument is that the District Court erred in concluding that the Constitution imposes the more rigorous requirement of greatest possible equality in the size of congressional districts, as measured by absolute deviation from ideal district size. The Government then does not dispute Montana's contention that the Constitution places substantive limitations on Congress' apportionment power, and that violations of those limitations would present a justiciable controversy. Where the parties differ is in their understanding of the content of these limitations. In short, the Government takes issue not with the existence of a judicially enforceable right, but with the definition of such a right.

When a court concludes that an issue presents a nonjusticiable political question, it declines to address the merits of [503 U.S. 442, 458]   that issue. See Gilligan v. Morgan, 413 U.S. 1, 112 (1973); Baker v. Carr, 369 U.S., at 197 ; see also Colegrove v. Green, 328 U.S. 549, 552 -556 (1946) (plurality opinion). In invoking the political question doctrine, a court acknowledges the possibility that a constitutional provision may not be judicially enforceable. 36 Such a decision is, of course, very different from determining that specific congressional action does not violate the Constitution. 37 That determination is a decision on the merits that reflects the exercise of judicial review, rather than the abstention from judicial review that would be appropriate in the case of a true political question.

The case before us today is "political" in the same sense that Baker v. Carr was a "political case." 369 U.S., at 217 . It raises an issue of great importance to the political branches. 38 The issue has motivated partisan and sectional debate during important portions of our history. Nevertheless, the reasons that supported the justiciability of challenges to state legislative districts, as in Baker v. Carr, as well as state districting decisions relating to the election of Members of Congress, see, e.g., Wesberry v. Sanders, 376 U.S. 1 (1964); Karcher v. Daggett, 462 U.S. 725 (1983), apply with equal force to the issues presented by this litigation. The controversy between Montana and the Government turns on the proper interpretation of the relevant constitutional provisions. As our previous rejection of the political question doctrine in this context should make clear, the interpretation of the apportionment provisions of the Constitution is well within the competence of the Judiciary. See [503 U.S. 442, 459]   Davis v. Bandemer, 478 U.S. 109, 123 (1986); Baker v. Carr, 369 U.S., at 234 -237; cf. Gilligan v. Morgan, 413 U.S., at 11 . The political question doctrine presents no bar to our reaching the merits of this dispute and deciding whether the District Court correctly construed the constitutional provisions at issue.

Our previous apportionment cases concerned States' decisions creating legislative districts; today we review the actions of Congress. Respect for a coordinate branch of Government raises special concerns not present in our prior cases, but those concerns relate to the merits of the controversy, rather than to our power to resolve it. As the issue is properly raised in a case otherwise unquestionably within our jurisdiction, we must determine whether Congress exercised its apportionment authority within the limits dictated by the Constitution. See INS v. Chadha, 462 U.S. 919, 940 -941 (1983); Powell v. McCormack, 395 U.S. 486, 521 (1969). Without the need for another exploration of the Baker factors, it suffices to say that, as in Baker itself and the apportionment cases that followed, the political question doctrine does not place this kind of constitutional interpretation outside the proper domain of the Judiciary.


In Wesberry v. Sanders, 376 U.S. 1 (1964), the Court considered the claim of voters in Fulton County, Georgia, that the disparity between the size of their congressional district (823,680) and the average size of the 10 districts in Georgia (394,312) deprived them of the right "to have their votes for Congressmen given the same weight as the votes of other Georgians." Id., at 3. This Court upheld the claim, concluding that Article I, 2, had established a "high standard of justice and common sense" for the apportionment of congressional districts: "equal representation for equal numbers of people." 376 U.S., at 18 . The constitutional command that Representatives be chosen "by the People of the several [503 U.S. 442, 460]   States" meant that "as nearly as is practicable, one man's vote in a congressional election is to be worth as much as another's." Id., at 7-8. Writing for the Court, Justice Black explained:

In subsequent cases, the Court interpreted that standard as imposing a burden on the States to "make a good faith effort to achieve precise mathematical equality." Kirkpatrick v. Preisler, 394 U.S., at 530 -531; see also Karcher v. Daggett, 462 U.S., at 730 .

Our cases applying the Wesberry standard have all involved disparities in the size of voting districts within the same State. In this case, however, Montana contends, and a majority of the District Court agreed, that the Wesberry standard also applies to apportionment decisions made by Congress, and that it was violated because of an unjustified variance between the population of Montana's single district and the ideal district size.

Montana's evidence demonstrated that, if Congress had used the method of the harmonic mean, sometimes referred to as the "Dean Method", instead of the method of equal proportions, sometimes called the "Hill Method", to apportion the districts, 48 of the States would have received the same number of Representatives, while Washington would have received one less - eight instead of nine - and Montana would have received one more. Under an apportionment undertaken according to the Hill Method, the absolute difference [503 U.S. 442, 461]   between the population of Montana's single district (803,655) and the ideal (572,466) is 231,189; the difference between the average Washington district (543,105) and the ideal is 29,361. Hence, the sum of the differences between the average and the ideal district size in the two States is 260,550. Under the Dean Method, Montana would have two districts with an average population of 401,838, representing a deviation from the ideal of 170,638; Washington would then have eight districts averaging 610,993, which is a deviation of 38,527 from the ideal district size. The sum of the deviations from the ideal in the two States would thus be 209,165 under the Dean Method (harmonic mean), while it is 260,550 under the Hill Method (equal proportions). More generally, Montana emphasizes that the Dean Method is the best method for minimizing the absolute deviations from ideal district size.

There is some force to the argument that the same historical insights that informed our construction of Article I, 2, in the context of intrastate districting should apply here as well. As we interpreted the constitutional command that Representatives be chosen "by the People of the several States" to require the States to pursue equality in representation, we might well find that the requirement that Representatives be apportioned among the several States "according to their respective Numbers" would also embody the same principle of equality. Yet it is by no means clear that the facts here establish a violation of the Wesberry standard. In cases involving variances within a State, changes in the absolute differences from the ideal produce parallel changes in the relative differences. Within a State, there is no theoretical incompatibility entailed in minimizing both the absolute and the relative differences. In this case, in contrast, the reduction in the absolute difference between the size of Montana's district and the size of the ideal district has the [503 U.S. 442, 462]   effect of increasing the variance in the relative difference 39 between the ideal and the size of the districts in both Montana and Washington. 40 Moreover, whereas reductions in the variances among districts within a given State bring all of the affected districts closer to the ideal, in this case a change that would bring Montana closer to the ideal pushes the Washington districts away from that ideal. 41   [503 U.S. 442, 463]  

What is the better measure of inequality - absolute difference in district size, absolute difference in share of a Representative, or relative difference in district size or share? Neither mathematical analysis nor constitutional interpretation provides a conclusive answer. In none of these alternative measures of inequality do we find a substantive principle of commanding constitutional significance. The polestar of equal representation does not provide sufficient guidance to allow us to discern a single constitutionally permissible course.

A State's compliance with Wesberry's "high standard of justice and common sense" begins with a good faith effort to produce complete equality for each voter. As our cases involving variances of only a fraction of one percent demonstrate, that goal is realistic and appropriate for state districting decisions. See Karcher v. Daggett, 462 U.S., at 730 -743. In this case, however, whether Montana has one district or two, its variance from the ideal will exceed 40 percent.

The constitutional guarantee of a minimum of one Representative for each State inexorably compels a significant departure from the ideal. In Alaska, Vermont, and Wyoming, where the statewide districts are less populous than the ideal district, every vote is more valuable than the national average. Moreover, the need to allocate a fixed number of indivisible Representatives among 50 States of varying populations makes it virtually impossible to have the same size district in any pair of States, let alone in all 50. Accordingly, although "common sense" supports a test requiring "a good faith effort to achieve precise mathematical equality" within each State, Kirkpatrick v. Preisler, 394 U.S., at 530 -531, the constraints imposed by Article I, 2, itself make that goal illusory for the Nation as a whole. [503 U.S. 442, 464]  

This common sense understanding of a characteristic of our Federal Government must have been obvious to the masters of compromise who framed our Constitution. The spirit of compromise that provided two Senators for every State and Representatives of the People "according to their respective Numbers" in the House must also have motivated the original allocation of Representatives specified in Article I, 2, itself. Today, as then, some compromise between the interests of larger and smaller States must be made to achieve a fair apportionment for the entire country.

The constitutional framework that generated the need for compromise in the apportionment process must also delegate to Congress a measure of discretion that is broader than that accorded to the States in the much easier task of determining district sizes within state borders. Article I, 8, cl. 18, expressly authorizes Congress to enact legislation that "shall be necessary and proper" to carry out its delegated responsibilities. Its apparently good faith choice of a method of apportionment of Representatives among the several States "according to their respective Numbers" commands far more deference than a state districting decision that is capable of being reviewed under a relatively rigid mathematical standard. 42   [503 U.S. 442, 465]  

The District Court suggested that the automatic character 43 of the application of the method of equal proportions was inconsistent with Congress' responsibility to make a fresh legislative decision after each census. 44 We find no merit in this suggestion. Indeed, if a set formula is otherwise constitutional, it seems to us that the use of a procedure that is administered efficiently and that avoids partisan controversy supports the legitimacy of congressional action, rather than undermining it. To the extent that the potentially divisive and complex issues associated with apportionment can be narrowed by the adoption of both procedural and substantive rules that are consistently applied year after year, the public is well served, provided, of course, that any such rule remains open to challenge or change at any time. We see no constitutional obstacle preventing Congress from adopting such a sensible procedure.

The decision to adopt the method of equal proportions was made by Congress after decades of experience, experimentation, and debate about the substance of the constitutional requirement. Independent scholars supported both the basic decision to adopt a regular procedure to be followed after each census and the particular decision to use the method of equal proportions. 45 For a half century, the results of that method have been accepted by the States and [503 U.S. 442, 466]   the Nation. That history supports our conclusion that Congress had ample power to enact the statutory procedure in 1941 and to apply the method of equal proportions after the 1990 census.

The judgment of the District Court is reversed.


[ Footnote 1 ] Article I, 2, originally provided that "Representatives . . . shall be apportioned among the several States . . . according to their respective Numbers, which shall be determined by adding to the whole Number of free Persons, including those bound to Service for a Term of Years, and excluding Indians not taxed, three fifths of all other Persons."

Section 2 of the Fourteenth Amendment modified this provision by establishing that "Representatives shall be apportioned among the several States according to their respective numbers, counting the whole number of persons in each State, excluding Indians not taxed."

[ Footnote 2 ] 55 Stat. 761-762; 2 U.S.C. 2a(a).

[ Footnote 3 ] 775 F.Supp. 1358, 1366 (Mont. 1991).

[ Footnote 4 ] Three States, California, Florida, and Texas, accounted for 14 of those gains; five States, Arizona, Georgia, North Carolina, Virginia, and Washington, each gained one seat. 2 App. 20.

[ Footnote 5 ] New York lost three seats; Illinois, Michigan, Ohio, and Pennsylvania each lost two seats; and Iowa, Kansas, Kentucky, Louisiana, Massachusetts, Montana, New Jersey, and West Virginia each lost one seat. Ibid.

[ Footnote 6 ] See ibid.

[ Footnote 7 ] The three state officials brought suit on behalf of all voters in Montana.

[ Footnote 8 ] Complaint  19.

[ Footnote 9 ] Id.,  28-29.

[ Footnote 10 ] Having granted summary judgment on the first claim, the District Court found it unnecessary to reach the merits of the claim relating to the automatic method of apportionment. 775 F.Supp., at 1366.

[ Footnote 11 ] Ibid.

[ Footnote 12 ] Id., at 1369 (quoting 2 J. Story, Commentaries on the Constitution of the United States 676 (1833)).

[ Footnote 13 ] Montana alleged that the "method of the harmonic mean" or the "method of smallest divisors" would yield a fairer result. Subsequent to the decision below, a District Court in Massachusetts rejected a challenge to Congress' adoption of the method of equal proportions. In that litigation, Massachusetts plaintiffs asserted that the superiority of another method, that of "major fractions," demonstrated that the method of equal proportions was unconstitutional. Massachusetts v. Mosbacher, Civ. Action No. 91-11234WD (Mass., Feb. 20, 1992).

[ Footnote 14 ] The first and second requirements are set forth explicitly in Article 1, 2, of the Constitution. The requirement that districts not cross state borders appears to be implicit in the text, and has been recognized by continuous historical practice. See 775 F.Supp., at 1365, n. 4; id., at 1368 (O'Scannlain, J., dissenting).

[ Footnote 15 ] Section 2, cl. 3, required an enumeration of the population to be made within three years after the first meeting of Congress and provided that, "until such enumeration shall be made, the State of New Hampshire shall be entitled to choose three, Massachusetts eight, Rhode Island and Providence Plantations one, Connecticut five, New York six, New Jersey four, Pennsylvania eight, Delaware one, Maryland six, Virginia ten, North Carolina five, South Carolina five, and Georgia three."

[ Footnote 16 ] See M. Balinski & H. Young, Fair Representation, Meeting the Ideal of One Man, One Vote 10-13 (1982) (hereinafter Balinski & Young).

[ Footnote 17 ] See id., at 16-22. President Washington's veto message read as follows:

3 Annals of Cong. 539 (1792).

[ Footnote 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, 2 Stat. 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, 1832, ch.91, 4 Stat. 516. See generally L. Schmeckebier, Congressional Apportionment 111-113 (1941).

[ Footnote 19 ] See Balinski & Young 73-75.

[ Footnote 20 ] Ibid.

[ Footnote 21 ] Act of June 25, 1842, 5 Stat. 491.

[ Footnote 22 ] Act of May 23, 1850, 24-26, 9 Stat. 432433. 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).

[ Footnote 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 3840; Chafee, Congressional Reapportionment, 42 Harv.L.Rev. at 1026.

[ Footnote 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.

[ Footnote 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:

[ Footnote 26 ] The five were the "method of smallest divisors," the "method of the harmonic mean," the "method of equal proportions," the "method of major fractions," and the "method of greatest divisors." 1 App. 17.

Each of the methods corresponds to a different formula for producing a "priority list." A priority list is the mechanical method used in modern apportionments to translate a particular method of apportionment into a particular assignment of Representatives. The technical process of forming the priority list proceeds as follows. First, one Representative is assigned to each State to satisfy the constitutional guarantee. Second, the population of each State is divided by a certain tabulated series of divisors. Third, the quotients for all the States are arranged in a single series in order of size, beginning with the largest quotient, for the 51st Member of the House. This forms the priority list. The series of quotients is different for each of the five apportionment methods. See Chafee, Congressional Reapportionment, 42 Harv.L.Rev. at 1029, n. 39.

The following are the divisors by which a State's population is divided under each method ("n" is the number of the State's next seat): [503 U.S. 442, 453]  

Smallest Divisors: n-1


Major Fractions: n ___


Greatest Divisors: n

Thus, the divisors for the second, third, fourth, and fifth Representative of a State are as follows:

Smallest Divisors: 1, 2, 3, 4

Greatest Divisors: 2, 3, 4, 5

See ibid. For example, the 1990 census indicated that the most populous States were California and New York. California had a population of 29,839,250, and New York had a population of 18,044,505. See 2 App. 20. Under the method of smallest divisors, the quotients are:

See 2 App. 53. Under the method of greatest divisors, the quotients are:

Under any method, the first 50 seats are assigned one to each State. If the method of smallest divisors is employed, the 51st seat is assigned to California, and the 52d seat is assigned to New York. Under the method of greatest divisors, however, California is assigned both the 51st and the 52d seats, because the quotient for its third seat is 9,946,417, which is higher than the quotient for New York's second seat, which is 9,022,252.

[ Footnote 27 ] The committee explained the test as follows:

[ Footnote 28 ] A person's "share" of a Representative is the reciprocal of the population of a person's district. For example, in an ideal district under the 1990 census, each person has a share of 1/572,466 of a Representative.

[ Footnote 29 ] The relative difference between two numbers consists of subtracting the smaller number from the larger number and then dividing the result by the smaller number. 1 App. 24 (Ernst Declaration).

[ Footnote 30 ] See ibid.

[ Footnote 31 ] See id., at 19. The committee considered only the extent to which each method favored the small or large States in comparison to the other methods. The committee did not attempt to determine absolute bias. Some scholars have asserted that, in absolute terms, the method of equal proportions favors small States over large States, and that the method of major fractions is the method with the least inherent bias between small and large States. See Balinski & Young 72-78. That contention has been disputed. See Massachusetts v. Mosbacher, Civ. Action No. 91-1123WD (Mass., Feb. 20, 1992), p. 57.

[ Footnote 32 ] 2 App. 35.

[ Footnote 33 ] U.S. Const., Art. I, 2, cl. 3.

[ Footnote 34 ] Ibid.

[ Footnote 35 ] See Brief for United States 24-34; Tr. of Oral Arg. 10-13.

[ Footnote 36 ] See Henkin, Is There a "Political Question" Doctrine?, 85 Yale L.J. 597, 599 (1976).

[ Footnote 37 ] See M. Redish, The Federal Courts in the Political Order 116-117 (1991).

[ Footnote 38 ] Not only is the composition of the House of Representatives implicated by the case, but also the composition of the electoral college that elects the President. That college include representation from each State equivalent to the sum of its Senators and Representatives. U.S. Const., Art. II, 1, cl. 2.

[ Footnote 39 ] See n. 29, supra.

[ Footnote 40 ] Under the Hill method (equal proportions), the relative differences between Montana's and Washington's districts and the ideal, respectively, are 40.4% and 5.4%, under the Dean method (harmonic mean) they are 42.5% and 6.7%. See 1 App. 27.

The absolute and relative differences between the actual average district size and the ideal district size in an apportionment using the Hill Method (Montana has 1 Representative, and Washington has 9 Representatives) are as follows:

Montana 803,655 231,189 40.4%

Washington 543,105 29,361 5.4%

Total Absolute Difference 260,550

The absolute and relative differences between the actual average district size and the ideal district size in an apportionment using the Dean Method (Montana has 2 Representatives, and Washington has 8 Representatives) are as follows:

Montana 401,828 170,638 42.6%

Washington 610,993 38,527 6.7%

Total Absolute Difference 209,165

The relative difference from the ideal is less both for Montana and for Washington in a Hill apportionment; the total absolute difference from the ideal is less in a Dean apportionment.

[ Footnote 41 ] Indeed, as Washington has more districts than Montana, it could be argued that deviation from ideal district size in Washington represents a more significant departure from the goal of equal representation than does a similar deviation in Montana. In his dissent in the District Court, Judge O'Scannlain noted the potential importance of taking account of the [503 U.S. 442, 463]   number of districts in a State, rather than merely the average size of a district. See 775 F.Supp., at 1371.

[ Footnote 42 ] Some evidence suggests that partisan political concerns may have influenced Congress' initial decision to adopt the equal proportions method in 1941. The choice of this method resulted in the assignment of an additional seat to Arkansas, a Democratic State, rather than to Michigan, a State with more Republican leanings. The vote to adopt equal proportions was along party lines (except for the Democrats from Michigan, who opposed the bill). See Balinski & Young 57-58; see also 775 F.Supp., at 1365. Nevertheless, although Congress has considered the apportionment problem periodically since 1941, it has not altered that initial choice. See Massachusetts v. Mosbacher, Civ. Action No. 91-11234-WD, (Mass., Feb. 20, 1992), pp. 40-42. Montana does not contend that the equal proportions method systematically favors a particular party, nor that its retention over a 50-year period reflects efforts to maintain partisan political advantage.

[ Footnote 43 ] See n. 25, supra.

[ Footnote 44 ] See 775 F.Supp., at 1366.

[ Footnote 45 ] In his article Congressional Reapportionment, written in 1929, Zechariah Chafee, Jr., wrote: